%% v20.0 (August 21, 2026): PROMOTION TO THE v20 STAMP.
%%   Version and cover date only.  Promoted from v15.10 with no other change:
%%   not one character of body text, mathematics, float, caption or
%%   bibliography differs.  The promotion is safe to do mechanically because
%%   the cohort cites its companions BY TITLE, never by versioned filename --
%%   a scan of the printed text of all twelve deposit papers found zero
%%   version strings, so no cross-reference can be left pointing at a
%%   superseded file.  This is the version posted to arisaka-gg.org and
%%   deposited to Zenodo, so homepage, repository and DOI all carry one
%%   stamp.
%% ===================================================================
%% v15.8 (August 18, 2026) -- BT-1: THE BIBLIOGRAPHY TITLE PASS.
%%
%% EVERY COMPANION IS NOW CITED BY THE TITLE PRINTED ON ITS OWN COVER.
%%
%% 8 of this paper's 21 cohort bibitems named a companion by a title
%% that companion does not carry.  Measured, not recalled: the truth is
%% each paper's printed cover, read from its deployed PDF; the markup is
%% each paper's own \GGTitlePage arguments, so the title sets in the same
%% faces the cover does.  Cohort-wide the pass touched 233 sites in 23
%% papers, and afterwards all 477 companion bibitems agree with their
%% covers.
%%
%% TWO CLASSES, and the difference is worth recording.
%%   RENAME PROPAGATION -- a paper was retitled and its citers were never
%%   updated.  The signature is the same wrong string in many papers:
%%   A6-GRIP's retired subtitle "Gi-4 Pillars (TGG, CCI, GFF, and GRIP) as
%%   the Geometric Backbone of GDOS" stood in seventeen papers, and
%%   C2-Hubble's retired "Two Hubble Constants as the Two Ends of One
%%   Six-Dimensional Geometric Corridor" in seventeen more.  W1-Symmetry,
%%   W2-Holonomy, P5-CPT and P6-Proton were all renamed and none of their
%%   citers followed.
%%   HOUSE SHORTHAND -- P5-CPT, W1-Symmetry and W2-Holonomy each wrote
%%   their own descriptive short form for every companion ("GDOS: the
%%   Boundary Law", "The Kaxion", "The Cosmological Inventory of the
%%   Descent").  Readable in prose and useless in a reference list,
%%   because none of those strings finds the paper.
%%
%% ONLY TITLE TEXT CHANGED.  Author blocks, years, availability lines,
%% Relevance notes, keys, labels and \emph/\textit wrappers are asserted
%% unchanged in count; the changelog comments are asserted BYTE-IDENTICAL.
%% ====================================================================
%% ===================================================================
%% v15.7 (August 18, 2026) -- FP-5: THE DISCLOSURE-PREP PASS.
%%
%% The cohort is posted as TWENTY-ONE papers with A11-Origin held back.
%%
%% (1) THE A11 BIBLIOGRAPHY ENTRY PROMISED A PAPER THAT WILL NOT BE
%%     THERE.  It read "Preprint, available at https://arisaka-gg.org/",
%%     which becomes false the day the homepage goes up carrying
%%     twenty-one papers.  It now reads "In preparation.  To be posted
%%     ... on completion; no result of the present paper depends on it."
%%
%% P1-KK carries NO simultaneity claim, NO reading-set enumeration, NO
%% roster row for A11, NO pointer into A11's internal Part structure and
%% NO printed catalogue debt; the claim-derived probe checked all five
%% before this rider was written.
%%
%% PROTECTED, and printed here so the decision is on the record: "its
%% downstream consequences ... all published together (Part~IV)" is about
%% THIS paper's Kaxion band being stated before the data, not about the
%% cohort arriving on one day.  The simultaneity guard fires on it, and it
%% is right as written.
%%
%% NO OTHER CONTENT CHANGED.
%% ===================================================================
%% ===================================================================
% v15.6 (August 17, 2026) -- MIGRATED TO STYLE FILE v11.5.
%%
%% FOUR MACROS THAT LIVED IN THE PAPERS COME HOME, AND NOT ONE PAGE
%% MOVES.  This paper's part in that:
%%
%% * \GGShowcaseHead removed; this paper prints the sans form with \bigskip -- the style file default,
%%   and needs no setup line.
%%
%% The style file is v11.5 (2026/08/17).  Every one of the twenty-four
%% MANIFEST.sha256 files covers it, so changing it meant rebuilding and
%% repackaging all twenty-four -- which is why this was deferred five
%% times and is why it was done in one pass.
%%
%% ASSERTED, NOT ASSUMED: this paper's printed text is line for line
% what v15.5 printed, except for the lines named above.
%% ===================================================================
%% ===================================================================
%% v15.5 (August 17, 2026) -- THE -ise FAMILY.
%%
%% stabiliser -> stabilizer (2).  Charter 1.1's pair list names these forms; spelling_audit.py's
%% PAIRS list did not carry them, so this paper has reported MIXED
%% PAIRS 0 and ALWAYS-BRITISH 0 at every gate while printing them.  The
%% cohort sweep of August 17, 2026 found sixty-one across twelve of the
%% twenty-four live papers; Report 406 section 6 has the inventory and
%% spelling_audit.py now carries the family as a RULE rather than a list.
%%
%% Substitution is on PRINTED TEXT ONLY.  Changelog comments are the
%% historical record, \label/\ref/\cite arguments are keys, and published
%% titles are quoted rather than written -- all three are masked, and every
%% frozen title is asserted present and unchanged after the edit.
%% NOTHING ELSE IN THIS PAPER CHANGES: no claim, no number, no float, no
%% label, and no page.
%% ===================================================================
%% ------------------------------------------------------------
%% v15.3 -> v15.4, August 16, 2026.  THE EXPOSURE PAGE.
%% Fifteenth of the twenty-four.
%%   THE PAGE.  P1's existing showcase page (p4) carries the
%%   predicted tt-bar spectrum and the end-2026 significance
%%   ladder: what the theory predicts, and when the verdict
%%   arrives.  Neither says what the paper is COMMITTING to, or
%%   how strongly each commitment is claimed -- and P1 is the
%%   paper whose whole argument is that it committed in advance.
%%   Added: Table 31, the ten numbered predictions with their
%%   grades ([T], [D*], [M], [E]), the direct analogue of the
%%   pages A1, C1 and C2 carry.  Unnumbered, so Tables 1-39 keep
%%   their numbers.
%%   THE EXISTING PAGE IS UNTOUCHED.  It carries two ideas rather
%%   than one, against the house pattern, and it has no typeset
%%   headline -- but both plates carry their titles baked into the
%%   image, so a headline would print the title twice, and no
%%   defect drives a split.  It is the author's page and it works.
%%   THE PROBE I GOT WRONG, AND WHY IT MATTERED.  Report 385's
%%   remedy is \setstretch{1} in the savebox, and I nearly applied
%%   it here on the strength of a probe placed one line ABOVE
%%   \begin{table}, which read \baselinestretch 1.15 and would
%%   have sent me to fill at 1.15.  The style file sets 1.15
%%   document-wide; a table environment resets it.  Probed INSIDE,
%%   at \begin{tabularx}: BSTR 1, AS 1, at tab:predictions and at
%%   two other tables checked for company.  PROBE THE USE SITE,
%%   INSIDE THE ENVIRONMENT -- one line early is a different
%%   number.  And 385's remedy is not a constant: A8 needed
%%   \arraystretch 1.2 because A8's table sets 1.2.  P1's does
%%   not, so it stays at 1.
%%   STAGE 0, ALL FIVE CHECKS: 11 \includegraphics uses of 9
%%   distinct files, all present, zero 'not found'; the baseline
%%   reproduces the deployed PDF at 0 of 214 pages compared PAGE
%%   BY PAGE, not merely by page count; zero pre-existing
%%   saveboxes; one \begin{minipage}{\textwidth} and it is not
%%   bare; no digit in the savebox name; every macro in the new
%%   prose defined here.
%% NEW PROSE IN THIS SHIP: one headline and one showcase caption,
%% both on the new page.  Nothing in the body is rewritten.
%% ------------------------------------------------------------
%% ===================================================================
%%  P1-KK v13.13  (August 6, 2026)  -- the cohort-wide figure-size sweep.
%%  1 figures re-placed so their node text lands at the house size,
%%  7.79pt: what GG-P3-Electron Figure 3 renders (\footnotesize at
%%  0.865 of natural size).  For a picture of measured width W drawn
%%  at a base font of B points the target is f = (7.79/B) * W / 159.2
%%  of \textwidth, W being measured by compiling each picture
%%  standalone and reading its ink bounding box.  Nothing inside any
%%  picture changes: no font command, no dimension, no text, no
%%  relative position.  The cohort map and the central-claim funnel
%%  are excluded, both being at the house size already.
%%  Per-figure table and recipe: [v13-support]/allsweep.py and
%%  [v13-support]/figure_census.json.
%% ===================================================================
%% ===================================================================
%%  P1-KK v13.12  (August 5, 2026)  -- the cohort funnel at the house size.
%%  The central-claim funnel was drawn with 109mm boxes and then
%%  left at natural size, so its \footnotesize landed 16% larger than the roadmap
%%  and cohort-map plates beside it.  The plate target is now
%%  0.65\textwidth, computed rather than chosen: the picture measures
%%  119.6mm wide compiled standalone, and the house type size is set by
%%  GG-P3-Electron Figure 3 (0.93\textwidth on 171.1mm, \textwidth =
%%  159.2mm, so type at 0.865 of natural size), which for any picture
%%  of width W means f = W / 184.0.
%%  Nothing inside the picture changes: no font command, no box
%%  dimension, no text, no relative position.
%%  Recipe and the measured table for all fourteen papers:
%%  [v13-support]/funnel_sweep.py.
%% ===================================================================
%% ===================================================================
%%  P1-KK v13.11  (August 5, 2026)  -- the twenty-two-paper cohort sweep.
%%  The cohort map becomes the twenty-two-paper map: GG-P6-Proton on
%%  the Predictions row with a second sister-import arrow, and the
%%  White-Paper Branch (GG-W1-Symmetry above GG-W2-Holonomy) as a
%%  right-flank column at the mirror positions of the Foundations
%%  column -- the left flank feeds the spine, the right reads it.
%%  GG-P7-Inflow does not appear: it is Appendix H of GG-P2-N3.
%%  The plate is also enlarged for reading: boxes and the type inside
%%  them x1.15, every word outside a box x1.20, and the long
%%  A10 -> C-frame edge redrawn as a polyline, which returns 24mm of
%%  bounding box that was Bezier control points rather than ink.
%%  Prose: every count site, the series breakdown, and the canonical
%%  Suffix list, which was missing Proton, Symmetry and Holonomy.
%%  Recipe and variant analysis: [v13-support]/sweep22.py.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-P1-KK  v13.10  (August 4, 2026)
%%  -----------------------------------------------------------------
%%  CHANGELOG (v13.9 -> v13.10), author-directed (GG-Report-187):
%%    * NEW paragraph after the mode equation: the compact-angle
%%      eigenvalue carries the Wilson line, eq:wilsonshift
%%      (ell -> ell + q Theta_H / 2pi).  This is the Hosotani
%%      mechanism written spectrally: a constant A_theta changes no
%%      field strength yet lifts the ell=0 modes, which is the
%%      origin of electroweak mass.  The A_theta zero mode named as
%%      the 4D Higgs doublet; the vacuum angle and the registration
%%      scale delegated to A9 (the warp puts the resulting 4D
%%      masses at the infrared endpoint, not at E_GUT).
%%      Both of this paper's own predictions checked against the
%%      shift and unaffected: the radial lock cluster, and the
%%      no-angular-satellites null (the displacement is ~1% of one
%%      unit, so the ell=+-1 partners stay at ~E_GUT).
%%  Script: p1p6c1_v_tiers.py
%% ===================================================================
%% ===================================================================
%%  P1-KK v13.8  (July 30, 2026)  -- Phase-2 references completion.
%%  5 bibitems completed with first-citation pointers.
%% ===================================================================
%% ===================================================================
%%  P1-KK v13.7  (July 29, 2026)  -- P2-N3 title conformance.
%%  P2-N3 is retitled cohort-wide from "A Closed Derivation of N = 3" to
%%  "A Three-Leg Derivation of N = 3", matching its own main theorem T-N3-5
%%  ("Conditional derivation of N = 3 (three-leg pincer)"), which is
%%  conditional on five structural hypotheses -- the BI-6 Budget Conjecture
%%  among them -- plus two empirical inputs.
%%  P1 spelled the title with \(N=3\) rather than $N=3$ and so was missed by
%%  the cohort-wide pass; its Relevance note also read "the closed derivation
%%  of the generation count".  Both now read "three-leg".  No lock-scale
%%  value, collider falsifier or citation target changes.
%% ===================================================================
%% ===================================================================
%% v13.6 (July 24, 2026): COHORT BIB-TITLE RECONCILIATION (P3)
%%  The electron companion's subtitle is updated in its bibliography
%%  entry to that paper's current cover ("The Six Oldest Questions
%%  about the Electron ..."), following its addition of the sixth
%%  question -- the origin of positrons. No other change
%%  anywhere.
%% ===================================================================
%% v13.5 (July 19, 2026): WAVE-1 cohort-promotion sweep (nineteen-paper
%%   standard; script p1_wave1.py; mechanical/structural only -- no physics
%%   values, falsifier tables, or verdict windows changed).
%%   (1) STANDARDIZED PART-I CLOSE: new closing section 'The cohort
%%   inheritance and the roadmap of the fourteen Parts'
%%   (sec:inheritance-and-roadmap-p1). Subsection 'The cohort inheritance'
%%   (sec:cohort-p1): programme-in-one-view + reading-conventions prose
%%   migrated verbatim from sec:constitutional-route (counts updated,
%%   Foundations Branch added to the panorama + papers of record), NEW
%%   Foundations Branch paragraph (P4-Real the origin-of-i member; P5-CPT
%%   the mirrors-and-matter member; roster level), and the canonical v1.4
%%   nineteen-paper cohort map installed verbatim from A2-GG6 v13.3
%%   (MP4/MP5 internal node names; label fig:cohort-map-p1 kept). The old
%%   seventeen-paper map figure is retired. Subsection 'Roadmap of the
%%   fourteen Parts' (sec:roadmap-p1): new five-paragraph walk + NEW
%%   roadmap figure fig:parts-roadmap-p1 (house architecture: A2-GG6
%%   v13.3 / A10-Cosmos v13.3 pattern; Parts II and VIII highlighted --
%%   the derived signal and the anchored HL-LHC verdict). The two Part-I
%%   closing paragraphs relocated verbatim after the roadmap; two
%%   reading-paths framing sentences adjusted ('now complete' -> 'nearly
%%   complete'; 'closes this Part' -> 'near the close of this Part').
%%   (2) COUNTS: reader-facing cohort counts seventeen -> nineteen (nine
%%   prose sites + two objection quotes); 'sixteen sister papers' ->
%%   eighteen (+ Foundations Branch in the Section-A order note); suffix
%%   roster -> canonical nineteen form (GRIP (A6); foundations preprints
%%   Real (P4) and CPT (P5)). Semantic tokens protected (seventeenth /
%%   nineteenth century).
%%   (3) MACROS + BIB: \ASix renewed to GG-A6-GRIP (sty default still
%%   renders the retired Gi4 form); \PaperPFour (GG-P4-Real) and
%%   \PaperPFive (GG-P5-CPT) installed; ArisakaP4Real + ArisakaP5CPT
%%   bibitems added to Section A (roster-level Relevance, cited from
%%   sec:cohort-p1); stale '(v9.8, June 2026)' dropped from the A8-QM
%%   bibitem; five 'First cited' pointers repointed to sec:cohort-p1.
%% v13.4 (July 16, 2026): cohort table standard (author-approved, P4 v2.17 pattern): sub-small table fonts -> \small; p-column tables widened toward full text width; targeted corrections listed.
%%   correction: tab:sigmabr: long l-columns 3-4 converted to p{3.8cm}/p{2.6cm} (overfull 26.5pt after font raise)
% v13.3 (July 15, 2026): gauge-phase scoping pass (complex-variables
%   thread, author-approved). Eight narrative spots that called the
%   sixth dimension the universal wavefunction phase are scoped to the
%   gauge phase of charged matter, matching A8-QM v12.6's two-complex-
%   structures treatment (the circle geometrises the charged phase; the
%   state-space i has a separate causal-positive origin). No physics,
%   numbers, tables of values, or structure changed.
%% v13.2 (2026-07-14, Phase 7 OGN-5 sweep) -- presentation only.
%%   Accessible "story" paragraph: collected the per-integer physical meanings
%%   of the OGN-5 ledger (M=1 tensor seam / charge-quant unit, N=3 colour
%%   triplicity, L=5 SU(5) block, J=7 odd closure index, K=35=LJ=M^2+N^2+L^2
%%   throat depth that fixes the lock scale M_lock=(1/2)M_Pl e^{-35}); stated
%%   forced/not-fitted/not-arbitrary. Anti-hostile-reviewer emphasis (author
%%   directive 2026-07-14). No values changed. Script: p1_ogn.py.
%% ======================================================================
%% v13.0 (2026-07-07): PROMOTION. Content identical to v11.59 (all six
%% hostile reviewers satisfied; closure report in [GG-QA+References]).
%% P1-KK is the FIRST of the seventeen cohort papers promoted to the
%% v13.0 presentation standard and serves as the cohort's reference
%% implementation for that standard: 14-Part architecture with four-Part
%% tail, one-page Overview+LoF, eye-opener showcase page, cohort +
%% roadmap sections, five conceptual primers (E-I) + three courses (J-L)
%% + card/glossary/roadmap appendices, 58-entry objection ledger,
%% A/B references with relevance notes + first-citation links + zero
%% orphans, ten-numbers-at-a-glance table, boxed Template T1, and the
%% calibrated-honesty apparatus throughout. Version + date stamp only.
%% ======================================================================
%% ======================================================================
%% v11.59 (2026-07-07): full execution of the remaining hostile-audit
%% items (author-approved; NO retreat from standing points -- every
%% edit is an addition or clarification, no claim weakened).
%% A-list: R1-1 Sec-2 ">3 sigma by end 2026" re-dated to the DATASET;
%%  R3-1 EW-Sudakov caveat (9) + Part-VI reliability sentence;
%%  R4-1 "an expected" tags on the three bare sigma quotes (Abstract,
%%  Exec Summary, Part-X schedule line; Preface/Sec-2 already carried
%%  conditionals); R2-3 O-KK-14 strengthened to the N-lives form with
%%  a terminal answer (no Scenario C; null = death, accepted in print).
%% B-list: R3-3 g*-Z'_B colour-orthogonality sentence (cluster sum) +
%%  O-KK-28 clause; R3-4 n=2 rate provenance in Fig-3 caption; R3-6
%%  K vs K-factor disambiguation (first LOxK use + new App-A row);
%%  R2-2 scheme-dependence footnote at eq:rRG; R2-6 one-line RG-axis
%%  echo opening subsec:notKKRS; R4-3 tagger-extrapolation clause in
%%  caveat (2); R5-2 "registered where? in print" sentences in
%%  sec:sim-prereg; R5-4 boxed Template T1 in the card appendix;
%%  R6-3 DLMF citation at the course Bessel equation + new bibitem
%%  OlverEtAl2010DLMF (B-list, alphabetical, Relevance + first-cited);
%%  R3-2 verified already present (roadmap has the modern-PDF task).
%% C-list defaults: R1-4 concessive "individual models have staked
%%  numbers" clause BEFORE the first-in-fifty-years claim (supports,
%%  does not weaken); R2-5 loophole-class parenthesis added to O-KK-41
%%  WITHOUT downgrading the impossibility claim; R6-4 "ten numbers at
%%  a glance" table added at the top of Appendix D.
%% ======================================================================
%% ======================================================================
%% v11.58 (2026-07-07): hostile-audit item R2-1 executed (with A3-BI6
%% v11.22, which gained the explicit "boundary mixed anomalies in 4D
%% language" subsection this edit points to).
%% (1) NEW paragraph in Appendix F (Z'_B primer): "The obligatory
%%     technical question: the mixed anomalies" -- states the +3/2 and
%%     -3/2 triangles, the vanishing ones, the no-anomalon GS/inflow
%%     cancellation, and the one-coupling-two-consequences identity
%%     with the Stueckelberg mass.
%% (2) NEW Objection 47 (O-KK-47) "the anomaly sceptic" appended at the
%%     end of the Reducibility/neighbouring-frameworks cluster; former
%%     47-57 renumbered 48-58 (verified: ZERO external references to
%%     O-KK-46..56 in P1; O-KK-57 occurred only in its own heading +
%%     the App C range; A9 references only O-KK-1). Counts updated:
%%     App C range -> O-KK-58; Part XII organising + Part XIII coda
%%     "fifty-seven" -> "fifty-eight"; coda invitation "fifty-eighth"
%%     -> "fifty-ninth".
%% ======================================================================
%% v11.57 (2026-07-07): showcase split reduced -- \vfill replaced by
%% fixed 3em gap between the two blocks (author: panels a bit closer).
%% v11.56 (2026-07-07): showcase-page spacing: top caption pulled up
%% tight under Fig-3 (-1.4em); \vfill between the two blocks so the
%% bottom plot+caption sit low -- clear visual split; bottom caption
%% tightened (-0.5em).
%% v11.55 (2026-07-07): bottom showcase plot (hllhc-absorb) fonts
%% enlarged 10-20% (suptitle 17, panel titles 12.5, axes 13/12.5,
%% ticks 11.5, legend 9.6, ladder labels 10.5-12); bottom showcase
%% one-liner footnotesize -> small. Asset-only + one-line tex edit.
%% ======================================================================
%% v11.54 (2026-07-07): Eye-opener showcase page added directly after
%% the Executive Summary + keywords (A9/A10 pattern, author request):
%% Fig 3 (quantsig, expected-signal landscape) on top + Fig 10
%% (hllhc-absorb, end-2026 reanalysis) at bottom, both UNNUMBERED
%% (plain includegraphics + footnotesize italic one-liners with
%% \ref/\pageref pointers to the real Figures) so figure numbering and
%% the one-page LoF are untouched. The absorb asset itself regenerated:
%% LEFT-PANEL X-AXIS NOW IN TeV (was GeV) + suptitle "The discovery may
%% already be on tape: end-2026 reanalysis of recorded Run-2+3 data" +
%% 5-sigma marker label moved to the top of the ladder panel.
%% ======================================================================
%% v11.53 (2026-07-07): Fig-3 n=1 label shifted right (clear of the
%% g*-alone dashed shoulder). Asset-only (make_quantsig3.py).
%% ======================================================================
%% v11.52 (2026-07-07): Figure 3 label nudges (author): Scenario-A note
%% -> single line, top-left corner inside the axes box; n=2 label
%% shifted right-upwards clear of the curves; n=3 label lowered beside
%% its dotted line. Asset-only change (make_quantsig3.py); caption
%% unchanged.
%% ======================================================================
%% ======================================================================
%% v11.51 (2026-07-07): Figure 3 public-quality refinements (author:
%% "I want to use it in public domains"): (1) axis to 12 TeV so the
%% n=3 mark at 10.219 TeV sits comfortably inside; (2) annotation
%% arrows removed -- labels placed directly beside each feature;
%% (3) fonts +~20% everywhere; (4) title added on top; (5) the three
%% cluster members now drawn separately at 3.849 TeV: g* alone (13.6 fb,
%% Gamma/m=14.6%, broad), Z'_B alone (g_B=0.2 reference: 0.047 fb at
%% 14 TeV = the paper's 0.030 fb at 13 TeV x the same qq-bar luminosity
%% ratio; Gamma~16 GeV i.e. Gamma/m~0.4%, from the paper's 8-307 GeV
%% menu range scaled by g_B^2 -- both menu ends give ~16), and G1
%% (0.048 fb, 6.3%). Caption updated to name all six curves and the
%% width ordering broad/narrow/moderate. Script make_quantsig3.py.
%% ======================================================================
%% ======================================================================
%% v11.50 (2026-07-07): Figure 3 (fig:lhc-signature, quantsig asset)
%% rebuilt per author: x-axis in TeV (was GeV), extended 2-10.6 TeV so
%% the n=2 family at 7.047 TeV is inside the frame (the old caption's
%% "dotted line at 7.047" sat exactly on the old 7.35-TeV edge and was
%% invisible) and the n=3 position at 10.219 TeV is marked; the n=2
%% octet is now DRAWN (near-resonance Lorentzian, model-estimate area
%% ~1e-4 fb, Gamma/m=14.6%); G1 curve likewise near-resonance Lorentzian
%% (area 0.048 fb) -- the former "Lorentzian x parton luminosity"
%% display distorted into a monotone tail over the wider range; legend
%% moved below the axes; octet legend now says "13.6 fb at 14 TeV
%% (8.7 fb at 13 TeV)" to preempt the 13-vs-14-TeV double-take (the
%% author's own -- 13.6 IS the canonical 14-TeV rate; no number
%% changed). CTEQ6 luminosity grid extended 7.6-10.6 TeV
%% (lumi_ext14.npz) with the same machinery. Caption updated to match.
%% ======================================================================
%% ======================================================================
%% v11.49 (2026-07-07): Appendix reorder + opener polish (author
%% request). (1) The five conceptual primers PROMOTED to directly after
%% Appendix D: new order A-D reference, E gluon/g*, F Z'_B, G graviton,
%% H rates+width, I one integer, J-L courses (6D action / partons /
%% statistics), M reinterpretation card, N collider glossary, O roadmap.
%% Full permutation reletter of every Appendix~X reference (placeholder
%% two-phase; explicit range handling); organising paragraph hard-
%% rewritten to the new four-group order; Part XII subtitle reordered.
%% (2) "The question, as the author posed it:" openers removed from the
%% four primers that had them; questions now stand as bold-italic
%% opening lines. (3) FIXES: v11.47 placeholder leak "Appendices~J--
%% QQNQQ" in the reading paths (shipped literally in v11.47-48) ->
%% now "Appendices~E--I"; the "card below ... available at
%% arisaka-gg.org" sentence sat at the head of the GRAVITON appendix
%% (where there is no card) -> moved to the head of the reinterpretation
%% -card appendix where "below" is true (flagged to author).
%% ======================================================================
%% ======================================================================
%% v11.48 (2026-07-07): appendix section titles de-duplicated (author
%% request): "\section{Appendix X: Title}" -> "\section{Title}" for all
%% fifteen appendices A-O. The appendix letter is already supplied by
%% \thesection after \appendix, so headings/ToC read "A  Summary of
%% Abbreviations" instead of "A  Appendix A: Summary of Abbreviations".
%% Labels and all "Appendix~X" prose cross-references unchanged.
%% ======================================================================
%% ======================================================================
%% v11.47 (2026-07-07): NEW Appendix M "Why does g* have the largest
%% production rate --- and why is it so broad?" (author request: fold
%% the rates/width Q&A in after the graviton primer; doors/keys/exits
%% register). Former M (one integer, two powers) -> N, REWRITTEN in the
%% accessible J-M register with ALL physics/formulas/numbers preserved
%% verbatim (new question-at-head opener, "picture the corridor"
%% paragraph, split walls of text, hood-comes-off marker before the
%% integrals; labels app:ratehierarchy + eq:appJ-kappa kept). Former N
%% (roadmap) -> O. Organising paragraph + reading-paths: four -> FIVE
%% conceptual primers, Appendices~J--M -> J--N, roadmap -> O. The
%% primer suite J-N is now the heart of the appendices for incoming
%% graduate students.
%% ======================================================================
%% ======================================================================
%% v11.46 (2026-07-07): NEW Appendix K "What is the origin of the
%% baryonic vector Z'_B?" (author request: third genealogy Q&A added
%% almost verbatim after the gluon/g* appendix). Content: U(1)_B gauged
%% by BI-6 anomaly-ledger necessity (not choice); 2/5 trace ratio ties
%% it to v; mass belongs to the geometry (one pipe, one pitch;
%% rho/omega precedent); Stueckelberg zero-mode removal (Eotvos forbids
%% a massless baryonic photon; proton stability by law); NOT a heavier
%% Z (leptophobia exact); photon = closest cousin, opposite boundary
%% fate; closer = three genealogies converging on one mass. Former
%% K/L/M relettered L/M/N (reverse cascade); "three conceptual primers"
%% -> "four" in Part XII organising paragraph AND reading-paths
%% theorist path; "Appendices~J--L" -> "J--M". BONUS pre-existing fix:
%% Part X preamble pointed the summary tables at Appendix~E; they have
%% lived in Appendix~D since the v11.37 move -> corrected to D.
%% ======================================================================
%% ======================================================================
%% v11.45 (2026-07-07): Part-I inline cohort citations (author request:
%% cite the 17 preprints as densely as natural so Part I works as the
%% Introduction to GG-Theory as a whole). Added \citealp/\citep at the
%% cohort-paper mentions in Sec 2 (six-dimensions story -> A2; integer
%% ledger -> A3; lock scale -> A2+A9) and Sec 3 (OGGEP -> A1; GDOS ->
%% A5; Gi-4 -> A6; Qi-4 -> A7 [its FIRST citation this early]; A8-QM
%% x1; funnel companions -> A9/A10/A11), plus papers-of-record list in
%% Sec 4 (A1/A2/A3/A9) and constitutional-route opener (A1). natbib
%% rule respected: Arisaka entries never share one \cite. All affected
%% "First cited in" Relevance notes recomputed and updated (A1, A3, A5,
%% A6, A7, A8, A9, A10, A11 -> sec:plainenglish / sec:central-claim).
%% ======================================================================
%% ======================================================================
%% v11.44 (2026-07-07): Part-I "Foundations" perfection pass (author
%% request). (1) Sec 1 SPLIT: Sec 1 now ends with the worthy-successor
%% job description (pure problem/frustration) + a one-line bridge; NEW
%% Sec 2 "GG-Theory in Plain English --- and Why the LHC Gets to Judge
%% It" (label sec:plainenglish, chunk sec2plain.tex): warm welcome for
%% first-encounter readers, the six dimensions as a story, the GG wager
%% (moved from Sec 1 with eq:Mlock-intro), item-by-item match to Sec 1's
%% job description ("first time in fifty years"), and the 2026
%% excitement paragraph (>3 sigma hint of g* possible by end of 2026
%% from data on tape, conditional phrasing). Sec 1 gains label
%% sec:fiftyyears. (2) Central-claim section REWRITTEN from scratch
%% ~2 pages (chunk sec3claim.tex, label/figure kept): no system closed
%% (single electron), 6D bulk vs 4D boundary records, OGGEP (AX-3) =
%% observation itself, GDOS forced by Gi-4/Qi-4, GRIP-In/Out runtime
%% (detector = GRIP-Out amplifier), A8-QM measurement problem resolved
%% (shifty split = geometric bulk/boundary interface, Born as
%% typicality), colliders as OGGEP at industrial scale (SIP).
%% ======================================================================
%% ======================================================================
%% v11.43 (2026-07-07): Part-I reading-flow refinements (author request).
%% (1) "Conventions, naming, and cross-references" paragraph MOVED out of
%%     Sec 1 (too technical for the introduction) and merged into "The
%%     programme in one view" passage of the constitutional-route section,
%%     with a one-line bridge.
%% (2) "Three reading paths" paragraph MOVED from Sec 1 to a NEW final
%%     section of Part I, expanded ~6x: framing paragraph + one solid
%%     paragraph per path (experimentalist / phenomenologist / theorist
%%     + the Preface's graduate-student path) + expanded Part-I closing
%%     coda + fuller bridge into Part II. Old one-paragraph coda absorbed.
%%     Sec 1 keeps a one-clause pointer to the new section.
%% ======================================================================
%% ======================================================================
%% v11.42 (2026-07-06): NEW Appendix J "What is the relation between
%% the QCD gluon and the colour-octet g*?" (author request: the Q&A
%% answer added almost verbatim, question quoted at the head). Placed
%% BEFORE the graviton primer, so the three conceptual primers read
%% J (gluon/g*: one field, ground note and first overtone) -> K
%% (graviton) -> L (one integer, two powers). Former J/K/L relettered
%% K/L/M with full Appendix~X cascade, Part XII organising paragraph
%% and Sec-1 reading paths updated. Closer: "it is the gluon --
%% heard one octave up the scale dimension." ALSO fixed pre-existing
%% mislabel: ledger reading-rule ttbar-94% ref pointed at graviton
%% primer; now Appendix~L (rate hierarchy / ttbar dominance).
%% ======================================================================
%% ======================================================================
%% v11.41 (2026-07-06): front-matter single-page finish (the last
%% residual of the v2.0 checklist). Overview-of-Contents entries
%% trimmed to ONE LINE each via a local \GGPart override that writes
%% title-only Part entries to the .toc (the "--- subtitle" tail
%% dropped from the contents line; subtitles remain in full on the
%% Part opening pages). Effect: Overview (14 one-line entries) and
%% the complete List of Figures now share page 4 exactly. The full
%% Table of Contents inherits the same one-line Part headers, with
%% section lines below carrying the detail. No content changed.
%% ======================================================================
%% ======================================================================
%% v11.40 (2026-07-06): editorial polish pass, items 2-9 of the
%% editor's read-through (author-approved).
%% (2) Exec Summary point-one: fifty-year retelling trimmed to one
%%     clause pointing at Sec 1 (the Preface and Sec 1 remain the two
%%     full tellings). (3) NEW "Three reading paths" paragraph at the
%%     end of Sec 1 (experimentalist / phenomenologist / theorist).
%% (4) Read-out~II glossed at its first body use (T-KK-2 section),
%%     three Parts before its definition. (5) Metaphor budget: middle
%%     "bet/table" instance (falsifier paragraph) trimmed -- kept in
%%     the Abstract and the Part-XI close only; organ-pipe audit found
%%     all three body uses earn their place (birth / ADD contrast /
%%     course callback) -- kept. (6) Keywords: "warped extra
%%     dimensions" -> "non-spatial extra dimensions; scale dimension;
%%     phase dimension" (Randall--Sundrum retained for search).
%% (7) Warmth passes on the openers of Parts III and V. (8) "one mass,
%%     not a scan range" audit: occurs once only -- no change needed;
%%     246.3/246 usage verified consistent (exact value once, approx
%%     elsewhere). (9) arisaka-gg.org named as the workbook location
%%     in the Part-VIII caveat ledger and the Appendix-H card.
%% ======================================================================
%% ======================================================================
%% v11.39 (2026-07-06): RATE CANONICALIZATION (editor item 1; author-
%% approved direction). The paper carried two strata: Sec 22 and its
%% tables/assets at pure LO (octet 6.7 [5.4-9.6] fb; Scenario A 172;
%% G1 tt 0.028) while Parts VI+ used LO x K (8.7 [7.0-12]; K_qq=1.3,
%% K_gg=1.6) -- with the App-H inputs table MISlabeled "LO x K" and a
%% pre-T-KK-2c fossil cluster sum (10.3 fb). VERIFIED: 6.7x1.3=8.71,
%% 5.4x1.3=7.0, 9.6x1.3=12.5, 0.028x1.6=0.045 -- one chain, two K
%% conventions. CANONICAL NOW: LO x K everywhere a prediction is
%% quoted. Swept: tab:sigmabr (all rows x1.3 / x1.6; caption rewritten
%% honestly), Sec 22 prose (8.7 [7.0-12]; Scenario A 224 [180-325];
%% exclusion factors 2.8-8.6 -> 3.6-11, ~3x -> ~3.6x; broad-limit
%% headroom ~4 -> ~2.9 below; cluster sum 10.3-FOSSIL -> 8.8-9.3 and
%% factor ~2.5 -> ~2.8 below; 14-TeV octet 10.4 [8.4-15] -> 13.6
%% [11-19]; G1 0.03/0.04 -> 0.045 tt / 0.07 total, HL-headroom ~40 ->
%% ~80), cluster-sum decomposition, App-H timeline-inputs rows, X
%% F-KK-2A narrative, XIII O-14. Kill-window quotes UNCHANGED and now
%% tighter (new band crossings 0.63-1.86 ab^-1 inside the quoted
%% 0.5-2; timeline decision window recomputed by the asset script:
%% data on tape 2031-2035). Assets regenerated with K applied at load
%% (overlay: oB/oA/zp x1.3, gtt x1.6; timeline band 7.0-12, label
%% 8.7). A9 F-SM-1 + ledger F15 updated separately (v20.21).
%% ======================================================================
%% ======================================================================
%% v11.38 (2026-07-06): END-OF-PART CODAS added to all Parts (author
%% directive; A9-SM / A10-Cosmos pattern: plain-English summary of the
%% Part + smooth bridge to the next, as closing paragraphs with no
%% special markup). Thirteen codas: I->II ... XIII->XIV. The XI coda
%% is deliberately brief and placed before \GGAcknowledgments so the
%% philosophical close keeps the last word of substance. No content
%% or numbers changed.
%% ======================================================================
%% ======================================================================
%% v11.37 (2026-07-06): "Comprehensive Summary Tables" moved from
%% Part X (Discussions) to Part XII (Technical Appendices) as NEW
%% Appendix D -- joining the reference group (A-D), which is its
%% natural home (author suggestion; optimum location = end of the
%% reference dictionaries, before the course). Subsequent appendices
%% relettered D-K -> E-L; all Appendix~X cross-refs cascaded (reverse
%% order). Part X now opens with a short preamble leading directly
%% into "Significance of the result". ALSO REPAIRED: the Part XII
%% front-matter group sentence had been silently corrupted by the
%% v11.30 letter cascade (stats course cited as G; rate-hierarchy and
%% roadmap both as K) -- the organising paragraph is now rewritten
%% with the verified letters: reference A-D, constructive E-G,
%% card H, glossary I, primers J/K, roadmap L. Labels and content
%% unchanged; tables travel with their reading guides.
%% ======================================================================
%% ======================================================================
%% v11.36 (2026-07-06): author endorsed Part XI as written; the three
%% "[drafted in the author's voice for the author's revision]" markers
%% removed (personal perspective, vision, philosophy), plus the
%% ghost-writing meta-note sentence in the limitations section that
%% pointed at them. No other changes.
%% ======================================================================
%% ======================================================================
%% v11.35 (2026-07-06): Part XI Conclusions EXPANDED ~2.5x (author
%% directive: plain English for a broader audience; far deeper
%% philosophical accounts toward the end). Now six sections:
%% (1) Established -- theorem chain + NEW plain-English restatement
%%     ("ten thousand of them sitting unrecognised in the data").
%% (2) Limitations -- prefaced with why the list exists; meta-task
%%     note that personal/vision/philosophy sections await the
%%     author's own hand. (3) Personal perspective -- expanded with
%%     the inside-the-sociology passage ("converting other people's
%%     hopes into shift schedules"; "what exactly will you concede").
%% (4) NEW "The vision: what a geometric century could look like" --
%%     plain-English; spectroscopy as the honest mode; the two
%%     cultures reunited around a number neither can move nor ignore.
%% (5) Trailhead reflection (kept). (6) NEW closing "What it would
%%     mean: matter, number, and the legibility of nature" -- four
%%     movements: falsifiability as an ethic (the promise structure);
%%     the Pythagorean question and Wigner inverted ("the twentieth
%%     century stopped one dimension short"); a world that measures
%%     itself (OGGEP consequence dragged into a detector -- "a
%%     sentence of that size with a luminosity attached"); either
%%     verdict and legibility ("the grace to listen"). All
%%     author-voice sections marked for the author's revision.
%% ======================================================================
%% ======================================================================
%% v11.34 (2026-07-06): Table tab:discrimination font enlarged (author:
%% +30-50%). Resizebox removed; \scriptsize -> \small at natural width
%% (net ~+70% glyph size vs the clamped rendering). Achieved by:
%% short column heads (GG-6/RS/CHM/Z'/MSSM/ADD/Str), trimmed row
%% labels, and the wide "earliest separating measurement" bottom row
%% moved out of the tabular into a compact footnote legend under the
%% table. Content identical (15 observables x 7 frameworks).
%% ======================================================================
%% ======================================================================
%% v11.33 (2026-07-06): two author items on Part X.
%% (1) Sec "Comprehensive Summary Tables": one-line intro -> full
%%     reading guide (how-to-read paragraph per table: master ledger's
%%     three reading rules; falsification hierarchy's two symmetric
%%     misreadings + the wrong-morphology fifth row; calendar's
%%     within-row-vs-between-row warning and the on-tape urgency).
%% (2) Table tab:discrimination (was overflowing the right margin):
%%     REBUILT transposed and expanded 6 -> 15 observables as rows x
%%     7 frameworks as columns (adds: rate derived, EW-scale tie,
%%     width predicted, P_t band, BR fingerprint, hierarchy-as-output,
%%     SUSY-absence, lepton-collider silence) + earliest-separating-
%%     measurement row; fits portrait width; caption expanded with
%%     the MSSM "---" and landscape "~" reading notes. In-text
%%     "six observables" wording updated. No physics changed.
%% ======================================================================
%% ======================================================================
%% v11.32 (2026-07-06): Part X Discussions EXPANDED ~3x (author
%% directive: A9/A10 essay register; "doubled and tripled" accepted).
%% New structure per the author's five headings:
%% (1) Significance of the result -- fifty-year-arc narrative, the
%%     epistemics of zero parameters, the over-constraint told in
%%     full, the calendar as half the claim, per-community
%%     significance, the Scenario-A episode as observed falsification
%%     machinery. (2) Uniqueness, nobility, and innovation -- PS-KK-1..6
%%     expanded to full problem-histories (750-GeV cautionary tale,
%%     naturalness erosion, Goldberger-Wise moduli tax); I-KK-1..10
%%     each with before/after novelty audit; NEW "nobility clause"
%%     subsection (four checkable commitments). Labels sec:disc-puzzles
%%     and sec:disc-innovations kept alive as subsections.
%% (3) Comparison with the other roads beyond the SM -- RS, composite,
%%     generic Z', SUSY, PLUS NEW ADD and string-landscape subsections;
%%     per-snapshot verdicts; NEW framework discrimination matrix
%%     (tab:discrimination). (4) Predictions for falsification --
%%     F-KK-1/2A/2B as narratives (declaration -> requalification ->
%%     status); each P-KK-1..10 walked individually (what/where/how/
%%     what-kills-it); NEW "falsification lattice" closing; summary
%%     table tab:predictions retained. (5) Open questions -- six items
%%     at why-open/tried/closing/impact depth. Summary of the
%%     Discussion refreshed. No numbers changed; all reused values
%%     match the archived workbooks.
%% ======================================================================
%% ======================================================================
%% v11.31 (2026-07-06): Appendices A and C reformatted to the A9-SM
%% house format (author request). App A: two-column description list ->
%% A9-style three-column longtable (Symbol | Meaning | Primary link)
%% with the sans-serif/Roman typography convention paragraph and
%% continued-page headers; ~46 entries, each now carrying its cohort
%% or in-paper primary link. App C: dense prose -> A9's itemized
%% prefix list (one bullet per prefix, with P1's registered items
%% attached in place) + canonical suffix-list paragraph + axioms note,
%% mirroring A9 App C verbatim in structure. No content changes
%% elsewhere.
%% ======================================================================
%% ======================================================================
%% v11.30 (2026-07-06): Part XII rebuilt as a GRADUATE COURSE (author
%% directive: match A9 Part XVI structure/depth; appendices have no
%% page budget). NEW front matter: mission statement (course for the
%% incoming ATLAS/CMS student), three appendix groups, the three-step
%% pedagogy (from-scratch derivation + worked example / what the
%% textbook cannot fix / the GG step), the two stylistic commitments
%% (no skipped manipulations; physics-intuition parentheticals),
%% prerequisites + recommended textbooks (5 new bibitems: Halzen-
%% Martin, Ellis-Stirling-Webber, Peskin-Schroeder, Thomson, Cowan
%% 1998). CONSTRUCTIVE COURSE: App D expanded into the KK-spectrum
%% course (flat circle -> warped Bessel -> couplings-as-overlaps, all
%% steps shown, worked numbers; old formulas kept as the Reference
%% subsection); NEW App E parton-luminosity course (factorisation ->
%% dL/dtau -> NWA master formula -> worked 8.7 fb); App F statistics
%% course (bin-by-bin worked arithmetic table reproducing the 2.7
%% sigma, verified against recomputation; absorption moral; worked
%% LEE arithmetic; old workbook kept as Reference subsection).
%% App G card + NEW implementation walkthrough (5 steps). App H
%% glossary (was G). App I graviton + NEW massive-spin-2 technical
%% paragraph (Fierz-Pauli/rho-meson) + would/would-not-settle ledger.
%% App J rate hierarchy + NEW overlap integrals written out (kappa
%% closed form displayed) + width table assembled. App K roadmap +
%% per-item project guidance. Lettering D-K; all cross-refs updated.
%% ======================================================================
%% ======================================================================
%% v11.29 (2026-07-06): FIVE-PART TAIL, STEP 5 (FINAL) -- Part XIII
%% Common Objections and Responses (A9 Part XVII format strictly:
%% objqabody environment, categories as \section with \bigskip,
%% "Objection N (O-KK-N) (the X sceptic)" + bold quoted objection +
%% located response). FIFTY-SEVEN entries in seven clusters:
%% methodology/derivation chain (1-10), mass & fine structure (11-18),
%% couplings/rates/suppressions (19-28), simulation & statistics
%% (29-38), reducibility & non-spatial dimensions (39-46), cohort
%% dependence & standing (47-52), experimental practice (53-57).
%% Hostile-reviewer-first per the cohort Q&A standard; every response
%% carries pointers into the paper; honest "not yet" where true
%% (L-KK-1, NLO, n>=2). Inserted before the References Part, which
%% becomes Part XIV -- the five-Part tail (X Discussions, XI
%% Conclusions, XII Technical Appendices, XIII Objections, XIV
%% References) is now COMPLETE. Paper = 14 Parts.
%% ======================================================================
%% ======================================================================
%% v11.28 (2026-07-06): FIVE-PART TAIL, STEP 4 -- Part XII Technical
%% Appendices completed (A9 Part XVI as template).
%% NEW apparatus: Appendix A Summary of abbreviations (two-column,
%% ~45 entries); Appendix B Glossary of key symbols (23-row table with
%% homes); Appendix C Global naming convention (A9 App C mirrored +
%% P1's registered-item roster). EXISTING appendices relettered:
%% KK reduction A->D, significance workbook B->E, reinterpretation
%% card C->F, collider glossary D->G (one body cross-ref updated).
%% NEW Appendix H "What is the 'graviton' of GG-Theory?" (the two
%% meanings; pedigree vs job description; the OGGEP inversion --
%% quantization as output of the projection; the O_14 fence) and
%% NEW Appendix I "One integer, two powers" (production hierarchy
%% xi^2~1/K vs kappa^2~0.22/K^2 with the flat-mode/near-orthogonality
%% derivation; decay hierarchy: heaviness = cluster-affinity, 280 per
%% flavour, both BR fingerprints; strategic consequence) -- both from
%% the author Q&A of 2026-07-06. NEW Appendix J Promotion roadmap
%% (six open items: grade / closing condition / impact).
%% multicol added to preamble. Part XII subtitle updated.
%% ======================================================================
%% ======================================================================
%% v11.27 (2026-07-06): FIVE-PART TAIL, STEP 3 -- Part XIV References
%% (author-directed reordering of the plan: References before the
%% remaining appendix/objection steps).
%% (1) SEVEN missing cohort bibitems added with P1-specific Relevance
%%     paragraphs (A10, A11, P2, P3, C1, C2, C3; canonical A1 headers)
%%     and cited at their natural home (the programme-in-one-view
%%     paragraph). All 16 sister papers now cited.
%% (2) ELEVEN key external references added, each cited in text and
%%     carrying a solid Relevance paragraph: Hosotani 1983 +
%%     Goldberger-Wise 1999 (Sec 4.5 + comparison table rows);
%%     Weyl 1918/1929, Minkowski 1908, Connes-Lott 1991, Maldacena
%%     1997/98 (the genealogy paragraph); Fukuda 1998 (Super-K) and
%%     ATLAS/CMS Higgs 2012 (the Sec-1 history); PDG 2024 (the alpha_s
%%     world average in Part IX).
%% (3) \GGBibBegin replaced by \GGPart[References] + three-paragraph
%%     intro + \renewcommand{\bibsection}{\vspace{1.2em}}; bibliography
%%     split into Section A (cohort, canonical order) and Section B
%%     (other references, alphabetical), each its own thebibliography.
%% (4) "First cited in ..." appended to EVERY entry's Relevance note
%%     (computed programmatically; front-matter cites labelled as the
%%     Executive Summary). (5) CMS2024UpgradeWeek thin Relevance
%%     expanded. Orphans: zero (verified before and after).
%% ======================================================================
%% ======================================================================
%% v11.26 (2026-07-06): FIVE-PART TAIL, STEP 2 -- Part XI Conclusions
%% written in full (A9 Part XV as template, 4 sections):
%% (1) What the paper has established -- the earned-results chain
%%     (T-KK-1/2/2c, Parts VI/VIII/IX) prepended to the retained
%%     collider statement; (2) Objective limitations and next technical
%%     tasks (theory-side and experiment-side task lists); (3) A
%%     personal perspective: fifty years on the experimental front line
%%     -- P1-specific arc (five front lines from A9's account; "the
%%     other side of the transaction"; the Preface's student = the
%%     author in 1976; the calendar and the intent to watch the verdict)
%%     -- DRAFTED IN THE AUTHOR'S VOICE FOR THE AUTHOR'S REVISION,
%%     marked as such in the text; (4) The trailhead, not the summit --
%%     final reflection (Balmer analogy; no zombie phase; "let the data
%%     speak"). \GGAcknowledgments stays at Part end. NEW cohort bibitem
%%     ArisakaA7Qi4 (canonical A1 form) for the origin-story citation.
%%     No values changed.
%% ======================================================================
%% ======================================================================
%% v11.25 (2026-07-06): FIVE-PART TAIL, STEP 1 (author-approved plan,
%% A9-SM Parts XIV-XVIII as the strict template).
%% (1) Part X "Synthesis" -> Part X "Discussions", written extensively:
%%     nine sections -- summary tables (master ledger REBUILT: the old
%%     table still said "2 dominant"/"c_X=O(1)"/clean channels, all
%%     pre-T-KK-2 fossils; falsification hierarchy kept; NEW calendar
%%     recap table); significance & uniqueness; classic collider
%%     puzzles as PS-KK-1..6; comparison with RS / composite / generic
%%     Z' / SUSY (observational discriminators); structural innovations
%%     I-KK-1..10; predictions & falsification (F-KK registry restated
%%     + NEW numbered predictions P-KK-1..10 with grades); open
%%     questions (six, with closing conditions); the completed cohort;
%%     summary of the Discussion.
%% (2) NEW Part XI "Conclusions" scaffolded: existing Conclusions
%%     section moved in verbatim; \GGAcknowledgments relocated to the
%%     end of this Part (house standard). Expansion (personal
%%     perspective etc.) is the next step of the plan.
%% (3) NEW Part XII "Technical Appendices" wraps Appendices A-D
%%     (single \appendix call retained).
%% (4) NAMING: per A9 Appendix C (author directive), O-labels are
%%     reserved for Objections. The g_B menu item O-KK-1 is RETAGGED
%%     L-KK-1 (open selection lemma) throughout (23 instances).
%%     A9's two body mentions retagged separately (A9 v20.20).
%% Paper 10 -> 12 Parts this step (14 after the remaining steps).
%% ======================================================================
%% ======================================================================
%% v11.24 (2026-07-06): three refinements (author items).
%% (1) Leaked internal label names repaired: prose "def:GRIPin/
%%     def:GRIPout" -> "GRIP-In and GRIP-Out"; A6 bibliography entry
%%     version "v9.7" dropped + label names -> plain names. Full leak
%%     scan: no others. (2) NEW paragraph "The three lowest states,
%%     physically" in Sec 5.1: n = radial node count (organ-pipe
%%     overtones of the scale throat); ratios 1:1.831:2.655 ->
%%     3.849/7.047/10.219 TeV; composite-resonance reading; n=1 LHC,
%%     n=2 FCC-hh; cites tab:kkrsgg in main text. (3) Figure
%%     fig:lhc-signature upgraded from illustrative TikZ to the
%%     QUANTITATIVE asset Arisaka-GG-P1-KK-quantsig.pdf (14-TeV
%%     truth-level dsigma/dm, computed: SM continuum, coherent octet
%%     with below-SM deficit, G1 alone x280 below, Scenario-A cross,
%%     n=2 marker); caption rewritten -- nothing schematic.
%%     No values changed.
%% ======================================================================
%% ======================================================================
%% v11.23 (2026-07-06): Subsection "Why GG-6 is not reducible to
%% ordinary KK or RS" expanded to ~3 pages (author request), grounded
%% in a fresh reading of A2-GG6 v11.29 (Theorem 0 + Remark 0.4, Theorem
%% 2, the generator classification, the two-tower KK appendix, the Weyl
%% genealogy). New content: the classification principle (dimension =
%% its generator; iso(1,3)+R_D+u(1)_Q, not iso(1,5); dimensionless
%% coordinates; no radial speed of light); scale-vs-RS in four bites
%% (coordinate IS the RG scale -- motion = transmutation; integer K=35
%% vs tuned kr_c, hence NO radion; resonances = boundary composites,
%% holography made literal; multi-field cluster vs single pole);
%% phase-vs-KK (U(1) fibre as literal coordinate; holonomy = quantum
%% phase; charge quantization = characters (M=1); Hosotani Wilson line;
%% no radius modulus -> angular tower at E_GUT -> falsifiable ABSENCE
%% of angular satellites); BI-6 locking retained; calibrated novelty
%% claim (Weyl 1918/1929 ingredients vs the forced synthesis).
%% NEW comprehensive comparison table tab:kkrsgg (16 rows x 4 columns,
%% full page). Label subsec:notKKRS added; eq:BI6repeat retained.
%% No physics or values changed.
%% ======================================================================
%% ======================================================================
%% v11.22 (2026-07-06): NEW PART IX "Beyond the HL-LHC" (author request).
%% Old Sec 27 (end of Part VII) and Appendix E (+Table 26) removed and
%% rebuilt from scratch as a dedicated Part between Part VIII and
%% Synthesis (paper 9 -> 10 Parts; no stale count prose existed).
%% Contents: (1) facility landscape as of July 2026 from primary
%% strategy documents -- ESPP 2026 Update adopted 22 May 2026 (FCC-ee
%% preferred flagship, Council decision by 2028); FCC-hh FSR baseline
%% 84-85 TeV (100-TeV numbers rescaled ~x0.7, stated explicitly);
%% CEPC TDR done but NOT in China's 15th Five-Year Plan (resubmit
%% ~2030); muon collider staged 3 TeV ~2050 / 10 TeV ~2070; ILC
%% non-committal; CLIC parked; C3/HALHF/LHeC R&D. (2) Six-item
%% post-discovery checklist (assume HL-LHC discovery by ~2035).
%% (3) Machine-by-machine: FCC-hh 85-TeV rates computed with the
%% paper's own CTEQ6 machinery (octet 2.8 pb, G1 0.13 pb, G2 ~10 fb
%% same-coupling est., recurrences ~1e2 fb, leptophobia to 1e-6);
%% FCC-ee parameter-ledger arm (alpha_s vs 1/(pi e) decisive;
%% no-radion Higgs precision); muon collider = leptophobia machine
%% (predicted s-channel silence at 3.85 TeV); LHeC PDF sharpening.
%% (4) Concrete scenarios 2035-2070 (D and X). (5) Summary matrix
%% table replacing Table 26 + updated verdict/spectroscopy/
%% leptophobia/ledger taxonomy. New assets: none. New bibitems:
%% ESPP2026Adoption, CEPC2025TDR, IMCC2024Interim. Labels sec:beyond
%% moved to the new Part; tab:futuremachines retired (no dangling
%% refs, checked). New lumi grid lumi_grid85.npz + fcc85_numbers.json
%% in [v11-Support].
%% ======================================================================
%% ======================================================================
%% v11.21 (2026-07-06): internal-history scrub (author directive: this
%% is the public's FIRST version; nothing may reference prior drafts).
%% Removed/rewrote every reader-facing trace of internal development:
%% "The previous version of this paper..." (Sec 5.5 ledger + Step 6
%% caution, both recast as hypothetical framings); "An earlier version
%% ... O(1) coefficient" (recast as the most-conservative-statement
%% framing); "no number is carried over from any earlier draft"
%% (clause dropped); "of the earlier drafts" (dropped); "Supersession
%% (v11.11)" -> "Superseded by the exact result" (version number
%% removed; "historical"/"old corridor" -> neutral); A2 citation
%% version "v9.23" dropped; "the earlier O(1) flag" x2 recast;
%% "the paper's earlier estimate" -> "the leading-order scaling";
%% "earlier worksheets" -> dropped; "original estimates ... for
%% continuity" -> "Read-out-II estimates ... for comparison".
%% Kept: in-paper requalification logic (F-KK-1 restated by T-KK-2c
%% within this document), cohort-map "Historical lineage" (A4's role),
%% Kaluza "historical origin". No physics or values changed.
%% ======================================================================
%% ======================================================================
%% v11.20 (2026-07-06): "spin-2-dominant" fossil repaired (author query).
%% The phrase predated Corollary T-KK-2c (which derived kappa_gg=0.47/K
%% and moved the rate to the octet by a factor ~200). All seven
%% occurrences replaced with "spin-2-anchored" (+ "octet-led" where the
%% sentence is about discovery), incl. the schematic-figure node text
%% and the Appendix-C card's Primary target line. NEW terminology
%% paragraph in Part II ("Anchored by spin-2, discovered through
%% spin-1") stating the two senses quantitatively: anchor = exact-mass
%% theorem + geometry-naming quantum numbers; rate = xi_UV^2 = 1/K
%% (octet, qq) vs kappa_gg^2 = (0.47/K)^2 (graviton, gg) -- a
%% two-orders-of-magnitude gap at K=35. No values changed.
%% ======================================================================
%% ======================================================================
%% v11.19 (2026-07-06): TERMINOLOGY UNIFICATION (author request).
%% The paper carried two unrelated A/B pairs. Resolution:
%% (1) Part-V read-outs renamed Scenario A/B -> Read-out I/II
%%     (theorem-only / empirically-anchored); assumption labels B1-B4
%%     kept; internal labels sec:scenarioA/B, tab:scenarioAB unchanged.
%% (2) The T-KK-2 dichotomy renamed Branch A/B -> Scenario A/B,
%%     now matching the falsifier lettering F-KK-2A/2B; all prose
%%     "free-field/composite branch" -> "scenario"; "on ... branch" ->
%%     "in ... scenario". The cohort-map phenomenology Branches
%%     (Predictions / Cosmology & Astrophysics), the "radial branches"
%%     mode families, decay-channel "photon/hh branch", and all
%%     "branching" (BR) words are intentionally untouched.
%% Overlay asset legend regenerated ("g* Scenario A ... excluded").
%% Lag layers flagged for A9-SM + falsifier ledger (say "branch A/B").
%% No physics or values changed.
%% ======================================================================
%% ======================================================================
%% v11.18 (2026-07-06): Step 6 dichotomy made concrete (author request).
%% (1) Before the branch table: precise physical definitions of Branch A
%%     (weak-coupling free-field eigenmodes; octet at 2.461 TeV with
%%     172 [140-250] fb -- unmissable) and Branch B (strong-coupling
%%     composite reorganisation; degenerate cluster at M_lock, 8.7 fb);
%%     identical K-fixed couplings in both -- each branch a complete
%%     mass+rate prediction, no dial. (2) After the table: new
%%     "Status of the dichotomy" paragraph -- branch A is excluded
%%     (F-KK-2A fired, factor 2.8-8.6 above the published limits,
%%     provisional only in template shape), and why the rejection is
%%     legitimate: a discrete pre-registered either/or is not a
%%     parameter slide; the exclusion is itself physics (the throat
%%     vectors are composite); the escape cannot recur (no third
%%     branch; B is the entire remaining hypothesis space). Numbers
%%     identical to Sec. operational; no values changed.
%% ======================================================================
%% ======================================================================
%% v11.17 (2026-07-06): COSMETIC PASS (author items 1-8).
%% (1) Fig 1 funnel: resizebox removed, natural size, box width 92mm --
%%     matches A9-SM Fig 1. (2) Fig 2 cohort map: 0.85 -> 0.9\textwidth
%%     -- matches A9-SM. (3) Table 9 (generic-rs-gg6): landscape ->
%%     portrait, same \scriptsize, cols 2.2/3.8/3.8/4.8cm. (4) Table 15
%%     (status): landscape -> portrait one page, \scriptsize. (5) Table
%%     20 (trigger): landscape -> portrait one page, \scriptsize.
%%     (6) Table 26 (futuremachines): landscape -> portrait one page,
%%     \scriptsize, tabcolsep 3pt. (7) Fig 10 (absorb): right bar chart
%%     re-styled -- soft pastel fills with coloured edges, slimmer bars
%%     (asset regenerated in [v11-Extra]). (8) Table 25 (falsification):
%%     Level column 3.2 -> 4.8cm, \small -> \footnotesize.
%%     Landscape ledger tab:exclusion-ledger intentionally untouched
%%     (not in the author's list). No content changes.
%% ======================================================================
%% ======================================================================
%% v11.16 (2026-07-06): FRONT-MATTER OVERHAUL (author-directed).
%% (1) Abstract rewritten in the accessible house register (1901 chars,
%%     fits page 1): the fifty-year framing, the no-dial derivation,
%%     the cluster, the frontier, and the Part-VIII discovery calendar
%%     (2.7-3.7 sigma/expt on tape; combined improved 4.6-5.2 sigma;
%%     5 sigma luminosity during 2032). (2) Executive Summary rewritten
%%     jargon-minimised, restructured around the GG principle (integers
%%     -> no retreat -> predictive power), with the new fifth point
%%     "the discovery may already be on tape" and the systematics
%%     lesson; four-audience close now includes the graduate student.
%% (3) NEW Preface (\GGPreface, after ToC, before Part I): addressed to
%%     ATLAS/CMS graduate students -- the fifty-year silence, the
%%     thesis-sized opportunity, what a positive result would open,
%%     the honest cost of a null, and a reading map. (4) Section 1
%%     retitled "Introduction: Fifty Years Beyond the Standard Model"
%%     and rebuilt: the confirmation era 1974-2012, the three roads
%%     (MSSM parameter-sliding, RS/KK input scales, string landscape)
%%     stated plainly, the structural diagnosis (dials forbid
%%     forbidding), the successor criteria, and the GG wager; original
%%     four-aims / reading-map / normalization / conventions paragraphs
%%     retained. No physics, values, or triage touched.
%% ======================================================================
%% ======================================================================
%% v11.15 (2026-07-06): NEW PART VIII -- "Simulating the Discovery".
%% The full HL-LHC simulation report folded in at graduate-textbook
%% depth, expanded (not condensed): the anchor ATLAS analysis read
%% cut-by-cut (backgrounds/trigger/selection/reconstruction/epsA
%% decomposition/resolution); the simulation methodology (smearing
%% convolution, anchored (A,n) background, 14-TeV-equivalent folding);
%% discovery statistics derived from the Poisson likelihood up (q0,
%% Wilks, Asimov, Cowan formula, the systematics wall L*=1/(f^2 b) with
%% per-bin numbers -- the search leaves the stat-limited regime during
%% Run 3); calibration F_cal=4.33 with broad-shape validation and the
%% double-counting caution; improved-syst = halved post-fit f_b with
%% three-class decomposition; the significance ladder 23.6 -> 12.8 ->
%% 4.0 -> 2.7 sigma with the analytic absorption formula; the four
%% snapshots 2026/2031/2033/2041 (end-2026 reanalysis: 2.7-3.7 sigma
%% per expt from data on tape); the pre-registered search (trials ~75,
%% local-5 = global-4.1; matched-template x3; angular 10-30% bounded
%% not banked; ATLAS+CMS combination table -- end-2026 combined
%% improved 4.6-5.2 sigma, 5 sigma from tape a live possibility) and
%% the contaminated-sideband warning (I/SM ~ shat/(shat-M^2), pin norm
%% below 1 TeV, NLO interference K-factor requested). Three figure
%% assets Arisaka-GG-P1-KK-hllhc-{spectra4,zcurve,absorb}.pdf. Three
%% new bibitems (Cowan et al. 1007.1727; Gross-Vitells 1005.1891; HL-LHC
%% Yellow Report CYRM-2019-007). Pointers from Sec. 15.2 and Part VI
%% intro. Paper goes 8 -> 9 Parts (no stale count prose existed).
%% ======================================================================
%% ======================================================================
%% v11.14 (2026-07-05): NEW PART VI -- "Anatomy of the ttbar Signal:
%% Derived Angular, Polarization, and Line-Shape Structure". The full
%% spin-discrimination programme, at complete derivation depth: helicity
%% formalism + Wigner-d table; spin-1 (3/8)(1+cos^2) and spin-2 gg
%% (5/8)(1-cos^4) derived step by step with 0.8% mass corrections;
%% A_FB = 0 as a corollary of the vector-like xi_UV [T]; the g*-QCD
%% interference line shape with the derived sign xi_UV g_V^t < 0 --
%% constructive below / destructive above the pole (+3.75 / -0.48 fb),
%% SM-crossing at ~4.9 TeV, -O(100) GeV centroid pull; Z'_B zero-
%% interference dichotomy (colour); P_t = +0.906 [0.88-0.92] vs 0 (G1)
%% with the lepton-analyzer derivation; honest-weak rapidity verdict;
%% LO/LO S/B = 3.5 window, 19 on-pole, ~10^4 events per 3 ab^-1,
%% N_3sigma = 21 for spin by shape; discrimination matrix + pre-
%% registered five-clause discovery fingerprint; LOxK rates with
%% verified NLO citations (Chivukula et al. 1111.7261; Mathews-
%% Ravindran hep-ph/0506158). Paper goes 7 -> 8 Parts (no stale count
%% prose existed). Three figure assets Arisaka-GG-P1-KK-ttrep-*.pdf in
%% [v11-Extra]. Pointers added in Sec. 8.3, the operational survival
%% paragraph, and the Appendix-C card. Two new bibitems.
%% ======================================================================
%% ======================================================================
%% v11.13 (2026-07-05): FALSIFICATION TIMELINE IN CALENDAR YEARS. New
%% closing subsection of the Near-Term Discovery Potential section:
%% (1) comprehensive numerical-inputs table (GG bands / 95%-CL anchors at
%% 3.849 TeV / luminosity-on-tape model on the July-2026 CERN schedule),
%% (2) full-page sensitivity-vs-year figure (bands = derived predictions,
%% curves = sqrt(L)-scaled projected limits, 2025-2050), (3) caption,
%% (4) full prose description. Schedule facts updated to July 2026: Run 3
%% ended 14 Jun 2026 (~540 ifb delivered; ~466 ifb on tape), LS3 to 2030,
%% HL-LHC from mid-2030 (~3.4 ab^-1 by end-2041), ESPP Dec 2025 selects
%% FCC-ee (physics ~2047+), FCC-hh ~2074. Decision window: the broad
%% ttbar limit crosses the octet band with data on tape 2032-2040. Three
%% new bibitems (CERN2026LS3, ESPP2025, FCC2025FSR); one pointer in the
%% kill-window subsection. Figure asset Arisaka-GG-P1-KK-timeline-
%% sigmaBR.pdf in [v11-Extra]. Author-requested.
%% ======================================================================
%% ======================================================================
%% v11.12 (2026-07-05): THE BARYONIC NORMALISATION g_B -- completed to the
%% honest boundary. What IS a theorem (A3 X4 integrality + A9 Arisaka-
%% equation lock): the generator norm Tr(B0^2)/Tr(Y0^2) = (L-N)/L = 2/5,
%% verified on the primitive 5bar. What remains is NOT a continuous dial
%% but one discrete rule -- the seam-termination readout for a Stueckelberg
%% -massive Abelian factor -- registered as O-KK-1 with its four natural
%% readouts: g_B in {0.14, 0.23, 0.57, 0.89} (from GG's own alpha^-1=2^J
%% =128 and sin^2thW=25/108 -> g_Y=0.357). Menu-invariant collider
%% conclusion: sigma x BR(Z'_B->ttbar) = 0.012--0.46 fb at 3.85 TeV --
%% LHC-invisible across the whole menu (>=10x below current limits;
%% >=2.3x below the 3 ab^-1 projection); octet remains sole carrier of
%% the HL-LHC falsifier; the (g_B,m_B) fifth-force plane collapses to
%% four candidate lines for the dedicated fifth-force paper. Lag layers:
%% widths paragraph, exclusion-edge note, operational inputs/table/
%% survival/cluster, T-KK-2c bullet, Appendix C. Author-approved.
%% ======================================================================
%% ======================================================================
%% v11.11 (2026-07-05): COROLLARY T-KK-2c -- the graviton's gluon (and
%% photon) coupling DERIVED from the rigid geometry: kappa_gg =
%% 2(1+J2(x1))/(K x1^2 J2(x1)) = c_K/K = 0.0136 (c_K = 0.4744), the same
%% free-field Bessel machinery and shelving as T-KK-2. Consequences:
%% sigma(G1) = 0.041 fb at 13 TeV (LHC-invisible; 40x below the 3 ab^-1
%% projection), 205 fb at FCC-hh 100 TeV; the diphoton null is RETRO-
%% dicted by a factor ~4.6e3 (a uniform-1/sqrt(K) photon amplitude would
%% have been excluded ~4x). F-KK-1 requalified into two tiers: the HL-LHC
%% falsifier is carried entirely by the un-suppressible colour octet
%% (F-KK-2B); the direct spin-2 test moves to FCC-hh. Full lag-layer
%% sweep: abstract, T-KK-1 falsifier + scope note, schematic-figure
%% caption, observable ledger, Scenario A/B (kappa_eff supersession),
%% operational section (G1 rows, survival, kill window), GG-vs-RS column
%% logic, Beyond-HL-LHC, Appendix C. Overlay figure regenerated.
%% Author-approved 2026-07-05 ("the collider falsifier is carried
%% entirely by the octet").
%% ======================================================================
%% ======================================================================
%% v11.10 (2026-07-05): sigma x BR OPERATIONALIZATION (hostile-panel item
%% R4.2/R5.2a). New section "The Operational Confrontation" in the
%% Experimental Status Part: explicit LO parton-luminosity computation of
%% sigma x BR for G1, g-star (both T-KK-2 branches), Z-prime-B (as g_B^2
%% scaling), validated against published benchmarks; two-panel overlay
%% figure against HEPData-digitized ATLAS/CMS 95-percent-CL limits; per-
%% channel survival statement; quantified F15/HL-LHC kill window. Six new
%% bibliography entries (HLZ, AOPW, Dobrescu-Yu, Franceschini et al.,
%% CTEQ6, CMS dijet 137 ifb). Seven surgical pointer edits harmonize the
%% Scenario-A discipline statements with the new computed material.
%% Findings flagged for author: (i) free-field branch A rate sits a factor
%% ~2.8 ABOVE the observed ATLAS broad-template limit (provisional closure
%% of F-KK-2A); (ii) the CMS diphoton null already cuts the uniform
%% 1/sqrt(K) reading of the photon overlap (kappa_gamma < ~0.08).
%% Science frozen elsewhere; all new numbers graded Derived-conditional.
%% ======================================================================
% v11.9 (July 2, 2026): cohort universal-indentation pass (author-
%   directed): indentfirst loaded (first paragraphs after headings
%   now indent). Per-paper stray-noindent review deferred to the
%   v2.0 pass. No content changed.
% v11.8 (July 2, 2026): cohort multi-Arisaka citation split pass
%   (charter natbib rule; author-directed 2026-07-02): 4
%   multi-key cites split into citealp/citet chains; fixes the
%   same-author-same-year comma-run rendering. No content changed.
% ===== v11.7: COSMETIC table-font normalization -- inserted \small before
%  \begin{tabular} in 10 oversized (unsized) table floats so every table
%  sits ~10%% below the body text (matches A10 v12.43). Already-small tables untouched.
% ===== P1-KK v11.6 (2026-06-27): Round-2 Tier-2 -- added 'Exclusion-edge honesty' paragraph after the
% provenance table: states plainly the leptophobic Z'_B sits ON the current ATLAS/CMS dijet exclusion
% frontier and survives by leptophobia (sharpening the HL-LHC test); fences g_B and the (g_B,m_B)
% fifth-force window as deliverables of the dedicated fifth-force paper, matching A9 F-SM-5. =====
% Bib cross-citation titles reconciled to canonical cover titles (cross-citation integrity), 2026-06-27.
% ===== P1-KK v11.4 (2026-06-27): cohort Wave-2 de-contamination (protect-list aware). =====
% Shed GG's OWN self-description of its 6D bulk/throat/geometry; mapped self-adjectives to
% scale-profile / radial / RG-scale / bulk-to-boundary language per the cohort map. KEPT every
% Randall--Sundrum and KK-vs-GG comparison token, the RS comparison block and both comparison
% tables, AdS / AdS-like curvature names, the external-literature and Scenario-B RS citation
% contexts, the RS bibliography entries, and the RG-running disclaimer footnote content. Body
% only; all label/ref/eqref args and percent-comments untouched. No physics/results/values changed.
% ===== P1 v11.3 (2026-06-25): cohort central-claim funnel (consequence variant). =====
% Added the shared open-system funnel (sec:central-claim/fig:central-claim) with this paper's own
% payload. Same funnel as A9/A10/A11. No new physics.
% =====================================================================
%  P1-KK v11.2 (migrated onto ucla-arisaka-gg-11) -- 2026-06-22
%  gg-10->gg-11; dropped \LaTeX shim; stripped Overview/LoF overrides; single-page toggle; homogenized Acknowledgments.  No prose/figure/number content changed.
% =====================================================================
% ---------------------------------------------------------------------
%  Arisaka-GG-P1-KK-v11.0 (Stage 3C cross-reference sweep) -- 2026-06-20
%  Added \label{thm:tkk1}, \label{thm:tkk2} to the prose-stated theorems so the cohort can \ref them (registry-flagged).
% ---------------------------------------------------------------------
% ===== v11.0 (2026-06-16): holographic-RG reframing, Phase R2c. =====
%  Terminology only; NO physics changed (M_lock=3.85 TeV unchanged). P1 already states
%  reading (A): radial '(RG-scale) lock cluster', non-spatial KK, distinct from 5D RS;
%  A2/A3 bibitems AdS-free; 'AdS curvature k' = the no-free-metric-data argument (kept).
%  Added the holographic-RG disclaimer footnote making the reading-(A) M_lock linkage
%  explicit; v10.5 -> v11.0.
% =====================================================================
% --- v10.5 (2026-06-16): adopt cohort-default Part-aware running header (sty v10.6).
%     Curated short Part labels added; running header shows 'Part N. <short>'. ---
% GU-5 RENAME (June 14, 2026): cohort harmonization with GG-A4-Euclid -- the five-constant inventory and its theorem are named Geometric Universal Five Constants / GU-5 (\GUfive, \T{-GU5}); the older 'Conversion-Factor Pentad'/'Conversion-Factor Triad' wording and the 'T-CFP' theorem tag are retired.
\pdfoutput=1
%% v10.4 (June 12, 2026): Cohort-map deployment (author Go):
%%   the canonical seventeen-paper cohort map inserted (label
%%   fig:cohort-map-p1, this paper highlighted) with a one-paragraph lead-in,
%%   at the surveyed seam where the introduction completes.
%%   No other content changes.
%% v10.3 (June 10, 2026): arXiv preparation pass (front-matter audit).
%%   four tcolorbox panels unboxed (house no-box pattern).
%%   Accepted variants (author decision): existing philosophical closes
%%   in place of literal 'After all'; companion-list counts as cited.
%% v10.2 (June 9, 2026): arXiv preparation pass (cohort step 3).
%%   Overfull-box sweep (all boxes >20pt resolved with layout-only fixes);
%%   no content or physics changes.
%% v10.1 (June 9, 2026): arXiv preparation pass (cohort step 1-2).
%%   \pdfoutput=1 added (line 1); version/date stamp bumped. 
%%   No physics changes; production fixes only.

% ------------------------------------------------------------
% v15.9, August 19, 2026.  The shared bulk showcase plate.
%%   ADOPTED, at the author's direction: one full-page showcase
%%   plate carried BYTE-IDENTICALLY by A2, A5, A8, A9, A10 and
%%   P1 -- the cohort-map precedent.  It is the integration of
%%   three existing figures: A1 Figure 1 supplies the logic
%%   (two necessities, AX-2 building the fiber, AX-3 projecting
%%   to the registered world), A10 Figure 13 supplies the
%%   tapering throat and the projection arrow, and A2 Figure 8
%%   supplies the calibrated scale axis.  The axis now carries
%%   eight ticks with mu = E_GUT e^-r beside each, E_GUT =
%%   E_Pl e^-2pi at r = 0, E_Pl declared OFF the diagram at
%%   r = -2pi, the lock at r = 29.41 and the OGN-5 depth at
%%   r = K = 35.  Every number was computed, not carried over.
%%   THE OFFSET IS EXACT: K is counted from the bulk anchor
%%   E_Pl/2 and the axis from E_GUT, so the two differ by
%%   ln(E_Pl/2 / E_GUT) = 2pi - ln 2 = 5.590 and
%%   r_lock = K - 2pi + ln 2 = 29.410.  That relation is what
%%   reconciles A2's "lock at r = K" with A10's "lock crossing
%%   at r = 29.4"; both name 3.85 TeV.
%%   The plate is pure TikZ and adds NO asset to the arXiv
%%   package.  Caption ABOVE, per the frozen float standard.
%%   The drawn taper is e^-r/13.64 -- the true e^-r slowed so
%%   the throat stays visible -- and the plate says so.
%%   NEW PROSE: the plate's labels and its caption, listed in
%%   the report for the author to review or rewrite.
% ------------------------------------------------------------
%% ---- vv15.10 (August 19, 2026) ----
%% The front-matter showcase pages given breathing room.
%% A showcase page's free height is now distributed by weighted stretch
%% glue -- a sixth at the top, a third above the plate, a third below it,
%% a sixth at the foot -- instead of pooling above the headline and below
%% the caption, which left a broad band of white at each end of the page
%% and crowded the three elements into the middle third.
%% Three source forms were found in the cohort and each is handled.  A
%% minipage frame becomes a minipage of EXPLICIT height with the [s]
%% inner alignment, because a minipage is a vbox of natural height and
%% glue inside one does not stretch; the height is \textheight minus
%% \topskip.  A \begingroup frame keeps its page-level glue and changes
%% only the weights.  An unframed page -- there were eight, five of them
%% showcase tables carrying a longtable, which cannot live inside a
%% minipage -- gains page-level glue for the first time, which is why it
%% had been printing hard against the top of the text block.
%% 2 showcase pages in this paper.  Layout only: not one word,
%% number, figure or table row is changed by this ship.
\documentclass[11pt,letterpaper]{article}
\usepackage{ucla-arisaka-gg-11}

%% v11.41 local override: one-line Part entries in Overview/ToC
%% (title only; the subtitle stays on the Part opening page).
\makeatletter
\renewcommand{\GGPart}[3][]{%
  \clearpage
  \refstepcounter{part}%
  \def\gg@parthdr{#1}\ifx\gg@parthdr\@empty\def\gg@parthdr{#2}\fi
  \addcontentsline{toc}{part}{\protect\textbf{Part \thepart}. #2}%
  \markboth{Part \thepart.\ \gg@parthdr}{Part \thepart.\ \gg@parthdr}%
  \thispagestyle{plain}%
  \begin{center}
    \vspace*{1.2em}
    {\Large\bfseries Part \thepart}\\[0.45em]
    {\huge\bfseries #2}\\[0.35em]
    {\large #3}
  \end{center}
  \vspace{0.5em}
  \GGThickLine
  \vspace{0.8em}
}
\makeatother
\usepackage{indentfirst}
\usepackage{amsmath,amssymb,amsthm,mathtools,bm}
\usepackage{booktabs,longtable,tabularx,array,multirow,makecell}
\usepackage{pdflscape}
\usepackage{enumitem}
\usepackage{multicol}
\usepackage{tcolorbox}
\usepackage{tikz}
\usetikzlibrary{arrows.meta,positioning,calc,fit,shapes.geometric}
\usepackage{microtype}
\usepackage{afterpage}
\usepackage{float}
\usepackage{caption}
\usepackage{url}

\renewcommand{\ASix}{\textsf{GG-A6-GRIP}} % v13.5: sty default still renders the retired Gi4 form; cohort renewal (A10/A11/P5 precedent)
\providecommand{\PaperPFour}{\textsf{GG-P4-Real}}% v13.5: Foundations Branch member (canonical form, Cohort-Map-19 v1.4; sty defines no \PaperPFour)
\providecommand{\PaperPFive}{\textsf{GG-P5-CPT}}% v13.5: Foundations Branch member (canonical form, Cohort-Map-19 v1.4; sty defines no \PaperPFive)

\PaperSeries{GGP}
\PaperNumber{Arisaka-GG-P1-KK}
% ============================================================
% v13.13 -> v15.0  Stage 1 of the v15 migration.  August 15, 2026.
%   * THE DEPLOYED v13.13 PDF WAS A STALE ARTIFACT -- section
%     3.1's screen flagged P1-KK and P1-KK is its first confirmed
%     positive since A9.  Rebuilt with the live v11.4 style this
%     source gives 200 pages against the deployed 199; the
%     deployed title page carries the six-line pre-v11.3 author
%     block.  sty_check items 1 and 2 pass, so this is section
%     3.1's case and not section 14.3's stop.  The 200-page
%     REBUILD is the baseline for every later comparison.
%   * a4paper -> letterpaper, the first conversion since A10.
%     Scoped BY MEANING: 31 occurrences of the word `landscape'
%     and exactly ONE \begin{landscape}; 20 absolute widths and
%     every one a TikZ `minimum width' inside a resizebox.
%   * NO booktabs work: 0 \hline, 0 \cline, 0 vertical rules.
%   * American spelling, total, in printed text: 544 substitutions
%     in 100 distinct forms, every one hand-classified.  The
%     heaviest load of the migration.  26 forms were new to the
%     cohort map; one, `promising', is American and is protected.
%   NO PROSE WAS WRITTEN IN THIS STAGE.
% ============================================================
%% ------------------------------------------------------------
%% Stage 2A, same day.  The float standard.
%%   The heaviest float debt since A9: 20 thin captions, 6 bare
%%   grids outside any float, and 2 of the 6 over the six-row
%%   unbreakable limit.
%%   The 20 captions were EXTENDED and none rewritten -- the
%%   author's sentence is unchanged byte for byte.
%%   The 6 grids split by which rule they break.  Four become
%%   table floats: tab:scenario-members, tab:parton-lumi,
%%   tab:syst-bins, tab:worked-significance.  Two are 23 and 6
%%   rows and become LONGTABLES: tab:appA-symbols and
%%   tab:open-roadmap.  Wrapping a 23-row tabular in a float
%%   would satisfy the bare-grid rule and break the unbreakable
%%   rule -- a float cannot split.  longtable is what section 8
%%   points at, and the paper already uses it four times, so it
%%   is this paper's own device and not an import.
%%   ALL 26 CAPTIONS ARE NEW PROSE, listed by label in the
%%   report.  None introduces a numerical claim; every number in
%%   one is already printed in the grid it describes.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 2C, same day.  Plain-English Part closes.
%%   11 closes, one per content Part (I-XI).  The three
%%   back-matter Parts get none, per A5's rule -- and all three
%%   are set with plain \GGPart here rather than
%%   \GGAppendixPart, so the marker scan looks for both.
%%   ALL ELEVEN ARE NEW PROSE IN THE AUTHOR'S VOICE and are
%%   listed by Part and by page in the report.
%%   \section[SHORT]{LONG}, so ONE Contents entry per close.
%%   A8, A9, A10, C1 and C2 emit two; C3 was the first written
%%   the right way and this follows it.
%%   Part XI ends with \GGAcknowledgments twenty characters from
%%   the end of the Part, and its close is placed before it.
%%   No \ref{part:...} appears in any of them.
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%% Stage 2E, same day.  Short caption forms, and the List of
%% Tables P1 never had.
%%   Every migrated paper -- A1-A10, C1, C2, C3 -- calls
%%   \GGListOfTables.  P1 called only \GGListOfFigures, with 51
%%   floats.  One is added after the LoF -- and the six grids
%%   Stage 2A promoted had no way to be listed anywhere before.
%%   A list entry uses the FULL caption unless a short form is
%%   given.  25 captions had none; each is now the FIRST SENTENCE
%%   of its own caption, verbatim, truncated at a word boundary
%%   if over 95 printed characters, never on a function word and
%%   never inside a parenthesis or a quote.
%%   The site scan now MASKS COMMENTS.  C3 forced it: the
%%   changelog line `clearpage-isolated minipage + \captionof.'
%%   was matched by the raw scan, which then took the next brace
%%   in the file -- inside \begin{minipage} -- and would have
%%   spliced a short form into the changelog had tidy() returned
%%   anything but None.  A9 carries the same line and escaped the
%%   same way.  Asserted after the run: no comment line altered.
%%   NO PROSE WAS WRITTEN IN THIS STAGE -- every short form is a
%%   prefix of text the caption already carried.
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%%   House band 7.40-8.18, so all three measurable plates are
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%%   lands near 8.08 is confirmed to the second decimal.  NOTHING
%%   IS PLACED FOR TYPE SIZE.
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%%   left an overfull \vbox of 33.84pt the A4 baseline did not
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%%   computing: 0.903 overfull, 0.86 still overfull, 0.84 clean.
%%   0.84 is below the measured house fraction 0.871 and still
%%   inside the band at 7.52pt, so page and rule agree.
%%   The 12 \includegraphics plates have no declared base font --
%%   external PDFs, nothing for the rule to read.  Their fractions
%%   were set against the A4 width and now print 3.7 %% wider;
%%   frozen decision 3 leaves above-house alone.  REPORTED.
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%%   (yellow!20 -> !13, orange!22 -> !14, orange!32 -> !16).
%%   palette_fix reached all four.
%%   AND ITS WIDTH CLAMP DID NO DAMAGE -- the first clean run in
%%   the migration, after eleven casualties across A9, A10, C1,
%%   C2 and C3.  Not luck: every previous casualty was a MARK
%%   drawn above 1.2pt, and P1 draws none.  Its four TikZ
%%   pictures use no weight to carry meaning and its other twelve
%%   plates are external PDFs the clamp cannot reach.  Asserted
%%   rather than eyeballed: the multiset of line widths above
%%   1.2pt is identical before and after.
%%   No comment was rewritten -- the A10 defect did not recur.
%%   NOTHING WAS REVERTED.
%%   NO PROSE WAS WRITTEN IN THIS STAGE.
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%% Stage 5, same day.  Integrity.  The same three, a FOURTH time.
%%   P1 carried 19 of the 22.  Missing: P6-Proton, W1-Symmetry,
%%   W2-Holonomy -- the same three C1, C2 and C3 were missing.
%%   Report 359 predicted it; four for four.  The remaining
%%   unmigrated papers should be assumed to carry the same hole.
%%   All three sit at roster level in the cohort map here.  Two
%%   have a substantive hook in P1's own physics and one does not,
%%   and the notes grade them that way:
%%     * W2-Holonomy is the mechanism the phase circle RUNS ON --
%%       the lock spectrum is shifted by a Wilson line, which is
%%       displacement by holonomy and not by symmetry.
%%     * W1-Symmetry is the background this paper's central
%%       NEGATIVE claim needs: no modulus, no radion, no
%%       landscape, no supersymmetry requirement, none of them
%%       following from a symmetry.
%%     * P6-Proton has NO hook in P1's physics -- all 21 of its
%%       `proton' occurrences are LHC beams.  It is the sibling on
%%       the Predictions Branch holding the OTHER standing
%%       falsifier of the same ledger: a resonance that must
%%       appear and a decay that must not.  Weaker, and said so.
%%   NEW PROSE: three Relevance notes and two short clauses.
%%   The entry form was READ from this file first: P1 uses a THIRD
%%   form again, neither C1's nor C2's.  On C2 that was caught
%%   only by rendering the bibliography page.
%%   REPORTED, NOT FIXED: P1's A6 key is ArisakaA6GRIP where five
%%   shipped papers use ArisakaA6Gi4.  C3 has the same divergence,
%%   so it is two papers.  A key never prints.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Gate repair, same day.  Two broken citations that PRINT.
%%   final_audit reported 2 sites with two Arisaka keys in one
%%   cite.  Both are \citealp, and earlier this session I
%%   classified \citealp as a false positive because section 7
%%   names \citep.  THE CLASSIFICATION WAS WRONG.  natbib
%%   compresses consecutive same-author keys, and because these
%%   bibitems carry a compound year field -- (2026, A5-GDOS) --
%%   the compression printed the first field and then emitted the
%%   separators for the second with no text:
%%       (Arisaka, 2026, A5-GDOS,,)   on two pages
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%%   Read off the built PDF, not inferred.
%%   Repaired into THIS PAPER'S OWN FORM, already used in
%%   section 16: \citealp{..}; \citealp{..}.  A coherence repair,
%%   not new prose.  NOTHING IS SAID DIFFERENTLY; two dropped
%%   references start printing.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Gate repair, same day.  `analyser' -> `analyzer', 4 printed
%% sites, and a hole in the guard that found none of them.
%%   Stage 1 swept 547 British spellings and reported 0 left.
%%   It was right about everything its rule could see.  The
%%   gg_spelling branch [A-Za-z]{3,}lys(?:e|ed|es|ing) carries
%%   every ending of the analy- family EXCEPT the agent noun, so
%%   `analyser' and `analysers' were invisible to every sweep
%%   this toolchain has run.  SECOND INSTANCE of this failure:
%%   the same file already records `licences' surviving a clean
%%   sweep of A9 because the noun branch had no plural.  The
%%   branch is fixed and the note is written beside it.
%%   Found by enumerating the analy- family by hand after
%%   spelling_audit reported a mixed pair whose 14 minority
%%   forms were all `analyses' -- correct American English, the
%%   C2-Hubble false positive again.  The audit was right and
%%   incomplete at the same time.
%%   \label{eq:tt-analyser} IS NOT RENAMED -- it never prints
%%   and renaming it breaks the cross-reference.
%%   COHORT: the same screen finds `analysers' x6 + `analyser'
%%   x4 in the SHIPPED A8-QM v15.1 and `analysing' in the
%%   SHIPPED A4-Euclid v15.37.  Both ridered separately.
%%   NO PROSE WAS WRITTEN -- four words respelled.
%% ------------------------------------------------------------
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%% Gate repair, same day.  The one new sparse page.
%%   Sparse-page scan against the A4 rebuild: 3 for 3, and the
%%   new one is p189, a TWO-LINE ORPHAN closing Part XII --
%%   running head, two lines, folio, verified by rendering.
%%   THE PROSE IS MINE, written at Stage 2C, so this is a
%%   14.1 repair and not a 14.2 question: no sentence of the
%%   author's is touched.  Both of the close's jobs survive --
%%   the reader who worked J through L owns every chain, and
%%   the hand-off to the objections.  What goes is the list of
%%   what the neighbouring appendices supply, which their own
%%   captions already say, and the gentle-to-hostile gloss,
%%   which Part XIII's opening says better.
%%   Measured after the attempt, per section 7, not reasoned
%%   about: on C2 a 155-character trim recovered ONE line.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Ship, August 15, 2026.  P1-KK v15.0.  Gate result and two
%% corrections to numbers written earlier in this same ship.
%%   GATE: 213 pages | errors 0 | undefined 0 | multiply 0
%%     overfull hbox 14, one above 20pt at 83.53 -- the SAME
%%     display the A4 baseline carried at 100.32, NARROWER on
%%     Letter, so nothing new.  overfull vbox 0 (A4 had 1).
%%     float_audit 0/0/0 | float_content_scan 0/0 |
%%     orphan_group_scan 0 | palette_audit 0 | stamp OK |
%%     falsifier 0 (cohort 15 unchanged) | printed British 0 |
%%     metadata British 0 | two-Arisaka cites 0 | P7 0
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%%     {30, 87, 129}: p189 is NEW and is REPORTED, see the
%%     revert banner above.  This line originally read "a
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%%     author's Part XII close stood rewritten.
%%   CORRECTION 1.  The Stage 3 banner above says `the 12
%%     \includegraphics plates'.  Measured at package time:
%%     11 uses of 9 distinct files.  The count was wrong; the
%%     decision it supports -- leave the external PDFs alone --
%%     is unchanged, because none of them declares a base font
%%     whatever their number.  The folder also holds
%%     ...-expected-signals-public.png, which NEITHER v13.13 nor
%%     v15.0 references; it is not in the arXiv package.
%%   CORRECTION 2.  Earlier in this session I classified
%%     final_audit's \citealp hits as a false positive because
%%     section 7 names \citep.  Wrong: two of them PRINTED a
%%     broken citation here.  Repaired above.
%%   p32, p143 and p189 are all LEFT.  p32 is the AUTHOR'S
%%   prose, pre-existing at Part I in the baseline.  p143 is a
%%   Part IX close of mine at a Part the baseline also left
%%   sparse; clearing it needs five printed lines out of a
%%   sixteen-line Plain-English introduction, a frozen
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%%   author's Part XII close and is a 14.2 question, asked in
%%   the report.  ALL THREE REPORTED, none silently accepted.
%% ------------------------------------------------------------
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%% Gate, same day.  The one new sparse page -- and a revert.
%%   Sparse-page scan vs the A4 rebuild: baseline {30 Part I,
%%   87 Part VII, 129 Part IX}; this build {32 Part I, 143 Part
%%   IX, 189 Part XII}.  p189 is NEW: a two-line orphan closing
%%   Part XII, verified by rendering.
%%   I TRIMMED IT AND THEN PUT IT BACK.  The trim was made on
%%   the ground that the paragraph was mine, written at Stage
%%   2C.  It is not.  I had run the ownership grep on the PART
%%   IX paragraph -- four pages earlier in the same scan -- and
%%   carried its answer across without re-running it.  The Part
%%   XII close is in v13.13 VERBATIM: the author's sentence,
%%   rewritten under 14.1 when 14.2 governed.  Reverted, and
%%   the trim is put to the author as this ship's one question.
%%   OWNERSHIP IS PER PARAGRAPH AND IS ASSERTED PER PARAGRAPH.
%%   Two orphans surfaced five lines apart in one scan; one was
%%   mine, one was his, and they looked identical.
%%   p189 SHIPS, REPORTED as a new sparse page against the
%%   baseline, not silently accepted.  p32 is the author's and
%%   pre-existing; p143 is a close of mine at a Part where the
%%   baseline was also sparse, and clearing it would cost five
%%   lines of a sixteen-line Plain-English introduction.
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%% v15.0 -> v15.1, August 15, 2026.  THE PART XII ORPHAN, CLOSED
%% ON THE AUTHOR'S WORD.
%%   THIS EDIT CHANGES THE AUTHOR'S PROSE AND WAS AUTHORIZED.  It was
%%   Report 360's one section 14.2 question, recommended and held.
%%   THE HISTORY, KEPT RATHER THAN LOST: at P1's own gate this same
%%   trim was made in the belief the paragraph was MINE, from Stage
%%   2C.  It is the author's, verbatim in v13.13.  Two orphan pages
%%   surfaced in one scan; the ownership grep was run on the first
%%   (0 hits -- mine) and its answer carried to the second without
%%   re-running it (1 hit -- his).  The trim was reverted and the
%%   paper shipped at 214 pages with the orphan intact.
%%   WHAT GOES: the list of what the neighboring appendices supply,
%%   which their own captions and the Contents already give; and the
%%   "gentle to the hostile" gloss, which Part XIII's own opening
%%   paragraph makes.  WHAT STAYS: both of the paragraph's jobs, the
%%   count of fifty-eight, the register, and the closing quotation --
%%   the author's own words, unmoved.
%%   167 characters out; page 189's orphan cleared; 214 -> 213 pages.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Same rider.  TABLE 2 RAN OFF THE BOTTOM OF PAGE 29.
%%   Found while gating the orphan trim, not by any gate item.
%%   The SHIPPED v15.0 carries three "Float too large for page"
%%   warnings and Report 360 does not mention them, because no gate
%%   item reads that warning.  Two are harmless: page 105 is 54.7 pt
%%   over and prints clean, page 144 is 9.8 pt over and prints clean.
%%   THE THIRD IS NOT.  tab:kkrsgg is 264.75 pt -- three and a half
%%   inches -- too tall, and page 29 shows it: the last rows run off
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%%   page number is printed INSIDE the table.
%%   WHY float_audit MISSED IT: its UNBREAKABLE-AND-LONG test read 0
%%   for this paper.  The table is a 17-row tabular in a table float,
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%% v15.1 -> v15.2, August 15, 2026.  FIGURE 9's CAPTION RAN OFF THE
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% v15.2 -> v15.3, August 15, 2026.
%%   THE A6 KEY IS NORMALIZED: ArisakaA6GRIP -> ArisakaA6Gi4.
%%   Measured across all twenty-two deployed sources: twenty use
%%   ArisakaA6Gi4 and two -- this paper and the other one named in the
%%   note above -- used ArisakaA6GRIP.  Twenty against two settles
%%   which side moves.  A key never prints, so not one character of
%%   this paper's printed text changes; the cover stamp is the only
%%   page that differs.  The comment above is left exactly as it was:
%%   it is the record of the divergence and rewriting it would erase
%%   the history.  ALSO FIXED, IN THE TOOL: final_audit's cohort key
%%   list contained A6Gi4 only, so it scored this paper as missing a
%%   cohort reference it was not missing.
%% ------------------------------------------------------------
\PaperVersion{v20.0}
\PaperDate{August 21, 2026}

\title{The GG-6 Lock Cluster: A Non-Spatial Kaluza--Klein Resonance near 3.85 TeV as the Standing LHC Falsifier of GG-Theory}
\author{Katsushi Arisaka}
\date{}

\newcommand{\mplred}{\bar M_{\mathrm{Pl}}}
\newcommand{\kappaeff}{\kappa_{\mathrm{eff}}}
\newcommand{\UB}{\mathrm{U}(1)_B}
\newcommand{\UQi}{\mathrm{U}(1)_{Qi}}
\newcommand{\GKK}{G_{\mathrm{KK}}}
\newcommand{\gKK}{g_{\mathrm{KK}}}
\newcommand{\BKK}{B_{\mathrm{KK}}}
\newcommand{\ttbar}{t\bar t}
\newcommand{\bbbar}{b\bar b}
\newcommand{\hh}{hh}
\newcommand{\llbar}{\ell^+\ell^-}
\newcommand{\Mpl}{M_{\mathrm{Pl}}}
\newcommand{\MIR}{\Lambda_{\mathrm{IR}}}
\newcommand{\Mfirst}{M_{\mathrm{first}}}
%% v11.2: \LaTeX shim dropped -- gg-11 Acknowledgments use literal 'LaTeX'.
\newcommand{\thcl}{\theta_{\mathrm{cluster}}}
\providecommand{\GA}{\ensuremath{\mathrm{GA}}}
\providecommand{\GGA}{\ensuremath{\mathrm{GGA}}}
\providecommand{\GRIPin}{\ifmmode\mathsf{GRIP\text{-}In}\else\textsf{GRIP-In}\fi}
\providecommand{\GRIPout}{\ifmmode\mathsf{GRIP\text{-}Out}\else\textsf{GRIP-Out}\fi}

\setlist[itemize]{topsep=3pt,itemsep=2pt}
\setlist[enumerate]{topsep=3pt,itemsep=2pt}
\captionsetup{font=small,labelfont=bf}

%% =====================================================================
%% Front-matter override (per author, June 4 2026; mirrors A1 R4):
%% Overview of Contents (Part titles only) and List of Figures share one
%% page -- Overview opens the page, LoF continues below it, page break
%% before the full Table of Contents.
%% =====================================================================
\GGfrontmatteronepagetrue
%% v11.2: \GGOverviewOfContents (centered) via gg-11; single-page toggle above.
%% v11.2: \GGListOfFigures (centered) via gg-11.

\begin{document}
\setlength{\tabcolsep}{6pt}
% ---- v13.9 (August 2, 2026) ----
% Significance section gains the closing paragraph 'The
% discovery, read on the anatomy: one resonance, two fibres':
% the lock cluster as a simultaneous reading of both non-spatial
% fibres --- identity (tower, winding charges, spin, branching
% fingerprint) as circle-side holonomy data, location
% (M_lock = M_Pl/2 e^-35) as interval-side descent; a discovery
% as the first laboratory reading of a non-spatial fibre. Part
% of the P-series completion sweep.


\GGTitlePage
{The \GG{} Lock Cluster}
{A Non-Spatial Kaluza--Klein Resonance near 3.85~TeV as the Standing LHC Falsifier of \GGTheory{}}
{}
\thispagestyle{empty}

\begin{abstract}
\noindent
Every particle discovered since the mid-1970s --- the $W$ and $Z$, the top, the Higgs --- was on the Standard Model's list before it was found.  This paper is about the first particle that is not on that list.  \GGTheory{} \emph{derives}, rather than fits, a new mass scale: the lock cluster at \(\Mlock=\tfrac12 M_{\mathrm{Pl}}e^{-35}\simeq3.849\)~TeV, fixed by the integer \(K=35\) of the \OGNfive{} ledger with no adjustable parameter anywhere in the chain.  The same number over-constrains the electroweak scale, \(v=\Mlock(2/5)^3\approx246\)~GeV: two known scales, one integer, nowhere to hide.

At that mass the theory places a three-member cluster --- a spin-2 graviton, a color-octet vector \(g^*\), and a leptophobic \(Z'_B\): the discrete modes of a \emph{non-spatial} (renormalization-group) extra dimension --- with derived couplings that are top-rich, dijet-friendly, and lepton-poor, so existing clean-channel limits do not touch it.  The un-suppressible member is the octet: \(\sigma\times\mathrm{BR}(\ttbar)=8.7\)~fb at 13~TeV, \(\Gamma/m=15\%\), with a theorem-fixed interference sign.  It sits exactly at the current ATLAS/CMS \(\ttbar\) exclusion frontier: the prediction is being tested \emph{now}.

A full simulation anchored to the ATLAS Run-2 analysis (Part~VIII) turns this into a calendar.  A dedicated reanalysis of data already on tape carries an expected \(2.7\)--\(3.7\sigma\) per experiment, and \(4.6\)--\(5.2\sigma\) for a combined ATLAS+CMS analysis with improved systematics --- five-sigma discovery from \emph{existing} data is a live possibility; failing that, single-experiment \(5\sigma\) luminosity is on tape during 2032.  Because mass, width, rate, sign, and angular shape are all pre-registered, the search pays no look-elsewhere penalty --- and a null result at \(\lesssim2\)~ab\(^{-1}\) retires the implementation outright.  The mass is the bet; the LHC is the table.
\end{abstract}

\begin{ExecutiveSummary}
The question behind this paper is simple: \emph{if \GG{} is right, what exactly should ATLAS and CMS see --- and how fast can they either find it or kill it?}  The answer is best given in seven points, in as plain a language as the subject allows.

\textbf{First --- the principle, and why it predicts.}\quad \GGTheory{} rests on one structural idea, \emph{geometry--gauge locking}: the energy scale at which nature is probed is treated not as a bookkeeping label but as a genuine geometric dimension\footnote{Consistent with the cohort reading: this ``six-dimensional bulk'' is the renormalization-group geometrization of the energy scale --- the radial fifth dimension is the renormalization-group axis $r=\ln(E_{\mathrm{GUT}}/\mu)$, not a Randall--Sundrum spatial dimension --- so the radial ``lock cluster'' near $3.85$~TeV is an RG threshold/lock spectrum rather than a tower of spatial Kaluza--Klein gravitons; see \citet{ArisakaA2GG6}.}, and the geometry and the gauge fields lock together at thresholds set by \emph{integers} --- the ledger \((M,N,L;J,K)=(1,3,5;7,35)\).  Integers are the entire content of the theory's freedom: there is no coupling to adjust, no mass to scan, no continuous dial of any kind.  This is the source of its unusual predictive power: a theory built on counting numbers cannot retreat, so everything it says is a commitment --- the property whose fifty-year absence \S1 recounts in full.

\textbf{Second --- one derived number, doubly spent.}\quad The most rigid output is the \emph{lock scale},
\begin{equation*}
\Mlock = \frac{M_{\mathrm{Pl}}}{2\,e^{35}} \simeq 3.849\,\mathrm{TeV},
\end{equation*}
where the \(35\) in the exponent is the integer \(K\) --- not a fitted quantity.  The same number is spent a second time: the electroweak scale, known since the 1980s, must equal \(\Mlock(2/5)^3\approx246\)~GeV, and does.  A reader should pause on what this over-constraint means: the theory does not merely predict something new at \(3.85\)~TeV; it ties that prediction to a number already measured, so it cannot slide the target without breaking what already works.

\textbf{Third --- a cluster of three, not a lone bump.}\quad At the lock scale the theory places three sibling states, degenerate in mass: a spin-2 graviton mode; a heavy gluon-like state (the color-octet \(g^*\)); and a heavy photon-like state that couples to quarks but shuns electrons and muons (the ``leptophobic'' \(Z'_B\)).  Their decay preferences are derived, not chosen: rich in top quarks, friendly to jets, poor in leptons and photons.  This is why the LHC's cleanest and most famous exclusion channels --- electron--positron pairs, photon pairs --- say almost nothing about it, and why the decisive channel is the top-quark pair.

\textbf{Fourth --- it is being tested now.}\quad The un-suppressible member of the cluster is the octet: its coupling to the colliding quarks is fixed by the same integer \(K\) and cannot be lowered.  The computed rate, \(\sigma\times\mathrm{BR}(\ttbar)=8.7\)~fb at 13~TeV with a broad width (\(\Gamma/m\simeq15\%\)), sits exactly at the frontier of the published ATLAS/CMS top-pair exclusions --- neither safely beyond reach nor yet excluded, because no published search template matches this shape: broad, interfering with the QCD continuum at a derived sign, enhancing the spectrum below \(3.85\)~TeV and depleting it above.  A dedicated reinterpretation of existing data is therefore the single most urgent request this paper makes.

\textbf{Fifth --- the discovery may already be on tape.}\quad Part~VIII simulates the search the way an analysis team would run it, anchored item by item to the published ATLAS Run-2 analysis and calibrated to its published limits.  The result: a dedicated reanalysis of the Run-2+3 data already recorded carries an expected \(2.7\)--\(3.7\sigma\) per experiment, and \(4.6\)--\(5.2\sigma\) for a combined ATLAS+CMS analysis with realistically improved background systematics --- \emph{five-sigma discovery from existing data is a live possibility}, and single-experiment \(5\sigma\) luminosity is on tape during 2032.  Because the mass, width, rate, interference sign, and angular shape are all specified in advance, the search pays none of the statistical penalties of a generic bump hunt: no look-elsewhere discount (worth \(0.8\sigma\) at threshold, or equivalently a doubling of the dataset), and none of the factor-three dilution of a mismatched search template.

\textbf{Sixth --- the fight is systematics, not statistics.}\quad The simulation's structural lesson: beyond Run~3 the search is no longer limited by how much data exists but by how well the background is known.  More luminosity alone moves the significance by only \(\sim\!1\sigma\) per \emph{tripling} of the dataset; halving the background uncertainties --- the standard, decomposable HL-LHC expectation --- is what crosses the discovery threshold.  The practical consequences for the experiments are spelled out, including one warning: the signal's own interference contaminates the low-mass sideband, so the background normalization must be anchored below \(\sim\!1\)~TeV.

\textbf{Seventh --- either answer is decisive.}\quad This is the sharpest collider falsifier the \GG{} program owns.  If the cluster is there, the correlated pattern --- one mass, three states, top-rich decays, the derived interference shape, and a second family near \(7\)~TeV --- is unmistakable, and the property measurements (spin, chirality) follow within two years of HL-LHC running.  If it is not there, a null result at \(\lesssim2\)~ab\(^{-1}\) does not wound the theory; it \emph{retires the current implementation outright}, and takes the electroweak-scale derivation down with it.  No parameter can be adjusted to escape, because there are no parameters.  The direct spin-2 confirmation, at its derived coupling, is then FCC-hh physics (\(\sim\!205\)~fb at 100~TeV).

Each community will read this wager in its own way.  \emph{A collider experimentalist} meets a concrete, pre-registered search: one mass, one width, one sign, one angular shape, shipped with a reinterpretation card and a trigger roadmap.  \emph{A graduate student} meets a thesis-sized analysis whose decisive data already exist (see the Preface).  \emph{A theorist} meets an extra-dimension signature whose scale is derived from an integer rather than fitted --- and a falsifier with no escape hatch.  \emph{A general physicist} meets a plain wager: one number, \(3.849\,\mathrm{TeV}\), staked against the next decade of the LHC.  The remainder of the paper makes that wager precise.
\end{ExecutiveSummary}

\GGKeywords{geometry--gauge locking; Kaluza--Klein particles; non-spatial extra dimensions; scale dimension; phase dimension; Randall--Sundrum; ATLAS; CMS; HL-LHC; spin-2 resonance; top-antitop resonances; trigger-level analysis; data scouting; falsifiability; open systems; OGGEP; GDOS}

%% ---- v15.4: the showcase headline, at the house default -------
%% \Large, which is the house default as of Report 391: every
%% showcase headline the author set himself is \Large or larger,
%% and nothing below that floor is used anywhere in the cohort.
%% v11.5 (August 17, 2026): THE DEFINITION IS IN THE STYLE FILE NOW.
%% The comment above is the record of why it was here -- the style
%% file is shared, and changing it meant rebuilding every arXiv
%% package.  That was done, so what is left is what this paper sets.
%% ---- v15.4: the savebox for the exposure page ----------------
%% FILLED AT THE IN-BODY USE SITE'S MEASURED SPACING, AND THE
%% FIRST PROBE READ THE WRONG NUMBER.  Probing one line ABOVE
%% \begin{table} reported \baselinestretch 1.15 -- which is the
%% surrounding text, not the table.  The style file sets
%% \setstretch{1.15} document-wide, and a table environment in
%% these papers resets it: probed INSIDE, at \begin{tabularx},
%% tab:predictions runs BSTR 1, AS 1, and so do tab:glance and
%% tab:status.  The front-matter fill site runs 1.15.  So the box
%% is filled at \setstretch{1} -- explicitly, so it does not
%% depend on where it is filled -- and \arraystretch is left at 1
%% because this table does not change it.  A8 needed 1.2 because
%% A8's table sets 1.2; that number is A8's, not a house value.
\newsavebox{\GGscPredP}
\sbox{\GGscPredP}{\setstretch{1}\small\begin{tabularx}{\textwidth}{>{\raggedright\arraybackslash}p{1.4cm} X >{\centering\arraybackslash}p{1.2cm}}
\toprule
\textbf{Id} & \textbf{Prediction} & \textbf{Grade}\\
\midrule
P-KK-1 & \(\sigma\times\mathrm{BR}(g^*\to\ttbar)=8.7\,[7.0\text{--}12]\)~fb at 13~TeV; \(13.6\,[11\text{--}19]\)~fb at 14~TeV; \(\approx2.8\)~pb at 85~TeV & [D\(^*\)]\\
P-KK-2 & Octet branching fingerprint \(\ttbar:b\bar b:jj\approx70:3.5:1\); \(\mathrm{BR}(\ttbar)=0.94\,[0.93\text{--}0.95]\) & [D\(^*\)]\\
P-KK-3 & Graviton pattern \(\ttbar:W_LW_L:Z_LZ_L:hh=0.69:0.15:0.08:0.08\), \(\Gamma/m=6.3\%\), \(\sigma(85~\mathrm{TeV})\approx0.13\)~pb & [D\(^*\)]\\
P-KK-4 & Interference sign: enhancement below the pole, deficit above; SM crossing at \(\simeq4.9\)~TeV; centroid pull \(-\mathcal O(100)\)~GeV & [T]\\
P-KK-5 & Top polarization \(P_t=+0.91\), band \([+0.88,+0.92]\); \(A_{FB}=0\) & [T]/[D\(^*\)]\\
P-KK-6 & Exact leptophobia: \(\mathrm{BR}(Z'_B\to\ell\ell)=0\); \(m_{Z'_B}=m_{g^*}\) exactly & [T]\\
P-KK-7 & No angular satellites at the TeV scale (the \(\theta\) tower sits at \(\sim\EGUT\)) & [T]\\
P-KK-8 & Second radial family at \(1.831\,\Mlock=7.047\)~TeV (position; rate estimate \(\sim\!10\)~fb at 85~TeV) & [T]/[M]\\
P-KK-9 & No radion; Higgs couplings Standard-Model-like apart from \(G_1\to hh\) & [T]\\
P-KK-10 & Lepton-collider silence: no \(s\)-channel cluster line in \(\mu^+\mu^-\); SM-like contact terms at 3~TeV & [T]\\
\bottomrule
\end{tabularx}}

%% ---------- v11.54 eye-opener showcase page (unnumbered) ----------
\clearpage
\begin{center}
\includegraphics[width=0.97\textwidth]{Arisaka-GG-P1-KK-quantsig.pdf}

\vspace{-1.4em}

{\footnotesize\itshape Every expected \GG{} collider signal, computed with no adjustable parameter: the three-member lock cluster at the derived $3.849$~TeV and its radial family at $7.05$ and $10.22$~TeV (Figure~\ref{fig:lhc-signature}, p.~\pageref{fig:lhc-signature}).}

\vspace{3em}

\includegraphics[width=\textwidth]{Arisaka-GG-P1-KK-hllhc-absorb.pdf}

\vspace{-0.5em}

{\small\itshape Why the verdict is near: a dedicated reanalysis of collisions already recorded carries an expected $2.7$--$3.7\sigma$ per experiment --- and the ladder shows exactly where the sensitivity goes (Figure~\ref{fig:sim-absorb}, p.~\pageref{fig:sim-absorb}; Part~\ref{part:hllhcsim}).}
\end{center}
\clearpage
%% ---------- end showcase ----------

%% ---- v15.4 showcase page 5: THE EXPOSURE -------------------
%% P1's existing showcase page says what the theory predicts and
%% when the verdict arrives.  Neither plate says what the paper is
%% committing to, or how strongly each commitment is claimed --
%% and P1 is the paper whose whole argument is that it committed
%% in advance.  Table 31 does, on one page, with a grade column.
%% The direct analogue of the pages A1, C1 and C2 carry.
%% This is the only P1 showcase page with a typeset headline: the
%% other two have their titles baked into the plate images, and a
%% table has no such title to inherit.
%% Unnumbered (\captionof*), so Tables 1-39 keep their numbers.
%% -----------------------------------------------------------
\clearpage
%% Self-filling showcase frame (Report 462): a minipage of EXPLICIT
%% height with the [s] inner alignment, so the interior glue stretches,
%% and 1 : 2 : 2 : 1 weights -- top, above the plate, below it, bottom.
%% The height is \textheight MINUS \topskip; see the report for why.
\noindent\begin{minipage}[t][\dimexpr\textheight-\topskip\relax][s]{\textwidth}
\vspace*{\stretch{1}}
\centering
\GGShowcaseHead{Ten Predictions,\\[3pt]
Each Graded Before the Data Arrive}
\par\vspace*{\stretch{2}}
{\setlength{\abovecaptionskip}{5pt}%
\small\captionof*{table}{\textbf{What this paper is willing to be wrong about.} Ten numbered predictions, each carrying the grade that says how strongly it is claimed: [T] a theorem, which falls with its theorem; [D\(^*\)] derived subject to the condition printed beside it; [M] one of a finite listed menu; [E] somebody else's number this paper consumes. Four grades and no unlabeled rows, because mixing the four is how a prediction list comes to look stronger than it is. The rates and the branching fingerprint are what the LHC can reach with data already on tape; the interference sign and the exact leptophobia are theorems, and one measurement retires either. Nothing here was written after the data. In full: Table~\ref{tab:predictions}, p.~\pageref{tab:predictions}.}}
\par\medskip
\par\vspace*{\stretch{2}}
\usebox{\GGscPredP}
\par\vspace*{\stretch{1}}
\end{minipage}
\clearpage
%% >>>>>>>>>>>>>>>>>> GG-BULK-ONE-DIAGRAM v1.0 >>>>>>>>>>>>>>>>>>
%% Shared showcase plate, byte-identical in A2, A5, A8, A9, A10
%% and P1.  Pure TikZ: it adds no asset to the arXiv package.
%% Every macro it defines is \def'd under the \GGbd prefix and
%% scoped to the picture, so it cannot clash with a host paper.
%% Caption ABOVE, per the frozen float standard.
\clearpage
%% Self-filling showcase frame (Report 462): a minipage of EXPLICIT
%% height with the [s] inner alignment, so the interior glue stretches,
%% and 1 : 2 : 2 : 1 weights -- top, above the plate, below it, bottom.
%% The height is \textheight MINUS \topskip; see the report for why.
\noindent\begin{minipage}[t][\dimexpr\textheight-\topskip\relax][s]{\textwidth}
\vspace*{\stretch{1}}
\centering
\GGShowcaseHead{A Six-Dimensional Bulk Whose Fifth Dimension Is Scale\\[2pt]
and Whose Sixth Is Phase}
\par\vspace*{\stretch{2}}
{\small\captionof*{figure}{\textbf{The whole program in one diagram, with the scale axis
carrying numbers.} The bulk $\cM_4\times I_r\times S^1_\theta$ is a fiber bundle: over every
point of the Lorentzian stage sit the two \emph{non-spatial} fibers, scale and phase. \AX{2}
(\GGEP) acts \emph{upward} and builds them; \AX{3} (\OGGEP) acts \emph{downward}, and
registration is the projection that erases $(r,\theta)$ --- the origin of quantum behavior.
The vertical axis is calibrated: $\EGUT=\EPl e^{-2\pi}$ at $r=0$, the \OGNfive{} depth
$r=K=35$ at the infrared boundary, and the lock at $r=29.41$, which is $K-2\pi+\ln 2$
exactly.}}
\par\medskip
\par\vspace*{\stretch{2}}
\resizebox{0.95\textwidth}{!}{%
\begin{tikzpicture}[
  font=\fontsize{11}{13.2}\selectfont,
  every node/.style={inner sep=2pt},
  box/.style={rounded corners=2pt, align=center, inner sep=6pt, line width=0.712pt},
  lbox/.style={rounded corners=2pt, align=left, inner sep=5pt, line width=0.712pt,
               font=\fontsize{8.8}{10.6}\selectfont},
  tick/.style={line width=0.712pt, black!70},
  slice/.style={draw=blue!60!black, fill=blue!16, line width=0.712pt},
  fiber/.style={draw=violet!65!black, line width=0.845pt},
  note/.style={font=\fontsize{8.8}{10.6}\selectfont, align=left},
  rnote/.style={font=\fontsize{8.8}{10.6}\selectfont, align=left, text width=4.85cm},
  sml/.style={font=\fontsize{8.8}{10.6}\selectfont},
  tny/.style={font=\fontsize{6.6}{7.9}\selectfont},
  rlab/.style={font=\sffamily\fontsize{9.6}{11.5}\selectfont},
  mulab/.style={font=\sffamily\fontsize{8.8}{10.6}\selectfont}
]
%% ---- geometry constants ------------------------------------------
\def\GGbdTop{21.0}          % y at r = 0
\def\GGbdSc{0.455000}       % cm per unit r  (15.9 cm for r = 0..35)
\def\GGbdXc{6.8}             % throat centre
\def\GGbdWz{4.05}             % drawn half-width at r = 0
\def\GGbdWk{13.64}           % drawn half-width = wz * exp(-r/13.64)
\def\GGbdXR{11.9}            % right annotation column
\def\GGbdY#1{\GGbdTop-\GGbdSc*(#1)}
\def\GGbdW#1{\GGbdWz*exp(-(#1)/\GGbdWk)}

%% ================= TOP: the two necessities ========================
\node[box, draw=black!60, fill=black!8, text width=15.8cm] (need) at (\GGbdXc,23.15)
 {\textbf{Two structures no local, causal, interacting, charged quantum theory can avoid}\\[2pt]
  a renormalization \emph{scale}\qquad and\qquad a gauge \emph{phase}};
\draw[-{Stealth[length=3.4mm,width=2.6mm]}, line width=0.956pt, black!75]
      (\GGbdXc,22.28) -- (\GGbdXc,21.42);
\node[right, note, fill=white, inner sep=2pt] at (\GGbdXc+0.30,21.88)
 {\AX{2} (\GGEP): \emph{geometrize} each necessity --- this \textbf{builds} the two fibers};

%% ================= THE THROAT =====================================
\filldraw[fill=blue!8, draw=blue!60!black, line width=0.845pt]
  plot[domain=0:35,samples=90,variable=\r] ({\GGbdXc-\GGbdW{\r}},{\GGbdY{\r}})
  -- plot[domain=35:0,samples=90,variable=\r] ({\GGbdXc+\GGbdW{\r}},{\GGbdY{\r}}) -- cycle;
\draw[draw=red!65!black,line width=0.956pt] ({\GGbdXc-\GGbdWz},{\GGbdY{0}}) -- ({\GGbdXc+\GGbdWz},{\GGbdY{0}});

\foreach \r in {0,10,20,29.41,35}{
  \pgfmathsetmacro{\w}{\GGbdWz*exp(-\r/\GGbdWk)}
  \pgfmathsetmacro{\y}{\GGbdTop-\GGbdSc*\r}
  \pgfmathsetmacro{\sk}{0.17+0.145*\w}
  \draw[slice] ({\GGbdXc-\w},\y) -- ({\GGbdXc+\w},\y) -- ({\GGbdXc+\w+\sk},{\y+0.45*\sk})
               -- ({\GGbdXc-\w+\sk},{\y+0.45*\sk}) -- cycle;
}
\node[blue!45!black, align=center] at (\GGbdXc,{\GGbdY{4.6}})
 {\textbf{the 6D bulk}\\[1pt] $\cM_4\times I_r\times S^1_\theta$};

%% ================= THE SCALE AXIS =================================
\draw[-{Stealth[length=3.2mm,width=2.4mm]}, line width=0.956pt, black!75]
      (0,{\GGbdY{-1.6}}) -- (0,{\GGbdY{38.2}});
\node[rotate=90, anchor=south, black!75] at (-4.05,{\GGbdY{17.5}})
 {$I_r$ : the \emph{scale} dimension\ \ $r=\ln(\EGUT/\mu)$};
\node[black!70, rlab, anchor=south east] at (-0.12,{\GGbdY{-2.1}}) {UV};
\node[black!70, rlab, anchor=north east] at (-0.12,{\GGbdY{38.6}}) {IR};
\node[anchor=south west, black!62, align=left,
      font=\sffamily\fontsize{7.4}{8.9}\selectfont] at (-4.30,{\GGbdY{-3.0}})
 {\textsf{boundary energy scale}\\ $\mathsf{\mu=E_{GUT}\,e^{-r}}$\ \ \textsf{[GeV]}};

\foreach \r/\lab/\mulab in {0/{0}/{$\mathsf{2.28\times10^{16}}$},
                            5/{5}/{$\mathsf{1.53\times10^{14}}$},
                            10/{10}/{$\mathsf{1.03\times10^{12}}$},
                            15/{15}/{$\mathsf{6.97\times10^{9}}$},
                            20/{20}/{$\mathsf{4.70\times10^{7}}$},
                            25/{25}/{$\mathsf{3.16\times10^{5}}$},
                            29.41/{29.41}/{$\mathsf{3.85\times10^{3}}$},
                            35/{35}/{$\mathsf{14.4}$}}{
  \draw[tick] (-0.17,{\GGbdY{\r}}) -- (0.17,{\GGbdY{\r}});
  \node[anchor=east, rlab, black!80] at (-0.30,{\GGbdY{\r}}) {$\mathsf{r=\lab}$};
  \node[anchor=east, mulab, black!62] at (-2.00,{\GGbdY{\r}}) {\mulab};
}

%% ---- the UV edge --------------------------------------------------
\node[anchor=north west, rnote, text=red!60!black] at (\GGbdXR,{\GGbdY{-1.1}})
 {\textbf{UV boundary}, $r=0$:
  $\mu=\EGUT=\EPl\,e^{-2\pi}=2.28\times10^{16}$ GeV.
  \emph{The geometry begins here.} $\EPl$ sits at
  $r=-2\pi$, off the diagram.};

%% ---- what a slice is -----------------------------------------------
\node[anchor=east, note, text=blue!45!black] at (4.85,{\GGbdY{12.6}})
 {each $\cM_4$ slice is not a surface\\
  in space --- it is the \emph{whole} 4D\\
  effective theory at the scale $\mu(r)$};
\draw[blue!45!black, line width=0.594pt] (4.95,{\GGbdY{12.6}}) -- ({\GGbdXc-\GGbdW{13.8}-0.05},{\GGbdY{13.8}});

%% ---- the phase fibre ---------------------------------------------
\coordinate (fp) at ({\GGbdXc+\GGbdW{9.6}-0.62},{\GGbdY{9.6}});
\fill (fp) circle (1.6pt);
\draw[fiber] (fp) ellipse [x radius=0.76, y radius=0.30];
\draw[fiber,-{Stealth[length=2.2mm]}] ($(fp)+(0.76,0)$)
      arc [start angle=0, end angle=310, x radius=0.76, y radius=0.30];
\draw[violet!65!black, line width=0.594pt] ($(fp)+(0.78,0.05)$) -- (\GGbdXR-0.15,{\GGbdY{9.6}});
\node[anchor=north west, rnote, text=violet!65!black] at (\GGbdXR,{\GGbdY{8.9}})
 {$S^1_\theta$ : the \emph{phase} dimension --- compact,
  unitary, one Abelian holonomy
  $\oint A_\theta\,d\theta\in U(1)$. Attached at
  \emph{every} $r$; one copy drawn.};

%% ---- the lock ------------------------------------------------------
\node[star, star points=5, star point ratio=0.45, draw=red!70!black,
      fill=red!16, minimum size=0.72cm, line width=0.712pt] (lk)
      at (\GGbdXc,{\GGbdY{29.41}}) {};
\draw[red!70!black, line width=0.594pt] (lk.east) -- (\GGbdXR-0.15,{\GGbdY{29.41}});
\node[anchor=west, rnote, text=red!70!black] at (\GGbdXR,{\GGbdY{29.41}})
 {\textbf{the lock}, at $r=29.41$:
  $\Mlock=\tfrac12\EPl\,e^{-K}=3.85$ TeV, the
  Kaluza--Klein lock cluster.};

%% ---- the IR boundary ------------------------------------------------
\draw[blue!45!black, line width=0.594pt] ({\GGbdXc+\GGbdW{35}},{\GGbdY{35}}) -- (\GGbdXR-0.15,{\GGbdY{35}});
\node[anchor=west, rnote, text=blue!45!black] at (\GGbdXR,{\GGbdY{35}})
 {\textbf{IR boundary}, $r=K=35$: the \OGNfive{} integer
  scale depth, $\mu=14.4$ GeV. \emph{The bulk ends here};
  the cosmological descent runs on to
  $r_{\mathrm{GGA}}\simeq64.5$.};

%% ---- the calibration box ---------------------------------------------
\node[lbox, draw=olive!60!black, fill=yellow!12, text width=4.35cm, anchor=north west]
      at (0.28,{\GGbdY{15.3}})
 {\textbf{Why the lock sits at $r=29.41$, not at $35$}\\[4pt]
  $K=35$ is counted from the \emph{bulk} anchor
  $\tfrac12\EPl$; this axis is anchored at $\EGUT$. They
  differ by\\[4pt]
  \hspace*{0.1cm}$\ln\!\bigl(\tfrac12\EPl/\EGUT\bigr)=2\pi-\ln 2=5.590$,\\[4pt]
  \hspace*{0.1cm}$r_{\mathrm{lock}}=K-2\pi+\ln 2=29.410$.\\[4pt]
  One energy, two readings. Exact, not fitted.};

%% ---- the honesty note about the drawing ------------------------------
\node[lbox, draw=black!50, fill=black!8, text width=4.35cm, anchor=south west]
      at (0.28,{\GGbdY{36.3}})
 {\textbf{The taper is compressed.} Drawn half-width
  $e^{-r/13.64}$: the true $e^{-r}$ slowed $13.64\times$
  so the throat stays visible. To scale, the slice at
  $r=35$ would be $6.3\times10^{-16}$ of the one at
  $r=0$.};

%% ================= OGGEP PROJECTION ==================================
\draw[-{Stealth[length=5mm,width=4mm]}, line width=2.0pt, red!70!black]
      (\GGbdXc,{\GGbdY{36.6}}) -- (\GGbdXc,{\GGbdY{40.9}});
\node[anchor=west, note, text=red!60!black] at (\GGbdXc+0.35,{\GGbdY{38.9}})
 {\AX{3} (\OGGEP): registration $=$ the projection $\pi$ that erases $(r,\theta)$};

%% ================= BOTTOM: the registered world ======================
\node[box, draw=blue!55!black, fill=blue!10, text width=15.8cm] (bdy) at (\GGbdXc,{\GGbdY{43.4}})
 {\textbf{The registered 4D world: open, and therefore quantum}\ \ (\GDOS)\\[2pt]
  superposition \& interference are the unread phase $S^1_\theta$\\[1pt]
  measurement \& irreversibility are the coarse-grained scale $I_r$};
\end{tikzpicture}}
\par\vspace*{\stretch{1}}
\end{minipage}
\clearpage
%% <<<<<<<<<<<<<<<<<< GG-BULK-ONE-DIAGRAM v1.0 <<<<<<<<<<<<<<<<<<

%% ---- end showcase page 5 -----------------------------------


\GGOverviewOfContents
\GGListOfFigures
\GGListOfTables
\GGTableOfContents

\GGPreface

\noindent This Preface is addressed, more than to anyone else, to a graduate student who is in --- or is deciding whether to join --- an ATLAS or CMS analysis group, and who is wondering what to do with the next five years.

Consider, first, where the field actually stands.  The Standard Model was completed as a structure in the mid-1970s, and every collider discovery since has been an item from its list: the $W$ and $Z$ bosons (1983) confirmed the electroweak theory of 1967; the top quark (1995) had been required, at roughly its measured mass, by precision fits; the Higgs boson (2012) was the last unchecked box of a mechanism written down in 1964.  These were monumental achievements of experimental physics --- and every one of them was a \emph{confirmation}.  No collider has ever observed a particle that was not on the Standard Model's list.  If you join an LHC experiment today, you join half a century after the last time the list itself grew.

Now consider what this paper puts on the table.  A theory with \emph{no adjustable parameters} --- its entire freedom is a handful of integers --- derives a new mass: \(3.849\)~TeV, from the Planck scale and the integer \(35\), full stop.  It derives what lives there (three sibling states), how they decay (top pairs, mostly), how wide the main state is (\(15\%\)), how it interferes with the ordinary background (constructively below, destructively above --- a theorem, including the sign), and how it is distributed in angle.  It even ties the new number to one you already know: the same chain forces the electroweak scale, \(246\)~GeV.  There is no version of this theory where the resonance is at \(4.5\)~TeV instead, or couples differently, or fades to an undetectable corner.  It is either there, or the theory's current implementation is dead.  You will not find many statements of that shape in the beyond-Standard-Model literature; \S1 explains why their absence is precisely the lesson of the last fifty years.

Here is what makes this concretely \emph{your} opportunity rather than an abstraction.  Part~VIII of this paper simulates the search at the fidelity of a real analysis, anchored to the published ATLAS Run-2 top-pair search, and its headline is remarkable: \emph{the data that decide this may already be on tape.}  A dedicated reanalysis of the existing Run-2+3 dataset carries evidence-level sensitivity per experiment (\(2.7\)--\(3.7\sigma\)), and a combined ATLAS+CMS analysis with realistically improved systematics reaches \(4.6\)--\(5.2\sigma\) --- discovery-grade --- from collisions already recorded.  The analysis this requires is thesis-sized, not collaboration-sized in its first pass: one channel (boosted top pairs, one lepton), published object definitions, one new template (the interference-aware line shape of Part~VI), one warning to respect (\S\ref{sec:sim-contamination}).  The discovery window --- 2026 through roughly 2033 --- is not some future generation's; it is the exact span of one PhD begun now.  Somebody's thesis is going to decide this question.  There is no reason it should not be yours.

And consider what a positive result would mean.  The Higgs discovery, for all its magnitude, closed a chapter: it completed a known structure.  A cluster at \(3.849\)~TeV would \emph{open} one: the first particle beyond the Standard Model in the history of collider physics, the first direct evidence that the dimensions of nature include the energy scale itself, and --- through the electroweak over-constraint --- the first explanation of \emph{why} the weak scale is what it is.  That is not an incremental measurement.  That is the kind of result the LHC was built hoping for.

Honesty requires the other half, and it is the half that makes this science rather than salesmanship: the theory may simply be wrong.  But notice what the search costs you in that case --- almost nothing.  The analysis that tests it is, bin for bin, the world's best measurement of the high-mass top-pair spectrum and the strongest limit on \emph{any} broad top-philic resonance; the interference-aware template you would build is reusable across every colored-resonance model; and a clean exclusion of a zero-parameter theory is a permanent result, not a parameter-scan press release.  A pre-registered prediction is decided either way.  That is the property that has been missing from this field for fifty years, and it is the property this paper exists to put back.

How to read the paper, if you are the student described above: start with Part~VIII (the simulated search --- it is self-contained and teaches the statistics from the Poisson likelihood up), then Part~VI (what the signal looks like and why, at full derivation depth), then \S\ref{sec:current-status} (the current limits) and \S\ref{subsec:killwindow} (the kill criterion).  Work backwards into the foundations (Parts~I--III) as curiosity, not obligation, dictates.  The theory chapters will still be there after your trigger shift.

\GGPart[Foundations]{Foundations}{From the GG-Constitution to a six-dimensional collider target}

\section{Introduction: Fifty Years Beyond the Standard Model}
\label{sec:fiftyyears}

\paragraph{The triumph, and the silence that followed.}
The Standard Model was assembled, in every essential, by the mid-1970s: the electroweak theory in 1967--68, quantum chromodynamics and asymptotic freedom by 1973, the third generation anticipated on paper in the same years.  What followed was one of the great runs of confirmation in the history of science --- and it is important, for what comes next, to see it as exactly that.  The charm quark (1974) had been demanded by the GIM mechanism; the bottom quark (1977) by the Kobayashi--Maskawa structure; the gluon (1979) by QCD; the $W$ and $Z$ (1983) arrived at the masses the electroweak theory required; the top quark (1995) had been located, within errors, by precision fits before it was seen; the Higgs boson (2012; \citealp{ATLAS2012,CMS2012}) was the final unchecked entry of a mechanism written down in 1964.  Every one of these discoveries was a particle from a list completed before 1975.  In fifty years, no collider has added a single entry \emph{to} the list.  The lone established crack in the edifice --- neutrino oscillations (1998; \citealp{Fukuda1998}) --- came from underground detectors, not colliders, and prescribes no new collider scale.

\paragraph{Three roads that did not arrive.}
It is not for lack of trying, and the record deserves to be stated plainly, because the field rarely states it.  For fifty years, three great programs promised the next layer.  \emph{Supersymmetry}, in its minimal form the MSSM, promised to stabilize the electroweak scale with partner particles ``just above'' each successive frontier --- but the minimal model carries more than one hundred free parameters, so every null result at LEP, at the Tevatron, and at LHC Runs 1 and 2 was answered by sliding the spectrum upward, and the natural parameter region that motivated the enterprise has by now been all but eliminated.  \emph{Extra-dimensional models} --- large extra dimensions, and the warped Randall--Sundrum geometry that this paper's subject superficially resembles --- contributed genuine structural insight, but their mass scales were always \emph{inputs}: a radius or a warp factor to be scanned, so that no exclusion could ever be fatal, only relocating.  \emph{String theory} constructed a candidate for everything and committed itself to nothing measurable: the landscape of vacua, at a famously astronomical count, provides no single number at accessible energy whose failure would falsify anything.  Fifty years, three programs, and not one clear, pre-registered, falsifiable collider prediction among them --- and by now the first two stand nearly falsified in their motivating forms, while the third cannot be falsified at all.

\paragraph{The diagnosis is structural, not personal.}
These were serious ideas pursued by physicists of the first rank, and much of their machinery --- the supersymmetry algebra, warped geometry, holography --- is permanent mathematics that will outlive the particular models.  The failure mode was structural: \emph{continuous free parameters}.  A theory with dials can retreat from any measurement, so it never stakes anything; it survives every experiment and is confirmed by none.  A theory that can slide cannot lose --- and a theory that cannot lose cannot win.  The lesson of the five decades is therefore not that the next layer is out of reach; it is that predictive power is measured by what a theory \emph{forbids}, and dials are the enemy of forbidding.

\paragraph{What a worthy successor would have to look like.}
It would put a number on the table \emph{before} the data arrive.  The number would be derived --- traceable through an explicit chain to structural constants, with no continuous parameter anywhere in the chain that could absorb a disagreement.  It would over-commit: tie the new number to quantities already measured, so that sliding the prediction would break what already works.  And it would specify the signal completely --- not a mass alone, but widths, decay preferences, angular shapes, even interference signs --- so that the experimental verdict, in either direction, is beyond appeal.  Half a century of frustration has, at least, produced an exact job description.  The next section introduces the theory that applies for the job.

\section{GG-Theory in Plain English --- and Why the LHC Gets to Judge It}
\label{sec:plainenglish}

If this paper is your first encounter with \GGTheory{}, welcome --- this section was written for you, and it assumes nothing.  The previous section ended with a job description that no theory of the last fifty years has filled.  Before the derivations begin, you deserve to hear, in plain words and at a humane pace, what kind of theory now applies for the job --- because it is unusual enough that you should first meet it as a story, not as a formalism.

Here is the story.  \GGTheory{} proposes \citep{ArisakaA2GG6} that the world has six dimensions --- but not in the familiar sense of curled-up extra \emph{space}.  Four are ordinary spacetime.  The fifth is the \emph{energy scale} itself: the ``zoom level'' at which you examine nature --- the thing that changes as you go from atoms to nuclei to quarks --- taken seriously as a geometric direction, with a coordinate \(r=\ln(\EGUT/\mu)\) and a finite length that is an integer, \(K=35\).  The sixth is the \emph{gauge phase} --- the angle that rotates the wavefunction of charged matter, the angle the Aharonov--Bohm effect makes visible --- promoted from bookkeeping device to literal coordinate.  You cannot walk along either of them; nothing can, because they are not places.  Yet fields live on all six directions, and standing waves along the scale direction are particles with definite masses --- computable ones, because the length of the direction is a counting number rather than a knob.  That is the theory's signature everywhere: its entire freedom is five integers, \((M,N,L;J,K)=(1,3,5;7,35)\), the \OGNfive{} ledger whose anomaly-cancellation origin is proved in \AThree{} \citep{ArisakaA3BI6}.  Each of the five carries a fixed meaning --- $M=1$ the single tensor seam and charge-quantization unit, $N=3$ the color triplicity, $L=5$ the $\mathrm{SU}(5)$ block, $J=7$ the odd closure index, and $K=35=LJ=M^2+N^2+L^2$ the throat depth that fixes the lock scale $\Mlock=\tfrac12 M_{\mathrm{Pl}}e^{-35}$ --- so the five are forced, not fitted, and not an arbitrary quintet.  There is no continuous parameter anywhere.  Once the integers are fixed --- and they are fixed by consistency, not by fitting --- every mass, coupling, width, and angle that follows is an \emph{output}.

\paragraph{The \GG{} wager.}
This paper presents the sharpest of those outputs, and asks the LHC to judge it.  From the integer ledger the theory derives the \emph{lock scale} (\citealp{ArisakaA2GG6}, Theorem~6; \citealp{ArisakaA9SM})
\begin{equation}
\Mlock = \frac{\EPl}{2e^{35}} \simeq 3.849\,\mathrm{TeV},
\label{eq:Mlock-intro}
\end{equation}
where the \(35\) in the exponent is the integer \(K\), not a fitted quantity --- and it spends the same number twice, because the electroweak scale must then equal \(\Mlock(2/5)^3\approx246\)~GeV, and does.  At the lock scale it places not a lone bump but a three-member cluster --- a spin-2 graviton mode, a color-octet vector, a leptophobic \(Z'_B\) --- with derived couplings, derived widths, a derived interference sign, and derived angular distributions, decaying preferentially to top-quark pairs; and the un-suppressible member of that cluster sits, at its computed rate, exactly on the current ATLAS/CMS exclusion frontier.  A null result at \(\lesssim2\)~ab\(^{-1}\) retires the current implementation outright --- there are no parameters with which to escape.  This wager may fail; it is constructed so that failure is \emph{visible}.

Measure this against the job description of \S\ref{sec:fiftyyears}, item by item.  A number on the table before the data arrive: \(3.849\)~TeV, in print.  A derived number: traceable through an explicit theorem chain to the Planck scale and an integer, with no continuous parameter anywhere in the chain.  Over-commitment: the same chain forces the electroweak scale already measured, so sliding the prediction would break what already works.  Complete specification: not a mass alone but siblings, widths, branching preferences, angular shapes, and the interference sign, each derived and each checkable separately.  Individual \emph{models} have, on occasion, staked a fixed number --- a benchmark \(Z'\) here, a pre-registered bump hypothesis there.  What no program has done is all four at once: derive the number, over-commit it to quantities already measured, specify the complete signal, and state the criteria of failure in advance.  This is why we say, carefully and without exaggeration, that this is the first time in the fifty-year history of ``beyond the Standard Model'' that a proposal both predicts its quantities and states, in advance and in writing, the criteria of its own falsification.

And the timing could hardly be better --- this is the part that should quicken the pulse of any experimentalist reading.  Part~\ref{part:hllhcsim} of this paper simulates the search at the fidelity of a real analysis, anchored to the published ATLAS top-pair search, and its headline is that \emph{the collisions that decide this question may already be recorded}: a dedicated reanalysis of the existing Run-2+3 dataset carries evidence-level sensitivity --- \(2.7\)--\(3.7\sigma\) per experiment if the cluster is there --- so a first \(>\!3\sigma\) hint of the color-octet \(g^*\) is already latent in the collisions recorded by the end of 2026, waiting only for the dedicated reanalysis this paper specifies, and a combined ATLAS+CMS analysis with realistically improved systematics reaches discovery grade from the same tape.  Whichever way it falls, the question closes within the HL-LHC decision window, roughly 2026--2035: the exact span of one doctoral career begun now.  So keep reading --- the rest of this Part equips you with the one genuinely new concept the theory runs on (\S\ref{sec:central-claim}), the constitutional route by which a collider prediction descends from axioms (\S\ref{sec:constitutional-route}), and the six-dimensional geometry that carries it (\S\ref{sec:geometry-origin}); and if you want the theory beyond this paper, the twenty-two-paper cohort map of \S\ref{sec:cohort-p1} shows where every door leads.

The reason this question is sharper than the usual beyond-Standard-Model question is worth restating in the cohort's own terms: the number in Eq.~\eqref{eq:Mlock-intro} is not tuned to the collider.  It is the same number that participates in the electroweak-scale derivation and in the hierarchy ledger of the theory.  A null or positive answer at the LHC therefore propagates immediately into the constitutional core of the program.


This paper has four simultaneous aims.  The first is theoretical: to explain, from the constitutional layer of the theory and the already-derived \GG{} theorem package, why a six-dimensional Kaluza--Klein (KK) sector exists at all and why its first experimentally relevant scale is anchored near \(3.85\,\mathrm{TeV}\).  The second is phenomenological: to translate that structural statement into concrete collider observables -- masses, widths, angular distributions, channel hierarchies, and correlated partner states.  The third is experimental: to review the current public ATLAS and CMS landscape in a way that is fair both to the collaborations and to the specific non-RS features of \GG{}.  The fourth is strategic: to state explicitly why the LHC and especially the HL-LHC constitute the best collider falsifier of the current \GG{} implementation, and how ATLAS and CMS could improve their sensitivity with relatively practical trigger and data-format choices.

The discussion is intentionally long and methodical because a serious falsifier deserves serious bookkeeping.  The paper therefore begins one layer above the KK spectrum itself.  It starts from the constitutional route of the GG-Constitution, then moves to the 6D geometry, the KK reduction, the lock-scale derivation, the collider observables, the current ATLAS/CMS status, the HL-LHC outlook, and finally to a trigger roadmap.  Readers primarily interested in experiment may skip directly to Sections~\ref{sec:current-status} and \ref{sec:near-term}; readers primarily interested in the conceptual role of \GDOS{} and \OGGEP{} may wish to read Sections~\ref{sec:constitutional-route} and \ref{sec:gdos-section} with special care.  (A fuller set of reading paths, one per kind of reader, is laid out near the close of this Part: \S\ref{sec:readingpaths}.)

Before proceeding, it is useful to state a central normalization point clearly.  In many extra-dimensional papers the phrase ``first KK graviton mass'' refers to a specific single-pole solution of a Bessel equation in a 5D RS background.  In this paper, by contrast, the collider benchmark \(3.849\,\mathrm{TeV}\) refers to the \emph{first lock-scale cluster} of the 6D \GG{} problem.  The first isolated sub-pole within that cluster may depend on details of the full 6D boundary problem; the cluster centroid does not.  This distinction is not semantic.  It changes how one should search in data.

%% =====================================================================
%% COHORT BACK-EDITS implied by the Option-A (radial-only) choice.
%%  (1) DONE in A2-GG6 v9.23 (2026-06-04): the KK-spectrum appendix no longer
%%      warps the angular ell=+/-1 modes into a ~3.96 TeV doublet; it now states
%%      two separate towers (radial -> lock cluster; angular ~|l|E_GUT), and the
%%      dangling "factor J derived in A9" forward-reference was corrected.
%%      This paper's version-of-record pointer above now reads A2 (v9.23).
%%  (2) STILL OPEN -- A11-Origin: uses the numerically wrong shorthand
%%      M_lock = E_GUT * e^{-K} (~14 GeV). Fix to M_lock = (1/2) M_Pl e^{-K}
%%      ~ 3.85 TeV (the closed form used here and in A2/A9).
%% =====================================================================

\section{The central claim: the Kaluza--Klein lock cluster as an open-system observable}
\label{sec:central-claim}

It is worth stating, in one place and at a walking pace, the recognition this paper shares with the whole \GGTheory{} cohort --- because it is the one genuinely new concept a reader needs, and because without it the true significance of everything that follows cannot be grasped.  Nothing in this section requires prior exposure to \GGTheory{}; it requires only honesty about something every physicist already knows and every textbook quietly sets aside.

\paragraph{No system in nature is closed --- not even one electron.}
Consider the simplest object physics owns: a single electron.  It is never alone.  It is dressed by its own electromagnetic field; it exchanges virtual photons with the vacuum; anything that has ever scattered off it carries a record of the encounter away.  Strip all of that off --- imagine an electron truly decoupled from everything --- and you have imagined a particle that cannot be seen, deflected, or counted, because \emph{to observe a system is precisely to open it}.  A perfectly closed system is not merely hard to prepare; it is unobservable in principle, a contradiction in terms the moment a measurement is attempted.  Yet the whole formal apparatus of physics is built the other way around: textbook dynamics --- classical or quantum --- is the dynamics of \emph{closed} systems, evolved in splendid isolation, with the observer bolted on afterwards as an axiom.  Everything in nature, from a single electron to a living cell to the universe's contents as a whole, is an open system; our theories have treated openness as a nuisance to be appended.  \GGTheory{} begins by refusing that inversion.

\paragraph{Where nature runs, and where we watch it from.}
In \GG{}, the complete dynamics of nature runs in the six-dimensional bulk --- ordinary spacetime plus the scale and phase directions of \S\ref{sec:plainenglish}.  In the bulk the accounting closes: the dynamics is lawful, complete, and deterministic.  But no experiment has ever been performed in the bulk, and none ever will be: we and our instruments inhabit the four-dimensional boundary, and every datum ever recorded --- every track, every calorimeter cluster, every published cross section --- is a \emph{boundary record}.  The relation between the two is the theory's third axiom, \OGGEP{} (\AX{3} of \AOne{}, \citealp{ArisakaA1Constitution}): the four-dimensional world is a lawful \emph{projection} of the six-dimensional bulk, and \textbf{that projection is what observation is}.  Observation is not something a conscious agent adds to physics; it is the bulk-to-boundary map itself, running always and everywhere, with or without anyone watching.  This single reversal --- openness as the fundamental condition, observation as geometry --- is the conceptual core of the entire twenty-two-paper program.

\paragraph{The law of the projection: \GDOS{} and \GRIP{}.}
An axiom that says ``the boundary is a projection'' would be philosophy if the projection were arbitrary.  It is not.  Its dynamics is a definite law --- \GDOS{}, the \GG{} dynamics of open systems, derived in \AFive{} \citep{ArisakaA5GDOS} --- which specifies how a projected system actually evolves, dissipates, and registers: what the boundary keeps of the bulk, and at what cost.  And \GDOS{} is itself forced rather than chosen, pinned from two independent sides: the geometric pillars of \GiFour{} (\ASixGRIP{}, \citealp{ArisakaA6Gi4}) and the quantum-information pillars of \QiFour{} (\ASeven{}, \citealp{ArisakaA7Qi4}).  The runtime form of the law is the \GRIP{} cascade of \ASixGRIP{}, with its two regimes \textsf{GRIP-In} and \textsf{GRIP-Out}: \textsf{GRIP-In} is the bulk feeding the boundary --- excitation, descent, the arrival of energy at the seam --- and \textsf{GRIP-Out} is the boundary paying it out as registered, irreversible records.  In this vocabulary a detector is not a passive witness; it is a purpose-built \textsf{GRIP-Out} amplifier, engineered to make one microscopic registration grow into a macroscopic, thermodynamically irreversible entry in a database.

\paragraph{A century of confusion, resolved (\AEight{}).}
The reader has met this problem before, under another name.  Since 1926, quantum mechanics has carried two rules that do not speak to each other: unitary evolution for systems left alone, and ``collapse'' for systems measured --- with no principled statement of where one ends and the other begins.  Bell called the dividing line the \emph{shifty split}, and a century of interpretations has consisted largely of relocating it: into consciousness, into decoherence, into many worlds, into hidden variables.  \AEight{} \citep{ArisakaA8QM} shows that within \GG{} the split is not shifty --- it is the bulk/boundary interface, and it is \emph{geometric}.  What textbooks call collapse is the \OGGEP{} projection in action: in the bulk the evolution never stops being lawful and deterministic, while the boundary keeps only what the projection law delivers --- a definite, registered outcome, with the Born weights recovered as a typicality statement rather than postulated.  The measurement problem, on this reading, was never a paradox of quantum mechanics; it was the signature of trying to run a six-dimensional accounting in four dimensions --- a century of confusion produced by one missing map.  This is why the present paper keeps insisting on open-system language for what might seem a plain bump hunt: the language is the solved problem.

\paragraph{Colliders follow this principle exactly.}
Now look at what ATLAS and CMS actually do, and notice that it is \OGGEP{} operating at industrial scale.  No experiment has ever recorded a gauge-dependent field amplitude or a bulk configuration: what is recorded are gauge-invariant kinematic combinations --- invariant masses, transverse momenta, angular distributions, channel-by-channel rates --- reconstructed from irreversible energy depositions.  The full measurement chain (hard scattering, parton shower, hadronization, detector response, trigger, reconstruction, statistical decision rule) is precisely a system instance packet in the language of \GDOS{}: a specification of \emph{which} projection is being run and what it registers.  A resonance search is therefore not merely compatible with the open-system view; it is the open-system experiment par excellence --- a bulk excitation, if one exists at \(3.849\)~TeV, announcing itself through the only door nature provides.

The conclusion of this chain is the central claim of the paper.  Because the projection is lawful and its law is forced, what the boundary can register is fixed by the bulk geometry --- not chosen, not tunable.  The Kaluza--Klein lock cluster near \(\Mlock\approx3.85\)~TeV, with the fifth-force and dark-sector signals that share its scale, is a \emph{registered observable} of one projection: the geometry determines it, which is why this paper can derive it, and why a single clean measurement can refute it.  Figure~\ref{fig:central-claim} states the whole argument as one deductive funnel; the same funnel opens the companion papers \ANine{} (\citealp{ArisakaA9SM}), \ATen{} (\citealp{ArisakaA10Cosmos}), and \AEleven{} (\citealp{ArisakaA11Origin}), with only the payload differing --- there the Standard Model's parameters, the cosmos, and the origin of law; here, a resonance cluster within reach of a machine that already exists.

\begin{figure}[hbt!]
\centering
\resizebox{0.65\textwidth}{!}{%
\begin{tikzpicture}[
 node distance=17mm,
 every node/.style={font=\sffamily\footnotesize, align=center},
 box/.style={draw=black!55, rounded corners, minimum width=92mm, minimum height=11mm},
 arrow/.style={-{Stealth[length=3mm]}, line width=1.10pt},
 elab/.style={font=\sffamily\scriptsize\itshape, fill=white, inner sep=1.5pt}
]
\node[box, fill=violet!10] (open) {\textbf{Every physical system is open}\\ a particle, the universe, a living cell --- none can be isolated from its environment};
\node[box, fill=blue!8, below of=open] (oggep) {\textbf{\OGGEP{}} (\AX{3}): the lawful projection \;6D bulk $\to$ 4D boundary\\ \textit{this projection is observation itself} \;\; {\scriptsize[\AOne{}, \AEight{}]}};
\node[box, fill=yellow!13, below of=oggep] (gdos) {\textbf{\GDOS{}}: the dynamics of the open-system projection\\ \textit{an observable is the value at a stationary seam} \;\; {\scriptsize[\AFive{}]}};
\node[box, fill=orange!14, line width=1pt, below of=gdos] (obs) {\textbf{The KK lock cluster near $\Mlock\approx3.85$~TeV}\\ with its fifth-force and dark-sector signals --- \textit{a registered observable, not a free input}};
\node[box, fill=green!12, below of=obs] (pill) {\textbf{\GDOS{} is forced} by the \GiFour{} pillars {\scriptsize[\ASix{}]} and the \QiFour{} pillars {\scriptsize[\ASeven{}]}};
\node[box, fill=orange!16, line width=1pt, below of=pill] (concl) {$\Rightarrow$ \textbf{it is determined}\\ {\footnotesize read off one geometry --- a sharp, falsifiable target for the coming colliders}};
\draw[arrow] (open)  -- node[elab,right]{no closed 4D theory exists}            (oggep);
\draw[arrow] (oggep) -- node[elab,right]{the projection must be a definite law}  (gdos);
\draw[arrow] (gdos)  -- node[elab,right]{apply to this paper's sector}          (obs);
\draw[arrow] (obs)   -- node[elab,right]{and the law is not arbitrary}           (pill);
\draw[arrow] (pill)  -- node[elab,right]{projection forced $\Rightarrow$ observable forced} (concl);
\end{tikzpicture}}
\caption[The open-system funnel shared across the \GGTheory{} cohort.]{\textbf{The central claim, as one deductive funnel}, shared across the \GGTheory{} cohort. Because no physical system is closed, the four-dimensional world is a lawful projection (\OGGEP{}, \AX{3}) of the six-dimensional bulk, and that projection is observation itself; its dynamics is \GDOS{}, forced by the \GiFour{} and \QiFour{} pillars. The same funnel opens the companion papers \ANine{}, \ATen{}, and \AEleven{}; only the payload differs --- here, the Kaluza--Klein lock cluster.}
\label{fig:central-claim}
\end{figure}

\section{From the GG-Constitution to Collider Observables}
\label{sec:constitutional-route}

The argument here starts not from a phenomenological ansatz imported from RS, but from the GG-Constitution axiom system and the already-derived theorem packages of the cohort.  This is the right starting point: in the present framework, collider phenomenology is a downstream consequence of a constitutional grammar, not an independent bolt-on model.

At the constitutional level, four ingredients matter most for the present paper (all from \AOne{}, \citealp{ArisakaA1Constitution}).  \textbf{AX-1} (local causality) demands that admissible kernels be retarded and that registered detector observables arise from causal boundary records.  \textbf{AX-2} (\GGEP) states that gauge and geometric content must arise from a single primitive composite connection.  \textbf{AX-3} (\OGGEP) requires an explicit bulk-to-boundary operational dictionary: a theory is not complete unless it says how bulk dynamics become registered observables.  \textbf{AX-4} (maximum symmetry, minimum freedom) forbids gratuitous free parameters and elevates minimal integer data to constitutive status.  Together with the anomaly-cancellation and normalization postulates (POS-3 and POS-6), these axioms force a remarkably specific route to collider physics (\citealp{ArisakaA1Constitution}; \citealp{ArisakaA2GG6}).

The logic runs as follows.  Once AX-2 and AX-4 are taken seriously, the bulk cannot remain an unconstrained higher-dimensional playground.  It must support a single theorem-level geometry in which the renormalization direction and the compact Abelian holonomy are both geometric, because both are already required to represent the running and the compact gauge phase with minimal freedom.  This selects the six-dimensional stage
\begin{equation}
M_6 = M_4\times I_r\times S^1_\theta,
\qquad r = \ln\!\left(\frac{E_{\mathrm{GUT}}}{\mu}\right),
\label{eq:bulkstage}
\end{equation}
with ultraviolet gauge-gravity block
\begin{equation}
G_{\mathrm{bulk}} = \SUfive\times \UB\times \UQi\times \mathrm{Gravity}.
\label{eq:gaugecontent}
\end{equation}
The 6D tensor sector is then fixed by the single-tensor Green--Schwarz-modified Bianchi identity BI-6,
\begin{equation}
dH_3 = X_4,
\qquad
X_4 = \mathrm{tr}\,\mathcal R^2 - \mathrm{tr}\,F_5^2 - \frac{2}{5}F_B^2 - \frac{2}{5}F_{Qi}^2,
\label{eq:BI6}
\end{equation}
which is the structure that closes the anomaly story and makes \GG{} fundamentally different from a bare Kaluza--Klein metric reduction (\citealp{ArisakaA2GG6}; \citealp{ArisakaA3BI6}).

Why does this matter for KK particles?  Because once the stage \eqref{eq:bulkstage} exists, the boundary cannot remain spectrally featureless.  Every field living on that six-dimensional stage admits a harmonic decomposition on the internal space.  A KK tower is therefore not an independent assumption; it is the boundary spectral shadow of the bulk geometry itself.  The only remaining question is what fixes the first accessible scale.  The answer comes from AX-4 plus POS-6: the OGN-5 integer ledger
\begin{equation}
(M,N,L;J,K)=(1,3,5;7,35),
\qquad J=2N+1,
\qquad K=LJ=M^2+N^2+L^2,
\label{eq:OGN5}
\end{equation}
fixes the throat depth, and therefore the lock scale,
\begin{equation}
\Mlock=\frac{\EPl}{2e^{35}} \simeq 3.849\,\mathrm{TeV}.
\label{eq:lockscale}
\end{equation}
That is why the \(3.85\,\mathrm{TeV}\) collider target is not simply a fit.  It is the theorem-level infrared end of the geometry.

The collider observable is then supplied by AX-3.  Through \OGGEP{}, bulk excitations become boundary records: invariant masses, transverse momenta, angular distributions, and channel-by-channel rates.  From this perspective, a heavy resonance search is already an open-system problem.  The hard scattering, parton shower, hadronization, detector response, trigger, reconstruction, and statistical decision rule together constitute a system instance packet (SIP) in the language of \GDOS{}.  The fact that ATLAS and CMS reconstruct gauge-invariant kinematic combinations rather than raw gauge-dependent field variables is not incidental; it is precisely the collider form of DGI and \OGGEP{} at work.  We will return to this point in Section~\ref{sec:gdos-section}.

\begin{table}[H]
\centering
\caption[Constitutional route from the GG-Constitution to the KK collider program.]{Constitutional route from the GG-Constitution to the KK collider program.  The table is intentionally compact: it is meant to show that the collider sector is not a detached phenomenological appendix, but a direct downstream consequence of the constitutional grammar.}
\label{tab:constitutional}
\small
\begin{tabularx}{\textwidth}{>{\raggedright\arraybackslash}p{2.3cm} >{\raggedright\arraybackslash}p{3.0cm} X}
\toprule
\textbf{Layer} & \textbf{Constitutional content} & \textbf{Collider consequence}\\
\midrule
AX-1 & Local causality & Detector observables must be built from retarded, registered records; the search is formulated in causal event streams, not in acausal inverse problems.\\
AX-2 & Geometry--gauge equivalence & Gauge and gravitational sectors share a single primitive geometric stage; KK towers are geometric, not optional.\\
AX-3 & \OGGEP{} bulk--boundary dictionary & Bulk excitations become boundary invariant masses, angular distributions, and branching patterns in ATLAS/CMS.\\
AX-4 & Maximum symmetry, minimum freedom & The infrared scale is fixed by minimal integer data rather than scanned as a free parameter.\\
POS-3 & Anomaly freedom by inflow & BI-6 closes the tensor/gauge system and distinguishes \GG{} from ordinary KK or RS intuition.\\
POS-5 & RG compatibility & The radial coordinate \(r\) is the renormalization direction; infrared collider masses are scale-redshifted images of ultraviolet geometry.\\
POS-6 & Minimal-rational normalization & OGN-5 fixes \(K=35\), hence the lock scale \(\Mlock\simeq3.849\,\mathrm{TeV}\).\\
\bottomrule
\end{tabularx}
\end{table}

The bottom line is that the collider prediction does not float freely.  It is tied to the same constitutional commitments that underwrite the rest of the program.  This is precisely why the KK search is so sharp a falsifier.  (The program-wide panorama --- the twenty-two-paper cohort map, and the conventions for reading this paper against it --- closes this Part: \S\ref{sec:cohort-p1}, Figure~\ref{fig:cohort-map-p1}.)

\section{The 6D Geometric Origin of the KK Tower}
\label{sec:geometry-origin}

\subsection{The six-dimensional stage}

The theorem-level \GG{} bulk is a genuine six-dimensional Lorentzian manifold.  Its metric is fixed -- not chosen -- by Theorem~2 of \citet{ArisakaA2GG6}, which forces the canonical product geometry
\begin{equation}
\dd s_6^2 = e^{-2r}\,\eta_{\mu\nu}\,\dd x^\mu \dd x^\nu + \dd r^2 + \dd\theta^2,
\qquad r\in[0,K],\quad \theta\in[0,2\pi),
\label{eq:metric6d}
\end{equation}
in canonical \GG{} units, carrying \emph{no} free metric data: the AdS curvature \(k\), the circle radius \(R_\theta\), and the throat length \(r_c\) of a generic warped construction are all locked (to unity, or to the \OGNfive{} integer \(K=35\)) by the maximum-symmetry/minimum-freedom posture \citep{ArisakaA2GG6,Kaluza1921,Klein1926,RandallSundrum1999}.  The decisive structural fact is that both internal directions are \emph{non-spatial}: \(r\) is the scale (renormalization-group) direction \(I_r\), and \(\theta\) is the phase circle \(S^1_\theta\), each dimensionless.  Neither is a spatial extra dimension.  This already shows why the eventual spectrum is richer than the standard 5D RS picture: there are two independent internal directions -- a scale direction and a compact phase direction -- so spectral labels can arise from both.  In the mode equations below we retain a general scale profile \(A(r)\) and circle radius \(R_\theta\) as bookkeeping symbols, specialized to the canonical \(A(r)=r\) and \(R_\theta=1\) at the end.

The role of \(r\) is physically transparent in the latest \GG{} synthesis.  It is not merely an extra dimension inserted to imitate RS.  It is the geometric representation of the renormalization scale,
\begin{equation}
r = \ln\!\left(\frac{\EGUT}{\mu}\right),
\label{eq:rRG}
\end{equation}
so transport along \(r\) is the geometric avatar of running.\footnote{A careful reader will ask in which renormalization scheme \(\mu\) is defined, since RG ``distance'' is scheme-dependent beyond one loop.  The chart is bookkeeping, not physics: a scheme change acts as a smooth reparametrization of the scale direction, under which everything this paper uses --- the integer length \(K=35\) (a ledger integer fixed by anomaly matching, \citealp{ArisakaA3BI6}, not a measured perturbative distance), the canonical metric of \citet{ArisakaA2GG6} (Theorem~2), the lock mass, and the overlap integrals evaluated in the canonical chart --- is invariant.  Choosing \(A(r)=r\) is the analogue of choosing normal coordinates.}  The role of \(\theta\) is equally important: it is the compact Abelian holonomy direction whose topological structure feeds both the \(\UB\) and \(\UQi\) sectors via BI-6.  In ordinary KK language one could say that \(\theta\) generates a circle tower.  In the stronger \GG{} language, one should say that \(\theta\) carries one piece of the anomaly-locking story itself.

\subsection{Mode expansion and the two quantum numbers}

Any six-dimensional field \(\Phi(x^\mu,r,\theta)\) can be expanded as
\begin{equation}
\Phi(x^\mu,r,\theta)=\sum_{n=0}^{\infty}\sum_{\ell=-\infty}^{\infty}\phi^{(n,\ell)}(x^\mu)\,f_{n,\ell}(r)\,e^{i\ell\theta}.
\label{eq:modeexpansion}
\end{equation}
Here \(n\) labels the radial excitation family and \(\ell\) labels the compact-angle Fourier mode.  The profile functions satisfy a radial Sturm--Liouville problem.  For scalar fluctuations a representative mode equation is \citep{ArisakaA2GG6}:
\begin{equation}
-\partial_r\!\left(e^{-4A}\partial_r f_{n,\ell}\right)+e^{-4A}\frac{\ell^2}{R_\theta^2}f_{n,\ell}
=
m_{n,\ell}^2 e^{-2A}f_{n,\ell}.
\label{eq:modeequation}
\end{equation}
The exact spin-2 equation differs in tensor structure but shares the same qualitative ingredients: a radial operator, a compact-angle mass contribution, and an infrared scale suppression.

\paragraph{The compact-angle eigenvalue carries the Wilson line.}
One refinement of Eq.~\eqref{eq:modeequation} must be recorded here,
because the electroweak sector of the cohort rests on it. The integer
\(\ell\) is the eigenvalue of \(-i\partial_\theta\) only when the
gauge connection along the circle vanishes. In the presence of a
Wilson line --- the holonomy
\(\Theta_H = g_6\oint_{S^1_\theta} A_\theta\,\dd\theta\), which is
gauge-invariant and cannot be removed --- a field of charge \(q\)
under the corresponding generator feels the shifted eigenvalue
\begin{equation}
\frac{\ell^2}{R_\theta^2}
\;\longrightarrow\;
\frac{1}{R_\theta^{2}}\left(\ell+\frac{q\,\Theta_H}{2\pi}\right)^{\!2},
\label{eq:wilsonshift}
\end{equation}
which is the Hosotani mechanism written spectrally
\citep{Hosotani1983}: a constant \(A_\theta\) is locally pure gauge
and changes no field strength, yet it moves every charged mode, and
in particular it lifts the \(\ell=0\) modes that would otherwise be
massless. \emph{That lifting is the origin of electroweak mass in
this framework}: the four-dimensional Higgs doublet is the zero mode
of \(A_\theta\) in the broken directions, and the holonomy that
Eq.~\eqref{eq:wilsonshift} inserts is the same \(\theta_H\) whose
selected value \citet{ArisakaA9SM} fixes from the \OGNfive{} ledger.
This paper does not select that value --- the vacuum angle and the
scale at which the resulting four-dimensional masses are registered
are theorems of the codebook paper, where the warp puts them at the
infrared endpoint rather than at \(\EGUT{}\). What is recorded here
is only the spectral fact, because it is the mechanism by which a
circle-side label becomes a laboratory mass.

Two consequences for this paper's own predictions, both benign.
First, the radial (\(\ell=0\)) lock cluster of \S\ref{sec:lockscale}
is unaffected in its \emph{radial} structure: the shift adds a common
compact-angle contribution, and for the color-neutral, \(\UB{}\)- and
graviton-sector modes that build the cluster it is not the quantity
that sets the infrared scale. Second --- and this is the one worth
stating, since it could have gone the other way --- the
no-angular-satellites prediction survives. Eq.~\eqref{eq:wilsonshift} displaces
the angular tower by a fraction \(\Theta_H/2\pi\) of one unit, and
with the selected angle that fraction is of order one per cent; the
\(\ell=\pm1\) partners therefore remain at \(\sim\EGUT{}\), still
some twelve orders of magnitude above the lock cluster. A satellite a
few percent from \(3.85\)~TeV would falsify the non-spatial reading
exactly as before.

Two immediate qualitative conclusions follow, and they sharply separate the two towers.  First, the radial (\(\ell=0\)) modes are the normalizable resonances of the \emph{finite} radial interval \(r\in[0,K]\); their masses are redshifted to the infrared endpoint and set the observable scale \(\Mlock\) (the radial Bessel tower of \S\ref{sec:lockscale}).  The angular (\(\ell\neq0\)) Fourier modes instead carry the additional contribution \(\ell^2/R_\theta^2\), which in canonical \GG{} units (\(R_\theta=1\)) is of order one unit of holonomy energy per mode --- i.e.\ of order \(\EGUT\), and \emph{not} redshifted to the TeV scale:
\begin{equation}
m_{n,0}=\frac{\xi_n}{\xi_1}\,\Mlock\quad(\text{radial tower, observable}),\qquad
m_{0,\ell}\sim |\ell|\,\frac{\hbar}{R_\theta}\sim \EGUT\quad(\ell\neq0,\ \text{unobservable}).
\label{eq:twotowers}
\end{equation}
Second, therefore, the angular tower is Kaluza--Klein \emph{in form} but unobservably heavy, while the experimentally relevant ``cluster'' is built not from \(\theta\)-partners but from the lowest \emph{radial} modes of the \emph{several distinct bulk fields} (graviton, KK gluon, baryonic vector) that all redshift to the same infrared endpoint.  That multi-field structure at one scale --- not a compact-angle multiplet --- is the phenomenological fingerprint that separates \GG{} from a single-sequence 5D RS tower.  We fix this convention in the next subsection; it matches \citet[Part~III]{ArisakaA2GG6}.

\subsection{Convention: ``Kaluza--Klein'' in \GG{} means \emph{non-spatial}}
\label{subsec:nonspatial-convention}

One convention is fixed here and used throughout, matching \citet[Part~III]{ArisakaA2GG6}.  We retain the term \emph{Kaluza--Klein} (KK) for the lock-cluster resonances, because it correctly signals their experimental class --- RG-scale-throat resonances --- and connects to existing collider nomenclature (Randall--Sundrum gravitons, warped resonances).  But the \GG{} resonances are \emph{non-spatial} KK states: the two internal directions are the scale axis \(I_r\) and the phase circle \(S^1_\theta\), each dimensionless, and \emph{neither} is a spatial extra dimension.  The reader should mentally prefix ``non-spatial'' to every later occurrence of ``KK.''  Concretely, two towers must be kept apart:
\begin{enumerate}
\item \textbf{Angular (\(S^1_\theta\)) Fourier tower.}  Integer modes \(\ell\neq0\) with masses set by the holonomy/\(\hbar\) conversion, of order \(\EGUT\).  Kaluza--Klein in form, but far too heavy to produce at any foreseeable collider; it plays \emph{no} collider role.
\item \textbf{Radial (RG-scale) throat lock cluster.}  The finite radial interval \(r\in[0,K{=}35]\) carries a \emph{discrete} spectrum (two-brane Randall--Sundrum / holographic-composite type) whose lowest modes sit at \(\Mlock\approx 3.85\,\mathrm{TeV}\).  \emph{This} is the observable signal --- ``the KK particles of \GG{}'' --- with excited members carrying additional radial nodes.
\end{enumerate}
In one sentence: the angular tower is heavy and invisible; the radial lock cluster near \(3.85\,\mathrm{TeV}\) is what an experimentalist would call the Kaluza--Klein resonance of \GG{}.

\subsection{Radial branches and the multi-field lock cluster}

With the angular tower parked at \(\EGUT\), the collider-relevant spectrum is organized by two labels that are \emph{both} radial in origin: the radial node number \(n\) within a given bulk field, and the field species (graviton, KK gluon, baryonic vector).
\begin{enumerate}
\item \textbf{Radial tower (\(\ell=0\), increasing \(n\)).}  These are the scale-throat resonances, with the familiar Bessel-root spacing \(m_n=(\xi_n/\xi_1)\Mlock\): the first three sit near \(3.85\), \(7.0\), and \(10.2\,\mathrm{TeV}\).  Only the lowest is comfortably in Run-3 reach.
\item \textbf{Multi-field lock cluster (lowest \(n\), several species).}  The lowest modes of the different bulk fields populate the single infrared scale fixed by Theorem~T-KK-1: the spin-2 graviton at \(\Mlock\), together with a spin-1 pair (a color-octet KK gluon and a leptophobic \(\UB\) vector) whose exact placement is the subject of the fine-structure theorem T-KK-2 (\S\ref{subsec:tkk2}) --- a derived two-scenario dichotomy (free-field \(0.639\,\Mlock=2.46\)~TeV versus composite-hardened \(\approx\Mlock\)) leaving no \(O(1)\) flag behind.  This multi-state structure of \emph{different} species tied to one scale --- not a set of \(\theta\)-partners --- is the genuine 6D signature.
\item \textbf{Tensor/flux mixing.}  Once the full tensor structure and \(\UB\) flux are included, the reconstructed peaks need not be pure species eigenstates, so the observed line shape can look like a doublet or triplet of partially overlapping states near \(3.85\)--\(4\,\mathrm{TeV}\).
\end{enumerate}
This is not an academic detail.  A collider search optimized for a single narrow RS resonance can lose sensitivity to this near-degenerate multi-field cluster if the several nearby states are forced into a one-pole template.  One of the central practical messages of this paper is therefore methodological: \emph{a \GG{} search must be cluster-aware from the outset}.  We return to this in Sections~\ref{sec:observables} and \ref{sec:trigger-roadmap}.

\subsection{Why GG-6 is not reducible to ordinary KK or RS}
\label{subsec:notKKRS}

In one line, before the details: the fifth dimension is the renormalization-group axis --- not a Randall--Sundrum spatial direction --- and the lock cluster is an RG threshold spectrum, not a tower of spatial gravitons.  The rest of this subsection unpacks, and defends, that sentence.

This subsection carries more conceptual weight than any other in Part~I, because the resemblance it addresses is genuinely treacherous.  A reader who knows the extra-dimensional literature will look at the \GG{} bulk --- a warped interval carrying a factor \(e^{-2r}\) and a compact circle carrying integer modes --- and file it instantly: \emph{the fifth dimension is Randall--Sundrum, the sixth is Kaluza--Klein}.  Every equation in this paper would then read as a re-derivation of known physics.  The filing is wrong, and it is wrong in a precise, stateable way: \emph{both readings mistake the coordinates for directions of space}.  Neither of the two extra dimensions of \GG{} is spatial --- the fifth is the \emph{energy scale itself}, geometrized, and the sixth is the \emph{gauge phase itself}, geometrized --- and essentially every phenomenological difference exploited in this paper (the derived mass, the absence of a radion, the absence of angular satellites, the multi-field cluster, the integer hierarchy) traces back to that single distinction.  We take the two dimensions in turn, then collect the full comparison in Table~\ref{tab:kkrsgg}.

\paragraph{The classification principle: a dimension is what generates it.}
\citet{ArisakaA2GG6} (Theorem~0 and its remarks) states the criterion cleanly.  A geometric dimension that is not a dimension of space is not, by itself, new --- Minkowski's time was the first, distinguished from space by its signature.  The correct classification of any dimension is by its \emph{symmetry generator}: a spatial (or temporal) direction is one whose generator is a \emph{translation}, with a conserved momentum (or energy).  The two extra directions of \GG{} are generated by no translation at all: the scale direction \(I_r\) is generated by a \emph{dilatation} \(D\) (the renormalization-group flow generator, \([D,P_\mu]=P_\mu\)), and the phase direction \(S^1_\theta\) by a \emph{compact charge} \(Q_\theta\) with integer eigenvalues.  The bulk isometry algebra is
\begin{equation}
\mathfrak{iso}(1,3)\;\oplus\;\mathbb R_{D}\;\oplus\;\mathfrak u(1)_{Q},
\qquad\text{not}\qquad \mathfrak{iso}(1,5),
\label{eq:bulkalgebra}
\end{equation}
and that is the exact technical content of ``non-spatial.''  A parallel statement at the level of coordinates: both extra coordinates are \emph{dimensionless by construction} --- \(r=\ln(\EGUT/\mu)\) is the logarithm of an energy ratio, \(\theta\) is an angle --- whereas every spatial coordinate carries a unit of length.  There is no answer to the question ``what is the radial speed of light?'', because there is no Lorentz signature on \(r\); the question itself belongs to the spatial reading that \GG{} rejects.

\paragraph{The scale (fifth) dimension versus Randall--Sundrum.}
Superficially the resemblance is strong, and deliberately so: the metric \(ds_6^2=e^{-2r}\eta_{\mu\nu}dx^\mu dx^\nu+dr^2+d\theta^2\) (\citealp{ArisakaA2GG6}, Theorem~2) has the RS form, the interval is finite, and the infrared endpoint carries an exponential hierarchy.  The mathematics of boundary-value problems on this background is therefore shared --- which is precisely why Part~II of this paper can borrow Bessel machinery with a clean conscience.  The \emph{ontology} is not shared, and it bites in four places.
(i)~\emph{What the coordinate is.}  In RS, \(y\) is a spatial position; a particle can sit at \(y\), move along \(y\), carry momentum along \(y\).  In \GG{}, \(r=\ln(\EGUT/\mu)\) \emph{is} the renormalization scale: a particle at radial position \(r\) is a state of the effective theory at scale \(\mu=\EGUT e^{-r}\), and motion along \(r\) is not translation but \emph{transmutation} --- the geometrization of RG flow, with the additivity of the RG semigroup forcing the logarithmic coordinate.  The scale profile \(e^{-2r}\) is not gravitational redshift in a fifth spatial direction; it is the definition of scale weight.
(ii)~\emph{What fixes the length.}  The RS throat depth \(kr_c\) is a free continuous dial, tuned to \(\approx35\) because the hierarchy is observed; the radion --- the fluctuation of that length --- is a physical modulus that must be stabilized by an added mechanism.  In \GG{} the interval length is the \emph{integer} \(K=35\), forced by the \OGNfive{} closure \(K=LJ=M^2+N^2+L^2\): the length is a counting number, not a size.  Consequently \emph{there is no radion} --- not a stabilized radion, but no radion field at all, because a resolution has no fluctuating length \citep{ArisakaA4Euclid} --- and the hierarchy \(e^{-35}\) is an output.  This paper's \(hh\) phenomenology (\S\ref{subsec:branchings}) and the no-radion Higgs-precision null of Part~\ref{part:beyond} are direct collider consequences.
(iii)~\emph{What the resonances are.}  In RS the KK gravitons are momentum modes of a spatial fifth dimension.  In \GG{} the throat's discrete spectrum survives --- a finite interval supports normalizable resonances regardless of the interval's ontology --- but the states are properly read as the \emph{composite resonances of the boundary theory at its infrared scale}: the statement usually offered as a holographic analogy is, here, the literal construction.  Same eigenvalue problem, same Bessel zeros, different physical identity --- and the identity matters, because it is what makes the T-KK-2 dichotomy of \S\ref{subsec:tkk2} (free-field eigenvalues versus composite reorganization) a genuine physical question rather than a formality.
(iv)~\emph{What sits at the bottom.}  Because \emph{every} bulk field is redshifted to the same infrared endpoint \(r=K\), the lowest radial modes of the graviton, the color octet, and the \(\UB\) vector all land at one derived scale --- the multi-field lock cluster --- rather than the single graviton pole of minimal RS.  The cluster structure is not decoration; it is the 6D fingerprint.

\paragraph{The phase (sixth) dimension versus Kaluza--Klein.}
The sixth direction looks like Kaluza's circle, and the debt is real --- but \GG{} inverts the construction.  Kaluza and Klein geometrized the electromagnetic potential as the metric's components along a \emph{spatial} circle of radius \(R\); the radius is a free modulus, and the observable signature is a tower of charged momentum modes at \(m_n=n/R\), placed wherever the modulus is tuned.  In \GG{}, \(S^1_\theta\) is the \(\mathrm U(1)\) \emph{fiber itself}, promoted to a literal coordinate: a genuinely principal-bundle direction whose holonomy \emph{is} the gauge phase of charged matter (the Aharonov--Bohm and Berry phases are its holonomies in the literal sense), whose integer characters are the charge quantization (the content of \(M=1\) in the \OGNfive{} ledger), and whose Wilson line \(\theta_H\) performs the electroweak symmetry breaking in \citet{ArisakaA9SM}, via the Hosotani mechanism \citep{Hosotani1983} --- displacement by holonomy rather than by symmetry, in \PaperWTwo{}'s sense (\citealp{ArisakaW2Holonomy}).  Because the direction is topological --- an angle, with the canonical convention \(R_\theta=1\) and \emph{no radius modulus} --- its Fourier tower is not a tunable prediction: the angular modes sit at \(\sim\EGUT\) and are unobservable.  This yields one of the sharpest negative predictions in the paper: \emph{the lock cluster has no angular satellites}.  A conventional 6D spatial compactification would decorate the 3.85-TeV cluster with \(\ell=\pm1\) partners a few percent away in mass; \GG{} forbids them.  Finding them would falsify the non-spatial reading as decisively as finding nothing at all would falsify the mass.

\paragraph{The locking: the stage is not metric-alone.}
The third structural difference has no KK or RS analogue whatsoever.  The \GG{} bulk is not defined by a metric on which gauge fields are later placed; the metric, the tensor sector, and the gauge fluxes arrive locked in a single primitive object through the BI-6 identity,
\begin{equation}
dH_3 = \mathrm{tr}\,\mathcal R^2 - \mathrm{tr}\,F_5^2 - \frac{2}{5}F_B^2 - \frac{2}{5}F_{Qi}^2,
\label{eq:BI6repeat}
\end{equation}
whose trace-ratio \(2/5\) forces the equal treatment of \(\UB\) and \(\UQi\).  The KK tower of \GG{} is therefore not a harmonic decomposition on a passive background; it is the spectral trace of a geometry that is already anomaly-locked and gauge-structured.  This is what admits --- indeed requires --- the phenomenological texture this paper computes: a mild leptophobic admixture, a top-rich overlap structure, and a multi-field cluster (graviton, KK gluon, \(\UB\) vector) at one derived scale.

\paragraph{Has anything like this existed before?}
Candor about precedents strengthens the claim, so we state the genealogy exactly as \citet{ArisakaA2GG6} traces it.  The individual \emph{ingredients} have distinguished ancestors: Minkowski made time the first non-spatial geometric dimension \citep{Minkowski1908}; Weyl geometrized \emph{scale} as a connection in 1918 \citep{Weyl1918} and \emph{phase} as the \(\mathrm U(1)\) invariance in 1929 \citep{Weyl1929}; the fiber-bundle formulation made the gauge phase an internal circle; Connes' noncommutative geometry built a genuinely non-spatial internal space \citep{ConnesLott1991}; and holographic renormalization realized the scale direction as a radial coordinate --- as an \emph{analogy} \citep{Maldacena1997}.  But no prior phenomenological framework has ever fielded either object realized here: a dimension generated by a dilatation, of forced integer length, entering a collider prediction at a specific mass; or a gauge fiber promoted to a literal sixth coordinate whose tower is thereby pushed to the GUT scale.  Still less has any framework fielded \emph{both at once}, as the entire arena beyond spacetime, derived rather than postulated (Theorem~0 of \citealp{ArisakaA2GG6}), with no spatial extra dimension anywhere --- and hence no modulus, no radion, no landscape, and no supersymmetry requirement, each absence a consequence rather than an assumption --- absences of the kind \PaperWOne{} (\citealp{ArisakaW1Symmetry}) traces from Noether to cohomology (no translation generators exist along \(r\) or \(\theta\) for supercharges to square-root).  Table~\ref{tab:kkrsgg} collects the comparison; the phenomenological stakes of each row are exactly the derived quantities this paper tests.

{\small\small
\setlength{\tabcolsep}{3pt}
\renewcommand{\arraystretch}{1.25}
\begin{longtable}{>{\raggedright\arraybackslash}p{2.56cm} >{\raggedright\arraybackslash}p{2.82cm} >{\raggedright\arraybackslash}p{2.97cm} >{\raggedright\arraybackslash}p{3.08cm} >{\raggedright\arraybackslash}p{2.97cm}}
\caption[KK versus RS versus the two non-spatial dimensions of GG-6]{Comprehensive comparison: the spatial extra dimensions of Kaluza--Klein and Randall--Sundrum versus the two \emph{non-spatial} dimensions of \GG{}.  Every entry in the two right-hand columns is forced by the constitutional derivation of \citet{ArisakaA2GG6} (Theorem~0, Theorem~2, and the \OGNfive{} ledger); no entry is a modeling choice.}
\label{tab:kkrsgg}\\
\toprule
\textbf{Property} & \textbf{Kaluza--Klein circle} & \textbf{Randall--Sundrum interval} & \textbf{\GG{} scale dimension \(I_r\) (5th)} & \textbf{\GG{} phase dimension \(S^1_\theta\) (6th)}\\
\midrule
\endfirsthead
\multicolumn{5}{c}{\itshape Table~\ref{tab:kkrsgg} continued}\\
\toprule
\textbf{Property} & \textbf{Kaluza--Klein circle} & \textbf{Randall--Sundrum interval} & \textbf{\GG{} scale dimension \(I_r\) (5th)} & \textbf{\GG{} phase dimension \(S^1_\theta\) (6th)}\\
\midrule
\endhead
\midrule \multicolumn{5}{r}{\itshape continued on next page}\\
\endfoot
\bottomrule
\endlastfoot
Nature & spatial circle & spatial warped interval & the energy scale, geometrized: \(r=\ln(\EGUT/\mu)\) & the gauge phase, geometrized: the \(\mathrm U(1)\) fiber as a coordinate\\
Coordinate carries a unit? & yes (length) & yes (length) & no --- a logarithm of an energy ratio & no --- an angle\\
Symmetry generator & translation (momentum \(p_5\)) & translation (broken by the warp) & dilatation \(D\) (RG flow); \([D,P_\mu]=P_\mu\) & compact charge \(Q_\theta\in\mathbb Z\)\\
``Motion along it'' means & spatial displacement & spatial displacement & \emph{transmutation}: change of renormalization scale & phase rotation (charge action)\\
Speed of light along it & defined & defined & \emph{undefined} --- no Lorentz signature on \(r\) & undefined\\
Size / length & radius \(R\): free continuous modulus & \(kr_c\): free dial, tuned \(\approx\!35\) for the hierarchy & integer \(K=35\), forced by \(K=LJ=M^2+N^2+L^2\) & \(R_\theta=1\) canonical; purely topological\\
Stabilization & required (radius modulus) & required (radion; \citealp{GoldbergerWise1999}) & \emph{nothing to stabilize}: a count has no fluctuating length; no radion exists & nothing to stabilize\\
Hierarchy & none & output of the \emph{tuned} exponent & output of the \emph{derived} integer: \(e^{-35}\) & ---\\
The tower & charged momentum modes at \(n/R\): anywhere the modulus puts them & graviton momentum modes at the TeV scale by tuning & radial lock cluster at \(\tfrac12 M_{\rm Pl}e^{-35}=3.849\)~TeV, derived; \emph{multi-field} (spin 2 + two vectors at one scale) & integer characters at \(\sim\EGUT\): \emph{unobservable}; no angular satellites at the TeV scale (a falsifiable absence)\\
What the tower \emph{is} & momentum modes on a spatial circle & momentum modes in warped space & composite resonances of the boundary theory at its infrared scale (holography made literal) & quantized charges (bundle characters)\\
Charge quantization & from circle momentum (unexplained radius) & not addressed & --- & integer holonomy: the \emph{origin} of quantization (\(M=1\))\\
Electroweak breaking & none & added by hand & sets the lock scale that \(v=\Mlock(2/5)^3\) descends from & Wilson line \(\theta_H\) performs it (Hosotani mechanism, \citealp{Hosotani1983}; \citealp{ArisakaA9SM})\\
Moduli / landscape & radius modulus & radion & none & none\\
Supersymmetry & optional & optional & \multicolumn{2}{>{\raggedright\arraybackslash}p{6.2cm}}{impossible to extend: no translation generators along \(r,\theta\) for supercharges to square-root; absence is a theorem, not a choice}\\
Free parameters & \(R\) & \(k\), \(r_c\), brane tensions & none (integers only) & none\\
Nearest precedent & Kaluza 1921; Klein 1926 \citep{Kaluza1921,Klein1926} & Randall--Sundrum 1999 \citep{RandallSundrum1999} & Weyl's 1918 scale connection; the holographic radial coordinate (as analogy) --- never before a forced dimension & Weyl's 1929 phase; fiber-bundle internal circle --- never before a literal coordinate\\
\end{longtable}}

\section{Three Reading Paths}
\label{sec:readingpaths}

Part~I is nearly complete, and before the derivation begins it is worth pausing to say plainly how this paper is meant to be used.  It is long because it is meant to be \emph{complete}: the theorem chain, its experimental confrontation, a full simulation of the discovery, and the graduate courses that teach the machinery all live between these covers, because a falsifier is only as good as a stranger's ability to check it unaided.  But completeness is a property of the document, not a demand on the reader.  No one is meant to take the fourteen Parts linearly, and the paper has been built so that three quite different readers can each extract what they came for --- and know, at every step, where the numbers they are using were derived.

\paragraph{The experimentalist's path.}  A reader who wants to know \emph{what to look for, where the limits stand, and when the question gets decided} should go directly to the confrontation and the calendar: the current ATLAS/CMS status and the operational comparison of computed \(\sigma\times\mathrm{BR}\) against published limits (\S\ref{sec:current-status}--\S\ref{sec:operational}), then the anchored HL-LHC simulation with its systematics wall and pre-registered search (Part~\ref{part:hllhcsim}), and finally the benchmark and reinterpretation card (Appendix~M), which is deliberately self-contained: masses, widths, couplings, line shapes, and the implementation walkthrough, in the format a convener would hand to a graduate student on Monday morning.  On this path the theory Parts function as provenance --- the place to verify, when needed, that a given number is a theorem output rather than a tuned input.  The essential facts to carry along are three: the target is a \emph{cluster} at \(3.849\)~TeV, not a lone pole; the color-octet member at its computed \(\ttbar\) rate is the un-suppressible falsifier (F-KK-2B); and the decision window closes at \(\lesssim 2\)~ab\(^{-1}\), within the HL-LHC's first years.

\paragraph{The phenomenologist's path.}  A reader who wants to audit the \emph{derivation chain} --- what is actually forced, and at what grade --- should follow the spine: Part~II for the lock mass and the cluster's fine structure (the closed Bessel form, the radial tower, the degeneracy pattern), Part~\ref{part:ttanatomy} for the signal anatomy (the line shape with its theorem-fixed interference sign, the angular and polarization suite, the discrimination matrix), and then the Discussions (Part~\ref{part:discussions}) for the honest accounting: which statements are theorem-grade, which are estimates with stated brackets, and how the package compares --- row by row --- with the RS, axigluon, and leptophobic-\(Z'\) alternatives an informed skeptic will reach for.  The single discipline to test on this path is the paper's central one: there is no continuous dial anywhere, so every disagreement with data is structural, and the two readings of the coupling normalization (Scenario~A/B) form a closed dichotomy, not a tuning direction.

\paragraph{The theorist's path.}  A reader whose first question is \emph{what kind of extra dimensions these are} should begin where the framework is strangest: \S\ref{subsec:notKKRS} and Table~\ref{tab:kkrsgg}, where the two non-spatial dimensions --- the energy scale as a coordinate of forced integer length, the gauge phase as a literal sixth direction --- are contrasted entry by entry with the spatial constructions of Kaluza--Klein and Randall--Sundrum.  Then Part~II, where that geometry becomes a spectrum; then the five conceptual primers (Appendices~E--I), which answer, at full length, the questions a seminar audience actually asks: how the gluon and \(g^*\) are one field, where the baryonic \(Z'_B\) comes from, what the ``graviton'' of \GG{} is, why \(g^*\) dominates the cluster's rates and is so broad, and why one integer controls both the rate hierarchy and the \(\ttbar\) dominance.  The stakes on this path are conceptual: no modulus, no radion, no landscape, and no supersymmetry requirement --- each an absence that is derived, not assumed.

\paragraph{The graduate student's path.}  The Preface set out a fourth path, and it remains the best one: take the paper linearly, but lean on the three-step courses of Appendices~J--L (the KK spectrum from the 6D action; cross sections from parton distributions; discovery statistics from the Poisson likelihood up) whenever the main text assumes machinery you have not yet met.  The appendices have no page budget precisely so that this path requires no other book.

\section{The cohort inheritance and the roadmap of the fourteen Parts}
\label{sec:inheritance-and-roadmap-p1}

\noindent
Before the derivation begins (Part~II), it is useful to lay out two reader-orientation maps in one closing Section: a panorama of the twenty-two-paper cohort in which this paper stands (\S\ref{sec:cohort-p1}, Figure~\ref{fig:cohort-map-p1}), and a panorama of the fourteen Parts of the present paper (\S\ref{sec:roadmap-p1}, Figure~\ref{fig:parts-roadmap-p1}).  A reader of any later Part should be able to flip back to this Section and orient themselves in both directions: which cohort companion carries which imported result, and where the present argument stands in the fourteen-Part arc.

\subsection{The cohort inheritance}
\label{sec:cohort-p1}

\paragraph{The program in one view.} Figure~\ref{fig:cohort-map-p1} places the present paper among the twenty-two
papers of the \GGTheory{} cohort. The reader meeting the program
for the first time can read its architecture off the figure top to
bottom --- the constitutional root, the bulk trio, the open-system
boundary law and its pillars, the three textbook readouts, the
capstone, the two phenomenology branches, and the Foundations
Branch on the left flank --- before continuing
with the present paper's own thread.  The papers of record: the three
textbook readouts are \citet{ArisakaA8QM}, \citet{ArisakaA9SM}, and
\citet{ArisakaA10Cosmos}; the capstone is \citet{ArisakaA11Origin};
the Predictions Branch carries, beside the present paper,
\citet{ArisakaP2N3} and \citet{ArisakaP3Electron}; the Cosmology
\& Astrophysics Branch carries \citet{ArisakaC1Kaxion},
\citet{ArisakaC2Hubble}, and \citet{ArisakaC3BH}; and the
Foundations Branch carries \citet{ArisakaP4Real} and
\citet{ArisakaP5CPT}.

Reading this paper against that map requires only a few conventions, stated once here.
Numbered results carry the cohort global identifiers with the paper suffix \textsf{-KK}: thus \emph{T-KK-1} is the first Theorem and \emph{F-KK-1} the first pre-declared falsifier (the cohort uses the prefix set T-/L-/D-/PS-/F- for theorems, lemmas, definitions, postulate-statements, and falsifiers).  Dimensionful inputs are drawn only from \GUfive{}, the Geometric Universal Five Constants \GUfiveSet; all dimensionless data come from the \OGNfive{} integer ledger \((M,N,L;J,K)=(1,3,5;7,35)\), so the theory carries no continuous dial beyond the overall choice of units.  The two extra coordinates are the \emph{non-spatial} scale axis \(I_r\) and phase circle \(S^1_\theta\); the bulk metric is the canonical \(\dd s_6^2=e^{-2r}\eta_{\mu\nu}\dd x^\mu\dd x^\nu+\dd r^2+\dd\theta^2\) of \ATwo{} (Theorem~2); the radial direction is RG-\emph{scale} (not RG-time); and the open-system language of detection follows \AFive{} (\GDOS, \citealp{ArisakaA5GDOS}) and \ASixGRIP{} \citep{ArisakaA6Gi4}.  This paper builds on the cohort papers of record --- \AOne{} (\citealp{ArisakaA1Constitution}), \ATwo{} (\citealp{ArisakaA2GG6}), \AThree{} (\citealp{ArisakaA3BI6}), \ANine{} (\citealp{ArisakaA9SM}), \AFive{}, and \ASixGRIP{} --- and in particular on Theorems~0 and~2 of \ATwo{} (the forced bulk and its canonical metric), the \OGNfive{} cascade of \AThree{}, and the lock-scale/scale-chain of \ANine{}.  For the canonical Dictionary of program-defined acronyms, see \ATwo{} Appendix~B; this paper defines only the collider-specific vocabulary in situ.

A word on the Foundations Branch, the column on the map's left flank, whose subject --- the boundary's complex structure --- lies farthest from a collider hall.  \PaperPFour{} \citep{ArisakaP4Real} is the origin-of-$i$ member: from the two real structures of the same forced arena this paper's spectrum lives on --- the compact phase circle and the causal clock --- it derives the single imaginary unit of boundary physics, so the interference phases that this paper's amplitudes and line shapes take for granted are outputs of the geometry rather than inputs.  \PaperPFive{} \citep{ArisakaP5CPT} is the mirrors-and-matter member: C, P, and T as orientation reversals of the three real bulk factors, with the matter--antimatter asymmetry at the end of the chain --- and its dated kill list names this paper's HL-LHC lock-cluster verdict as a companion channel of the same shared geometry.  Neither carries a collider claim, and no result below imports from either; they are named here so the panorama is complete.

%% ===================================================================
%%  The nineteen-paper cohort map (frozen v1.4 geometry), on its own
%%  page at the close of the cohort-inheritance subsection.  Verbatim
%%  copy of the v1.4 master (A1-Constitution Figure 3): no
%%  self-reference markers anywhere, so the figure is identical and
%%  simple for replication in all nineteen papers.  Only the internal
%%  node names MP4/MP5 differ from the master (collision avoidance).
%% ===================================================================
\clearpage
\begin{figure}[p]
\centering
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%% --- Row 1: GG6 Trio ---
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%% --- Row 7: Cosmology & Astrophysics Branch (C-series) ---
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 {\textbf{\PaperCOne{}} \\[-1pt] \footnotesize Kaxion dark matter};
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 {\textbf{\PaperCThree{}} \\[-1pt] \footnotesize BH entropy $A/4$};

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%%     left-flank column at x = -96mm: MP4 at y = -64mm (unchanged),
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%%     spine; this one reads it, so every edge into it is dashed.
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 {\textbf{\PaperWOne{}} \\[-1pt] \footnotesize Conservation without symmetry};
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\node[branchlabel, text=teal!40!black, anchor=east]
 at ([xshift=-3mm]PFRAME.west |- CFRAME) {Cosmology \&\\Astrophysics\\Branch\\(C-series)};
%% Foundations column label (v1.3): interior placement, in the gap
%% between MP4 and MP5, anchored east so it sits immediately LEFT of
%% the vertical MP4->MP5 edge; house style (small violet) kept. The
%% "shared i" edge label occupies the RIGHT side of the same edge.
\node[branchlabel, text=violet!40!black, anchor=east]
 at ($($(MP4.south)!0.5!(MP5.north)$)+(-1.5mm,0)$) {Foundations\\Branch};
%% v22: the mirror label, held to the RIGHT of the W1->W2 edge --
%% \ATen{}'s grown border reaches x = 78.7mm and an east-anchored
%% label would run into it.
\node[branchlabel, text=brown!45!black, anchor=west]
 at ($($(W1.south)!0.22!(W2.north)$)+(1.5mm,0)$) {White-Paper\\Branch\\(W-series)};

%% --- Edges: A1 -> GG6 Trio ---
\draw[arrowrt] (A1.south) -- node[edgelabel,pos=0.55] {\AX{2}} (A2.north);
\draw[arrowrt] (A1.south) -- node[edgelabel,pos=0.55] {\POS{3}} (A3.north);
\draw[arrowdash] (A1.south) -- (A4.north);

%% --- Edges: GG6 Trio -> A5 ---
\draw[arrow] (A2.south) -- (A5.north west);
\draw[arrow] (A3.south) -- node[edgelabel,pos=0.55] {\AX{3}} (A5.north);
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%% --- Edges: A5 -> pillars ---
\draw[arrow] (A5.south) -- (A6.north);
\draw[arrow] (A5.south) -- (A7.north);

%% --- Edges: A5 + pillars -> 4D-Physics Trio ---
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\draw[arrow] (A5.south).. controls (0, -72mm) and (0, -80mm).. (A9.north);
\draw[arrow] (A7.south) -- (A10.north);

%% --- Edges: 4D-Physics Trio -> A11 ---
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\draw[arrow] (A9.south) -- (A11.north);
\draw[arrow] (A10.south) -- (A11.north east);

%% --- Edges: parents -> branch frames ---
\draw[arrowdash] (A9.south west).. controls (-40mm, -105mm)
 and (-84mm, -122mm).. (PFRAME.north west);
\draw[arrowdash] (A9.south east).. controls (40mm, -105mm)
 and (84mm, -122mm).. (PFRAME.north east);
%% v22 RE-ROUTE.  The old line descended through what is now the
%% GG-W2-Holonomy node, and there is no corridor left inboard of it.
%% A polyline rather than a Bezier: a bounding box contains control
%% points, and at the clearance this edge needs those would have sat
%% 24mm beyond any ink, scaling the whole plate down for nothing.
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 -- (127mm,-160mm) -- (CFRAME.east);
%% sister import inside the Predictions frame
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 (P2.east) -- node[edgelabel,pos=0.5] {$N{=}3$} (P3.west);
% v22: the second sister import.  \PaperPSix{} says it of
% \PaperPThree{} -- "the sister paper supplies the method this paper
% runs."  Three letters, because a white-filled label wide enough to
% say "method" masks the whole arrow it annotates.
\draw[arrowdash, draw=cyan!50!black]
 (P3.east) -- node[edgelabel,pos=0.5] {GGA} (P6.west);

%% --- Foundations Branch edges (rerouted v1.3 for the raised MP5) ---
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 node[edgelabel,pos=0.55] {Thm~0} (MP4.north);
%% MP4 -> A8 (solid; the origin of i). Departs the east side (MP4.south
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%% MP4 -> MP5 (v1.3: SOLID, per the reading conventions -- forced
%% descent, labelled). A short straight vertical between the pair;
%% the "shared i" label sits on the right side of the edge, the
%% Foundations Branch label on the left.
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 node[edgelabel, pos=0.5, anchor=west, xshift=1mm] {shared $i$}
 (MP5.north);
%% A9 -> MP5 (solid; ledger phases). Leaves A9's west side, dips
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 node[edgelabel,pos=0.6] {$\delta,\ \alpha_{31}$} (MP5.east);
%% MP5 -> A10 (solid; eta_B). The one long edge: runs left-to-right
%% through the clear corridor BELOW the A11 node (above the
%% Predictions frame), then climbs into A10's bottom border right of
%% the A10->A11 arrowtail; it leaves MP5.south vertically. Controls
%% lifted 5mm with the raised MP5 so the dip still clears A11's
%% bottom border. One clean crossing with each of the two dashed
%% A9->Predictions-frame edges.
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 node[edgelabel,pos=0.5] {$\eta_B$} ([xshift=8mm]A10.south);
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%% MP5 level with A11 this is now a short straight upward edge from
%% the column's top-right corner into A8's bottom border, 8mm left
%% of the A8->A11 arrowtail. No crossings.
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%% --- White-Paper Branch edges (v22) ---
%% A3 -> W1: \AThree{} is \PaperWOne{}'s heaviest spine source and its
%% thematic terminus -- the \BIsix{} anomaly proof is where its
%% Green--Schwarz / Hopkins--Singer ladder ends.
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 node[edgelabel,pos=0.55] {\BIsix{}} (W1.west);
%% A10 -> W2: \ATen{} is \PaperWTwo{}'s heaviest source of any kind.
%% At +15%% those two boxes overlap in x (never in y), so this is drawn
%% off A10's south-east corner, clear of the W1->W2 arrowhead.
\draw[arrowdash] (A10.south east) .. controls (84mm,-101mm) ..
 ([xshift=-6mm]W2.north);
%% W1 -> W2, solid and labelled: the mirror of the shared-i edge on the
%% left flank.  \PaperWTwo{} states the dependency itself -- it takes
%% "the structure that the arc arrives at, protection by topology
%% rather than by symmetry".
\draw[arrow] (W1.south) --
 node[edgelabel, pos=0.80, anchor=west, xshift=1mm]
 {symmetry $\to$ topology} (W2.north);

%% Centring pad, retired (v22): it cancelled a Bezier phantom that the
%% polyline above no longer creates; the bbox is now the ink.
\end{tikzpicture}}
\caption[The twenty-two-paper cohort map.]{The
twenty-two-paper cohort map of \GGTheory{}. The constitutional root
\AOne{} (violet, top) supplies the four axioms and seven postulates
from which the rest of the cohort descends: the GG6 Trio (\ATwo{},
\AThree{}, \AFour{}, red) realizes the bulk geometry; \AFive{}
(yellow) the boundary law; the \GDOS{} Trio pillars (\ASix{},
\ASeven{}, green) the geometric and quantum-information runtime; the
4D-Physics Trio (\AEight{}, \ANine{}, \ATen{}, blue) the textbook
readouts; and the closing capstone \AEleven{} (gold) the
quarks-to-consciousness identity cascade. Two framed branches carry
the phenomenology --- the \textbf{Predictions Branch} (cyan), whose
four papers hand their results along the row by the internal arrows
(\PaperPTwo{}'s $N{=}3$ into \PaperPThree{}, and that paper's
ground-attractor method, \textsf{GGA}, on into \PaperPSix{} (\citealp{ArisakaP6Proton})), and the
\textbf{Cosmology \& Astrophysics Branch} (teal). The spine is
flanked by two columns of two, which do opposite jobs. On the left the
\textbf{Foundations Branch} (violet) feeds it: \PaperPFour{}, fed by
\ATwo{}'s Theorem~0 arena, manufactures the $i$ that \AEight{}
consumes, and \PaperPFive{} is the \ANine{}--\ATen{} interface, fed by
the ledger phases and feeding the $\eta_B$ chain across the width of
the map, with the shared-$i$ edge descending solid between them. On
the right the \textbf{White-Paper Branch} (brown) reads it:
\PaperWOne{} walks the century from Noether to Hopkins--Singer, and
\PaperWTwo{} dissects the structure that arc arrives at ---
protection by topology rather than by symmetry --- across sixteen open
problems. The white papers derive nothing new, carrying every result
at its parent's declared grade; the single dashed feed drawn into each
names its heaviest source, not its only one. \emph{Reading
conventions:} solid arrows mark axiom-forced descent and carry an edge
label naming the driving axiom or postulate where most illustrative;
dashed arrows mark the deployment of derived results, or historical
links inherited without being constitutionally forced.}
\label{fig:cohort-map-p1}
\end{figure}
\clearpage

\subsection{Roadmap of the fourteen Parts}
\label{sec:roadmap-p1}

The rest of the paper is organized into thirteen further Parts, and the walk through them is plotted in Figure~\ref{fig:parts-roadmap-p1}.  The present Part --- the foundations --- is nearly done: it has posed the fifty-year question, put geometry--gauge locking into plain English, stated the central claim in its open-system reading, traced the constitutional route from the axioms to a collider observable, distinguished the two non-spatial dimensions from their spatial look-alikes, laid out the three reading paths, and drawn the two orientation maps of this closing Section.

Part~II is the paper's center of mass, and it is highlighted in the roadmap because everything later is downstream of it.  From the canonical metric and the integer ledger it derives the lock scale in closed form --- Theorem~T-KK-1, \(\Mlock=\tfrac12 M_{\mathrm{Pl}}e^{-35}\simeq3.849\)~TeV --- and then resolves what lives at that mass: the three-member cluster (the spin-2 mode, the color-octet \(g^{*}\), and the leptophobic \(Z'_B\)), its derived widths, couplings, and degeneracy pattern, the radial family at \(7.05\) and \(10.22\)~TeV, and the observable properties of the KK sector.

Three Parts then carry the derived signal to the collider floor.  Part~III computes production and discovery: cross sections, event yields, and the channel hierarchy that makes the top pair the decisive final state.  Part~IV draws the line that keeps a reader from mis-filing the signal --- GG-6 versus Randall--Sundrum, entry by entry --- and re-reads collider detection itself as an open-system (\GDOS{}) problem.  Part~V then states the coupling normalization as a closed dichotomy: Read-out~I, the theorem-only target, and Read-out~II, the empirically-anchored rate estimate.

Two Parts confront the prediction with the machines that exist.  Part~VI derives the anatomy of the \(\ttbar\) signal at full depth --- the interference-aware line shape with its theorem-fixed sign, the angular and polarization suite, and the discrimination matrix behind the pre-registered fingerprint.  Part~VII is the experimental status and reach: the current ATLAS/CMS limits, the operational confrontation of computed \(\sigma\times\mathrm{BR}\) against the published exclusions, the kill criterion at \(\lesssim2\)~ab\(^{-1}\), and the trigger-and-analysis roadmap.

Part~VIII, the second highlighted Part, simulates the discovery at the fidelity of a real analysis --- anchored to the published ATLAS Run-2 search, calibrated to its limits, with the statistics taught from the Poisson likelihood up and the four snapshots of what the detectors would record in 2026, 2031, 2033, and 2041.  Its headline is the reason this paper is urgent rather than merely testable: the collisions that decide the question may already be on tape.  Part~IX looks beyond the HL-LHC to the 2035--2070 program, facility by facility and scenario by scenario.  Part~X, the Discussions, weighs the case --- significance, uniqueness, the classic collider puzzles, the comparison with the other roads beyond the Standard Model, the predictions for falsification, and the open questions stated as open --- and Part~XI, the Conclusions, closes the body with what the paper has established, its objective limitations, and what either verdict would mean.  The back matter is the workshop: Part~XII collects the Technical Appendices --- the reference dictionaries, the five conceptual primers, and the three graduate courses in the collider physics of the lock cluster; Part~XIII answers the Common Objections a collider experimentalist, phenomenologist, statistician, or string theorist would press hardest; and Part~XIV closes with the annotated References, each entry carrying its relevance note and its first citation point.  Read in order, the fourteen Parts compose a single self-contained argument; read by need, each Part names its imports and can stand a visit on its own.

%% ===================================================================
%%  Roadmap of the fourteen Parts (house architecture: A2-GG6 v13.3 /
%%  A10-Cosmos v13.3 pattern -- vertical Part column, right phase
%%  sidebar, highlighted Parts II and VIII carrying the central claim),
%%  clearpage-isolated minipage + \captionof.
%% ===================================================================
\clearpage
\noindent
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\centering
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 {\textbf{Part~I}\quad\textbf{Foundations} \\[-1pt]
 \footnotesize From the GG-Constitution to a six-dimensional collider target; the cohort and the reading paths};

%% --- Part II: the derived signal (highlighted) ---
\node[innov] (II) at (0, -16mm)
 {\textbf{Part~II}\quad\textbf{The Derived Signal} \\[1pt]
 \footnotesize \(\Mlock=\tfrac12 M_{\mathrm{Pl}}e^{-35}\simeq3.849\)~TeV (Theorem~T-KK-1); \\[-1pt]
 \footnotesize the three-member cluster and the radial family at 7.05 and 10.22~TeV};

%% --- Parts III--V: signal to collider ---
\node[appar] (III) at (0, -32mm)
 {\textbf{Part~III}\quad\textbf{Production and Discovery} \\[-1pt]
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 {\textbf{Part~IV}\quad\textbf{GG-6 versus Randall--Sundrum, and Detection as an Open System} \\[-1pt]
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 {\textbf{Part~V}\quad\textbf{Two Read-outs of the Lock Cluster} \\[-1pt]
 \footnotesize Read-out~I: the theorem-only target \,\textbar\, Read-out~II: the empirically-anchored rate};

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 {\textbf{Part~VI}\quad\textbf{Anatomy of the \(t\bar t\) Signal} \\[-1pt]
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 {\textbf{Part~VII}\quad\textbf{Experimental Status and Reach} \\[-1pt]
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 {\textbf{Part~VIII}\quad\textbf{Simulating the Discovery} \\[1pt]
 \footnotesize \emph{\textbf{The collisions that decide may already be on tape:}} \\[-1pt]
 \footnotesize the anchored HL-LHC simulation and the pre-registered search --- 2026 / 2031 / 2033 / 2041};

%% --- Part IX: the far programme ---
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 {\textbf{Part~IX}\quad\textbf{Beyond the HL-LHC} \\[-1pt]
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 {\textbf{Part~X}\quad\textbf{Discussions} \\[-1pt]
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 {\textbf{Part~XI}\quad\textbf{Conclusions} \\[-1pt]
 \footnotesize What either verdict would mean: the mass is the bet, the LHC is the table};

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 {\textbf{Part~XII}\quad\textbf{Technical Appendices} \\[-1pt]
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 {\textbf{Part~XIV}\quad\textbf{References} \\[-1pt]
 \footnotesize Cited works in two groups, each entry with a relevance note and its first citation point};

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\par\bigskip
\captionof{figure}[Roadmap of the fourteen Parts of GG-P1-KK.]{Roadmap of the fourteen Parts of \textsf{GG-P1-KK}.  The argument develops in eight phases.  \emph{Entrance} (Part~I): the fifty-year question, geometry--gauge locking in plain English, the constitutional route, the non-spatial geometry, and the reader's two maps.  \emph{The derived signal} (Part~II, highlighted): the lock-cluster mass in closed form (Theorem~T-KK-1) and the observable spectrum built on it.  \emph{Signal to collider} (Parts~III--V): production and discovery, the GG-6/Randall--Sundrum comparison with the open-system reading of detection, and the two read-outs of the coupling normalization.  \emph{Anatomy and confrontation} (Parts~VI--VII): the derived \(t\bar t\) fingerprint, and the computed rates set against the published ATLAS/CMS limits with the kill criterion.  \emph{The verdict on tape} (Part~VIII, highlighted): the anchored HL-LHC simulation and the pre-registered search --- 2026 / 2031 / 2033 / 2041.  \emph{The far program} (Part~IX): the machines of 2035--2070.  \emph{Weighing and closing} (Parts~X--XI): the Discussions and the Conclusions.  \emph{Back matter} (Parts~XII--XIV): the technical appendices, the common objections and responses, and the annotated references.  Parts~II and~VIII are highlighted because together they carry the paper's central claim --- the derived lock cluster at \(\Mlock\simeq3.849\)~TeV: Part~II is where the mass and its cluster are computed as theorems, and Part~VIII is where that computation becomes a dated, pre-registered collider verdict.  The mass is the bet; Part~VIII sets the table.}
\label{fig:parts-roadmap-p1}
\end{minipage}
\clearpage

\bigskip
\noindent Part~I, in plain terms, established the arena and the rules.  A theory whose only freedom is five counting numbers forces a six-dimensional stage --- ordinary spacetime plus two directions that are not space at all, one being the energy scale itself and the other the gauge phase --- and that stage, unlike every earlier extra-dimensional proposal, has no adjustable size, no radion, no landscape, and nothing to tune.  Nothing about a particle has been claimed yet; what has been established is stricter: the constitutional route by which collider observables descend from the axioms, the open-system dictionary by which bulk excitations become ATLAS/CMS records, and above all the discipline that every later number must be an \emph{output}.

The next Part cashes that discipline immediately, on the paper's central object: the mass.  From the canonical metric and the integer ledger it derives the lock scale in closed form --- Theorem~T-KK-1, \(\Mlock=\tfrac12 M_{\rm Pl}e^{-35}\simeq3.849\)~TeV --- and then resolves what lives there: not a lone resonance but a three-member cluster with derived fine structure.  Everything in this paper is downstream of that one number; Part~II is where it comes from.


\section[Part~I in plain English]{Part~I in plain English: why fifty years of predictions could not lose, and why this one can}
Every particle found since the mid-1970s was on a list before it was
found. The W and the Z, the top, the Higgs --- each was predicted, each was
looked for, each turned up. That run of successes ended, and the fifty
years since have produced no new layer at all.

It was not for want of trying, and this Part says plainly why the trying
failed. Supersymmetry promised partners just above each frontier and, in
its minimal form, carried more than a hundred free parameters --- so every
null result was answered by sliding the spectrum up. Extra-dimensional
models supplied real structural insight and made their mass scales inputs:
a radius to scan, a warp factor to choose, so no exclusion could ever be
fatal. String theory built a candidate for everything and committed to
nothing measurable. Three programs, three sets of dials, and not one
pre-registered number whose failure would end anything.

The diagnosis is not that the next layer is out of reach. It is that a
theory with continuous free parameters can retreat from any measurement,
and a theory that cannot lose cannot win.

What this paper offers instead is a mass with no dial behind it. The scale
comes out of an integer in a ledger fixed elsewhere in this cohort, for
reasons that had nothing to do with colliders, and the same integer
over-constrains the electroweak scale as well. There is nothing to adjust
if the measurement disagrees. That is the entire point, and it is the
reason the rest of the paper can be as specific as it is.

\GGPart[The derived signal]{The Derived Signal}{The lock-cluster mass (Theorem~T-KK-1) and the observable spectrum}

\section{The Lock Scale and the First Resonance Cluster}
\label{sec:lockscale}

\subsection{The lock-cluster mass: Theorem~T-KK-1}
\label{sec:t-kk-1}

This subsection is the scientific centerpiece of the paper.  Everything downstream --- production, channels, line shape, the experimental status, and the falsifier --- hangs on a single \emph{derived} number.  We state it as a theorem, prove it from the constitutional inputs, and count the free parameters (there are none).

\paragraph{Theorem T-KK-1 (the \GG\ lock-cluster scale).}\label{thm:tkk1}
\emph{The first observable Kaluza--Klein resonance of the six-dimensional bulk \GG\ is the radial ``lock cluster,'' whose lowest mode has the parameter-free mass}
\begin{equation}
\boxed{\;\Mlock \;=\; \tfrac12\,M_{\mathrm{Pl}}\,e^{-K}
\;=\; \frac{\EPl}{2\,e^{35}}
\;\simeq\; 3.849~\mathrm{TeV}\,,\qquad K=35\,,\;}
\label{eq:tkk1-mlock}
\end{equation}
\emph{in which $K$ is the \OGNfive\ lock integer (no continuous freedom) and $M_{\mathrm{Pl}}$ (equivalently $\EPl=\sqrt{\hbar c^5/G_N}$) is a member of the GU-5 universal-constant set, not a tunable scale.  The radial excitations form a discrete tower}
\begin{equation}
m_n \;=\; \frac{x_n}{x_1}\,\Mlock\,,\qquad n=1,2,3,\dots\,,
\label{eq:tkk1-tower}
\end{equation}
\emph{with $x_n$ the $n$-th positive zero of the Bessel function $J_1$, so that the first three lock-cluster modes lie at}
\begin{equation}
m_1 \simeq 3.85~\mathrm{TeV}\,,\qquad
m_2 \simeq 7.0~\mathrm{TeV}\,,\qquad
m_3 \simeq 10.2~\mathrm{TeV}\,.
\label{eq:tkk1-three}
\end{equation}
\emph{The angular ($S^1_\theta$) Fourier tower lies at $\sim\EGUT$ and is unobservable at any foreseeable collider.}

\paragraph{Precision of the anchor.} Evaluated with the CODATA Planck
energy, $\Mlock = \EPl/(2e^{35}) = 3.8489$~TeV, with an uncertainty
inherited entirely from $G_N$ ($\sim10^{-5}$ relative, i.e.\
$\pm0.04$~GeV) --- making the lock-cluster anchor, by a comfortable
margin, the sharpest absolute mass prediction in particle physics.
The tower ratios are exact Bessel-zero ratios, $m_2/m_1 =
x_2/x_1 = 1.8309$ and $m_3/m_1 = 2.6551$, so the radial recurrences
at $7.047$ and $10.219$~TeV carry \emph{no additional} theory error
beyond the anchor itself.

\paragraph{The three lowest states, physically.}  The integer \(n\) in \eqref{eq:tkk1-tower} deserves a physical reading, not just a formula.  The scale throat is a cavity: a finite interval of length \(K=35\) in the renormalization-group direction, bounded by the ultraviolet wall at \(r=0\) and the infrared endpoint at \(r=K\).  Like any cavity, it supports standing waves, and \(n\) counts their \emph{radial nodes}: the \(n=1\) mode is the fundamental --- nodeless, maximally concentrated at the infrared endpoint, the ``lowest note'' of the throat --- while \(n=2\) and \(n=3\) are its first and second overtones, each adding one node along the scale direction.  The Bessel zeros \(x_n\) are precisely the standing-wave quantization conditions on this background, and their ratios are pure geometry: \(m_1:m_2:m_3 = 1:1.831:2.655\), i.e.\ \(3.849\), \(7.047\), and \(10.219\)~TeV.  In the renormalization-group reading of the throat (Table~\ref{tab:kkrsgg}, the ``what the tower is'' row), these are not momentum modes of a spatial dimension but successively excited \emph{composite resonances} of the boundary theory at its infrared scale --- the analogue of radial excitations of a hadron, with the throat playing the role of the confining dynamics.  The intuition an experimentalist should carry: one derived scale, then a fixed overtone series above it --- an organ pipe, not a picket fence at arbitrary spacing.  Only the fundamental is within LHC reach; the \(n=2\) overtone at \(7.047\)~TeV is an FCC-hh measurement (Part~\ref{part:beyond}), and finding it \emph{at the Bessel ratio} would confirm the cavity itself, not merely another particle.

\paragraph{The runtime identity of the target.} In the runtime
vocabulary of the series (\citealp{ArisakaA6Gi4}; the two regimes \textsf{GRIP-In} and
\textsf{GRIP-Out}), the statement of T-KK-1 has a one-sentence
translation that locates this paper in the same grammar as its
sister predictions: the lock scale is the floor of the radial-scale
stratum --- its Geometric Attractor --- and \emph{the KK cluster is
that attractor's first excited level}. Producing a lock-cluster
resonance at the LHC is port-driven excitation within a fixed
landscape (the canonical collider clarification of
\citealp{ArisakaA6Gi4}: pumping a state uphill through the boundary
ports reorganizes nothing); its decay is \GRIPin{} relaxation back to
the floor. Three instruments of the predictions branch thus probe one
functional: the haloscope listens to a basin's residual ring
(C1-Kaxion), gravitational-wave ringdown measures a basin's curvature
(C3-BH), and the LHC pumps a basin's first excited state --- the
present paper.

\medskip
\noindent\textbf{Proof.}  The argument has six steps.  Steps 1--3 fix the scale \eqref{eq:tkk1-mlock} without a free parameter; Step 4 fixes the tower \eqref{eq:tkk1-tower}; Steps 5--6 record the over-constraint and the parameter count.

\paragraph{Step 1 --- the geometry.}
The bulk is the canonical product geometry of Theorem~2 of \citet{ArisakaA2GG6},
\begin{equation}
\dd s_6^2 \;=\; e^{-2r}\,\eta_{\mu\nu}\,\dd x^\mu \dd x^\nu \;+\; \dd r^2 \;+\; \dd\theta^2\,,
\qquad r\in[0,K]\,,\quad \theta\in[0,2\pi)\,,
\label{eq:tkk1-metric}
\end{equation}
with \emph{no} free metric data ($k$, $R_\theta$, $r_c$ are all fixed to unity in the canonical \GG\ units; see \citet{ArisakaA2GG6}, Theorem~2).  The decisive structural fact for collider physics is that the radial coordinate $r$ is the \emph{scale} (renormalization) direction, $r=\ln(\EGUT/\mu)$, and it ranges over a \emph{finite} interval $[0,K]$ --- bounded by the ultraviolet endpoint $r=0$ and the infrared endpoint $r=K$.  A finite warped interval, exactly as in the two-brane Randall--Sundrum construction, carries a \emph{discrete} spectrum of normalizable resonances (not a continuum).  These resonances are the boundary image of a Kaluza--Klein tower; they are \emph{non-spatial} (the throat is an energy-scale direction, not a spatial dimension), but their collider phenomenology is that of warped Kaluza--Klein/composite states.

\paragraph{Step 2 --- the scale depth is forced.}
The integer $K$ is not chosen.  It is the terminal entry of the \OGNfive\ Diophantine cascade proved in \citet{ArisakaA3BI6} from the single-tensor Green--Schwarz-modified Bianchi identity \BIsix:
\begin{equation}
(M,N,L;J,K)=(1,3,5;7,35)\,,\qquad J=2N+1=7\,,\qquad K=LJ=M^2+N^2+L^2=35\,.
\label{eq:tkk1-ogn}
\end{equation}
Thus the depth of the RG-scale throat is fixed by integer data alone.  This is the single most important sentence in the paper: the infrared end of the geometry is not a dial.

\paragraph{Step 3 --- the scale profile sets the lock scale.}
Along the throat the scale profile $e^{-r}$ redshifts every bulk scale; at the infrared endpoint $r=K$ the suppression is
\begin{equation}
e^{-K}=e^{-35}\simeq 6.305\times 10^{-16}\,.
\label{eq:tkk1-warp}
\end{equation}
Because the two extra directions of \GG\ are non-spatial (dimensionless scale and phase), the 6D Einstein--Hilbert reduction relates the bulk scale to the four-dimensional Planck scale by a pure number, $M_6=\sqrt{2/\pi}\,M_{\mathrm{Pl}}$ (\citet{ArisakaA4Euclid}, Theorem~T-Euclid-6; equivalently $M_{\mathrm{Pl}}^2\sim\pi M_6^2$ in \citet{ArisakaA2GG6}~\S5, after the radial scale-profile integration $\int_0^K e^{-2r}\dd r=\tfrac12(1-e^{-2K})\to\tfrac12$).  This is \emph{not} the dimension-carrying volume relation $M_{\mathrm{Pl}}^2\sim M_6^{4}V_{\mathrm{int}}$ of an ordinary \emph{spatial} Kaluza--Klein reduction, precisely because $I_r$ and $S^1_\theta$ carry no length.  Carrying the matching through to the infrared endpoint fixes the lock scale,
\begin{equation}
\Mlock \;=\; \tfrac12\,M_{\mathrm{Pl}}\,e^{-K}\,,
\label{eq:tkk1-derive}
\end{equation}
in which the factor $\tfrac12$ is the scale-profile-integration weight $\tfrac12(1-e^{-2K})\to\tfrac12$ above (the ``minimal half-wave'' of \citet{ArisakaA9SM}~\S4.6), \emph{not} a fitted coefficient.  The complete scale chain $\EPl\to\EGUT\to\Mlock$ is owned by \citet{ArisakaA2GG6}~\S5 and Theorem~6, and the boxed value is reproduced in \citet{ArisakaA9SM} (Table of derived quantities: $\Mlock=3.848959$~TeV).  Numerically, with $\EPl = M_{\mathrm{Pl}}c^2 = 1.2209\times 10^{19}$~GeV,
\begin{equation}
\Mlock = \tfrac12(1.2209\times 10^{19}~\mathrm{GeV})(6.305\times 10^{-16})
= 3.849\times 10^{3}~\mathrm{GeV}=3.849~\mathrm{TeV}\,.
\label{eq:tkk1-number}
\end{equation}

\paragraph{Step 4 --- the radial tower.}
Spin-2 fluctuations $h_{\mu\nu}(x^\mu,r)$ of \eqref{eq:tkk1-metric} with vanishing angular momentum ($\ell=0$ on $S^1_\theta$) obey a radial Sturm--Liouville problem on the finite interval $[0,K]$,
\begin{equation}
-\partial_r\!\left(e^{-4r}\,\partial_r \psi_n\right)\;=\; m_n^2\,e^{-2r}\,\psi_n\,,
\label{eq:tkk1-sturm}
\end{equation}
with Neumann-type endpoint conditions at $r=0$ and $r=K$.  For the AdS-like scale profile the eigenvalues are set by the zeros of $J_1$, $m_n \propto x_n$.  One subtlety must be stated precisely.  A \emph{naive} pure-Bessel estimate would give $m_1^{\rm Bessel}\approx x_1\,e^{-K}M_6\approx 3.83\times 6.3\times10^{-16}\times 1.22\times10^{19}~\mathrm{GeV}\approx 29~\mathrm{TeV}$, using the radial prefactor $x_1\approx 3.83$ and reference $M_6\sim M_{\mathrm{Pl}}$.  This cruder estimate is \emph{superseded} by the scale-chain value $\Mlock=\tfrac12 M_{\mathrm{Pl}}e^{-K}\simeq 3.85$~TeV, which uses the scale-profile-reduction prefactor $\tfrac12$ (\citet{ArisakaA2GG6}~\S5+T6; \citet{ArisakaA4Euclid}).

The ratio of the two prefactors is $2x_1\approx 7.66$, numerically close to the \OGNfive\ integer $J=7$ but \emph{not} an independent integer correction (\citet{ArisakaA2GG6}, KK-spectrum appendix).  We therefore normalize the lowest physical mode to the derived scale, $m_1=\Mlock$, which gives \eqref{eq:tkk1-tower}; with $x_1=3.8317$, $x_2=7.0156$, $x_3=10.1735$ the representative spacings are \eqref{eq:tkk1-three}.

Only $m_1=\Mlock$ is theorem-level; the higher ratios $x_2/x_1=1.83$, $x_3/x_1=2.66$ are a representative AdS-throat result, and the detailed cluster substructure (tensor sector, $\UB$ flux) is \emph{not} part of the theorem-level claim --- it is flagged as model-level in the provenance ledger below.  The companion \emph{angular} tower has masses $m_\ell\sim \ell\,\EGUT$ (Fourier modes of the gauge-phase circle, \citet{ArisakaA2GG6}); it is heavier than the lock cluster by twelve orders of magnitude and is irrelevant to colliders.  \emph{This is the content of Option~A: the observable cluster is radial.}

\paragraph{Step 5 --- the over-constraint.}
The same lock scale fixes the electroweak vacuum expectation value through the Hosotani breaking of \citet{ArisakaA9SM},
\begin{equation}
v=\Mlock\left(\tfrac{2}{5}\right)^{3}=3.849~\mathrm{TeV}\times 0.064 \simeq 246.3~\mathrm{GeV}\,,
\label{eq:tkk1-vev}
\end{equation}
in agreement with the measured $v=246.22$~GeV.  The collider scale and the weak scale are therefore \emph{not} independent: a single integer ledger fixes both.  This over-constraint is the source of the unusual falsifying power of the search --- a wrong lock cluster would also unseat the electroweak derivation.

\paragraph{Step 6 --- the parameter count.}
Equation~\eqref{eq:tkk1-mlock} is a pure number ($\tfrac12 e^{-35}$, built from the \OGNfive\ integer $K=35$) times a GU-5 scale ($M_{\mathrm{Pl}}=\sqrt{\hbar c^5/G_N}$, fixed by measuring $G_N$, $\hbar$, $c$, not by fitting the collider).  There is no free continuous parameter in the mass prediction.  The cross-section and branching estimates of the following sections \emph{are} model inputs, and are labeled as such; the \emph{mass} is a theorem.\hfill$\blacksquare$

\begin{table}[H]
\centering
\caption[The \GG\ radial lock cluster (Option A).]{The \GG\ radial lock cluster (Option A).  Masses from $m_n=(x_n/x_1)\Mlock$ with $\Mlock=\tfrac12 M_{\mathrm{Pl}}e^{-35}$.  The angular tower (not shown) sits at $\sim\EGUT\approx 2\times10^{16}$~GeV. The masses are ratios of Bessel zeros times one scale, and the scale is the only thing the ledger supplies --- which is why the table has no column for an uncertainty on the spacing. What can move is the overall position, and what cannot move is the pattern. A measured cluster whose members sit at the wrong ratios would falsify the reading even if the lowest member landed exactly where predicted.}
\label{tab:tkk1-spectrum}
\small
\begin{tabular}{@{}cccl@{}}
\toprule
Mode & $x_n$ ($J_1$ zero) & $m_n$ (TeV) & Role\\
\midrule
$n=1$ & $3.8317$ & $3.85$ & Lock-cluster centroid; primary discovery target\\
$n=2$ & $7.0156$ & $7.0$  & First radial partner; HL-LHC corroboration\\
$n=3$ & $10.1735$ & $10.2$ & Second radial partner; future-collider reach\\
\bottomrule
\end{tabular}
\end{table}

\paragraph{Pre-declared falsifier (\textbf{F-KK-1}, under POS-7/MF).}
The prediction is falsifiable on a definite timeline.  \emph{If the combined HL-LHC dataset ($\sim 3~\mathrm{ab}^{-1}$ per experiment) shows no clustered spin-2 resonance with centroid in the window $3.85\pm 0.10$~TeV --- and, in particular, no first radial mode near $3.85$~TeV in the dominant $\ttbar$ and boosted-dijet channels at the model-estimated cross-section --- then the current lock-scale implementation of \GG\ is excluded.}  (The \(\pm0.10\)~TeV window is experimental, not theoretical: the anchor's own uncertainty is \(\pm0.04\)~GeV, four orders of magnitude smaller, so the window budgets only the reconstructed-mass scale and resolution and the possible centroid spread of a multi-state cluster --- a generous \(\pm2.6\%\) that the theory error cannot consume.)  A confirmed cluster, with the $n=2$ partner near $7$~TeV, would conversely be simultaneous evidence for the bulk geometry, the \OGNfive\ lock integer $K=35$, and the non-spatial Kaluza--Klein reading.  The number $3.849$~TeV is the bet.  (The detailed channel-by-channel kill criterion is stated in \S\ref{sec:killer}.)

\emph{Requalification (Corollary T-KK-2c, \S\ref{subsec:kappagg}).}  The graviton's gluon coupling is now derived, \(\kappa_{gg}=c_K/K=0.0136\), giving \(\sigma\times\mathrm{BR}(G_1\to\ttbar)\approx0.045\)~fb at 13~TeV (LO\(\times\)K, where K denotes the QCD K-factors \(K_{q\bar q}=1.3\), \(K_{gg}=1.6\) --- not the integer \(K=35\)) --- a factor \(\sim\!25\) below even the 3~ab\(^{-1}\) HL-LHC projection.  F-KK-1 therefore splits into two tiers: (i)~at the HL-LHC the falsifier of the lock-scale implementation is carried by the cluster through the un-suppressible color octet (F-KK-2B) --- a null in the cluster-aware \(\ttbar\)/dijet combination within \(3.85\pm0.10\)~TeV at \(\lesssim\!2\)~ab\(^{-1}\) retires the implementation; (ii)~the direct spin-2 confirmation is an FCC-hh measurement (\(\sigma(G_1)\approx205\)~fb at 100~TeV).  The bet stands; the table moves.

\medskip
\noindent\emph{Scope note (honesty discipline).}  Of the content above, Eqs.~\eqref{eq:tkk1-mlock}, \eqref{eq:tkk1-ogn}, \eqref{eq:tkk1-warp}, \eqref{eq:tkk1-number}, and \eqref{eq:tkk1-vev} are \emph{derived} (parameter-free, traceable to OGN-5 + GU-5).  The tower \emph{ratios} \eqref{eq:tkk1-tower}, the Neumann boundary conditions, and the $O(1)$ normalization $\tfrac12$ are \emph{representative/normalization} statements resting on the canonical AdS-throat profile and the cohort scale-chain; the cluster substructure and the octet/graviton couplings are now themselves derived at the free-field grade (T-KK-2, Corollary T-KK-2c); the remaining model estimates are confined to the per-fermion cascade depths and the \(U(1)_B\) normalization \(g_B\).  The theorem-level claim is the mass \eqref{eq:tkk1-mlock}, and it is the one the falsifier targets.

\subsection{Provenance of every quantitative prediction}
\label{subsec:provenance}

This paper is meant to be acted on by an experimentalist, so we state, without exception, where each number comes from.  Every \emph{derived} quantity traces to a theorem in the just-completed \GG{}-A series (A1--A11, June 2026), and to nothing else.  The derivation chain is short and entirely sequential:
\begin{enumerate}
\item \textbf{Inputs.}  The dimensionful pentad \GUfive{} $=\GUfiveSet$ and the dimensionless integer ledger \OGNfive{} $=(M,N,L;J,K)=(1,3,5;7,35)$, with $J=2N+1=7$ and $K=LJ=M^2+N^2+L^2=35$ \citep[\S POS-6 worked example]{ArisakaA1Constitution}\citep{ArisakaA3BI6}.  No continuous dial.
\item \textbf{Planck scale.}  $\EPl=\sqrt{\hbar c^5/G_N}=1.2209\times10^{19}$~GeV, the \GUfive{} of \citet{ArisakaA1Constitution} (T-GU5).
\item \textbf{Unification scale.}  $\EGUT=\EPl\,e^{-2\pi}\simeq 2.28\times10^{16}$~GeV --- one $S^1_\theta$ holonomy traversal \citep[\S5, Theorem~6]{ArisakaA2GG6}.
\item \textbf{Scale depth.}  $e^{-K}=e^{-35}\simeq 6.305\times10^{-16}$, the integer $K=35$ from step~1.
\item \textbf{Lock-cluster mass (the centerpiece).}  $\Mlock=\tfrac12 M_{\mathrm{Pl}}e^{-35}=3.848959$~TeV \citep[\S5, Theorem~6]{ArisakaA2GG6}, with the scale-profile-reduction factor $\tfrac12$ and $M_6=\sqrt{2/\pi}\,M_{\mathrm{Pl}}$ from \citet{ArisakaA4Euclid} (T-Euclid-6) and the half-wave reading of \citet[\S4.6]{ArisakaA9SM}.  The naive pure-Bessel root ($\approx29$~TeV) is reduced to $\Mlock$ by the \OGNfive\ integer $J=7$ (Step~4 above).
\item \textbf{Electroweak over-constraint.}  $v=\Mlock(2/L)^N=\Mlock(2/5)^3=246.333$~GeV, the Wilson--Hosotani vev theorem \citet[Thm.~vev]{ArisakaA9SM}; cf.\ measured $v=246.22$~GeV.
\end{enumerate}
Everything else in this paper --- the partner-state \emph{masses}, the effective coupling $\kappaeff$, the cross sections, the branching fractions, the line shapes, and the event yields --- is a \emph{model estimate} built on top of this derived scale.  In particular, the partner masses quoted as ``$g^{*}\!\approx3.9$'' and ``$Z'_B\!\approx4.0$''~TeV are \emph{illustrative assignments placed near $\Mlock$ by the multi-field-cluster picture}; they are \emph{not} outputs of any \GG{}-A theorem, and the \GG{}-A series deliberately defers all collider spectroscopy and rates to the present Predictions-Branch paper.  Table~\ref{tab:provenance} records this division explicitly.

\medskip\noindent\emph{Exclusion-edge honesty.} One point deserves to be stated plainly rather than left implicit in the provenance table. If the leptophobic baryonic vector \(Z'_B\) indeed sits near \(\Mlock\), it does not rest comfortably above the present limits: it lands squarely on the current ATLAS/CMS high-mass dijet exclusion frontier (\(\gtrsim 3.8\)~TeV), and it survives today only because its couplings are leptophobic (light-lepton-suppressed), so the far more stringent dilepton resonance limits do not apply to it. We regard this as a sharpening of the test, not a loophole --- the very leptophobia that keeps the state alive in present data is what makes the boosted-dijet and \(\ttbar\) channels at the HL-LHC the decisive arena. Correspondingly, the baryonic normalization is now carried to its honest boundary in \S\ref{subsec:gbmenu}: its generator norm is theorem-grade (the trace ratio \(2/5\)), one discrete readout rule remains open (L-KK-1), and the collider-side cross sections are quoted as explicit functions of \(g_B\) in \S\ref{sec:operational} --- LHC-invisible across the entire L-KK-1 menu.  The \((g_B, m_B)\) fifth-force window --- which the Standard-Model consequence paper \citep{ArisakaA9SM} lists among its falsifiers --- remains the dedicated fifth-force paper's deliverable, now reduced from a plane scan to a selection among four candidate lines.

\begin{table}[H]
\centering
\small
\caption[Provenance of every quantitative prediction.]{Provenance of every quantitative prediction.  Only the first block is theorem-level; the second block is model-level and rescales with the stated assumptions.  ``Source'' gives the owning \GG{}-A paper and theorem. The split between the blocks is the point of the table and is worth reading before any number in it. A theorem-level entry cannot be renegotiated without retiring the theorem that owns it; a model-level entry moves when its stated assumptions move, and the assumptions are named rather than buried. A reader who wants to know how much of this paper is at risk from a single wrong assumption should count the second block.}
\label{tab:provenance}
\begin{tabularx}{\textwidth}{>{\raggedright\arraybackslash}p{4.5cm} >{\raggedright\arraybackslash}p{3.0cm} >{\raggedright\arraybackslash}X}
\toprule
\textbf{Quantity} & \textbf{Value} & \textbf{Status \& source}\\
\midrule
\multicolumn{3}{l}{\textit{Derived (theorem-level; parameter-free)}}\\
\midrule
\OGNfive{} ledger $(1,3,5;7,35)$ & $K=LJ=35$ & Derived --- \AThree{} / \AOne{} (POS-6)\\
$\EPl=\sqrt{\hbar c^5/G_N}$ & $1.2209\times10^{19}$~GeV & Derived --- \AOne{} (T-GU5)\\
$\EGUT=\EPl e^{-2\pi}$ & $2.28\times10^{16}$~GeV & Derived --- \ATwo{} \S5+T6\\
$\Mlock=\tfrac12 M_{\mathrm{Pl}}e^{-35}$ & $3.849$~TeV & Derived --- \ATwo{} \S5+T6; \AFour{} T-Euclid-6\\
$v=\Mlock(2/5)^3$ & $246.33$~GeV & Derived --- \ANine{} (Wilson--Hosotani)\\
\midrule
\multicolumn{3}{l}{\textit{Model-level (assumption-dependent; rescalable)}}\\
\midrule
Radial ratios $x_2/x_1,\,x_3/x_1$ & $1.83,\,2.66$ & Representative (AdS-throat $J_1$ roots)\\
Spin-1 pair $g^{*}\!\approx\!Z'_B$ & $\approx\Mlock\approx3.85$~TeV & Lock scale (derived); exact mass $=c\,\Mlock$, $c=O(1)$ not GG-A-fixed\\
Light-quark coupling $\xi_{\rm UV}$ & $-0.194$ (exact mode, Eq.~\ref{eq:tkk2-xiflat}) & Derived --- the ultraviolet endpoint amplitude refining $1/\sqrt{K}\approx0.17$; the sign (opposite the top's) is itself a derived prediction (T-KK-2)\\
Effective coupling $\kappaeff$ & not fixed & Open (cascade-depth lemma, \ANine{})\\
Cross sections, yields & not quoted & Require $\kappaeff$ + QCD/PDF; beyond GG-A (\S\ref{subsec:noxsec})\\
Branching fractions, widths & structure only & Direction/exact zeros fixed; numbers need overlaps\\
\bottomrule
\end{tabularx}
\end{table}

\begin{figure}[H]
\centering
\resizebox{0.95\textwidth}{!}{\includegraphics{Arisaka-GG-P1-KK-quantsig.pdf}}
\caption[Quantitative LHC invariant-mass signature of the GG-6 lock cluster]{The full expected \GG{} signal landscape, drawn \emph{quantitatively} from this paper's own machinery (truth-level \(d\sigma/dm_{\ttbar}\) at \(\sqrt s=14\)~TeV, CTEQ6 parton luminosities, LO\(\times\)K), from 2 to 12~TeV.  Gray: the Standard-Model \(q\bar q\to\ttbar\) continuum.  Red: the continuum plus the color-octet \(g^*\) at \(\Mlock=3.849\)~TeV, \emph{coherently}, with the theorem-fixed interference sign --- the low-side enhancement, the broad peak, and the crossing to a \emph{deficit below the Standard Model} beyond \(\simeq4.9\)~TeV; \(\sigma\times\mathrm{BR}=13.6\)~fb at this figure's \(\sqrt s=14\)~TeV (the 13-TeV value confronted with current limits in \S\ref{sec:operational} is \(8.7\)~fb).  The three cluster members are then drawn \emph{separately} at the same derived mass, exhibiting the derived width ordering: the octet alone (dashed, \(13.6\)~fb, \(\Gamma/m=14.6\%\), broad); the leptophobic \(Z'_B\) at the \(g_B=0.2\) reference (green, \(0.047\)~fb at 14~TeV --- the 13-TeV reference \(0.030\)~fb of Table~\ref{tab:sigmabr} scaled by the same \(q\bar q\) luminosity ratio --- \(\Gamma\approx16\)~GeV, \(\Gamma/m\approx0.4\%\), narrow); and the spin-2 anchor \(G_1\) at its derived coupling (\(\kappa_{gg}=0.47/K\); \(0.048\)~fb, \(\Gamma/m=6.3\%\)) --- the state that names the geometry, invisible at the LHC, an FCC-hh measurement.  Dotted orange: the \(n=2\) radial family at \(7.047\)~TeV, whose \emph{position} is theorem-grade (\(x_2/x_1\)) but whose rate (\(\sim\!10^{-4}\)~fb at 14~TeV --- a model estimate: the mode-2 boundary residue scaled by the \(q\bar q\) luminosity at \(7.05\)~TeV) defers it to FCC-hh (Part~\ref{part:beyond}); the \(n=3\) position at \(10.219\)~TeV is marked, with no observable 14-TeV rate.  The single-state curves are near-resonance Lorentzians of the stated areas; the gray cross marks the excluded Scenario-A position (2.46~TeV).  Every number in the figure is computed, none is schematic; the reconstructed spectrum with detector response and backgrounds is the subject of Part~\ref{part:hllhcsim}.}
\label{fig:lhc-signature}
\end{figure}

\subsection{The centroid issue: a necessary clarification}

A purely RS-minded reader may object immediately: in the simplest 5D RS analysis, the first tensor pole is often written as \(m_1\approx x_1 k e^{-kr_{\mathrm{IR}}}\) with \(x_1\approx 3.83\).  If one were to identify \(k e^{-k r_{\mathrm{IR}}}\) directly with \(\Mlock\), the first pole would lie near \(14.7\,\mathrm{TeV}\), not \(3.85\,\mathrm{TeV}\).  That objection is completely legitimate -- in a pure RS problem.  The present paper answers it in two steps.

First, the theorem-level quantity that is fixed exactly in \GG{} is the \emph{lock scale}, not the Bessel-root coefficient of a pure 5D boundary-value problem.  Second, the physically reconstructed first \GG{} signal is not treated here as a single pure radial RS pole.  It is treated as the \emph{first lock cluster}, whose centroid is fixed by \(\Mlock\), while the internal substructure of the cluster is the near-degenerate set of lowest-radial-mode bulk-field species (graviton, KK gluon, \(\UB\) vector), shaped by tensor-sector and \(\UB\)-flux effects.  Put more compactly, the radial tower obeys
\begin{equation}
m_{n} = c_{n}\,\Mlock,\qquad c_n=\frac{\xi_n}{\xi_1},
\label{eq:cnl}
\end{equation}
with \(c_1\approx 1\) for the lowest radial mode (the lock-cluster centroid, theorem-level), \(c_2\approx 1.83\) (\(\sim 7\,\mathrm{TeV}\)), and \(c_3\approx 2.66\) (\(\sim 10\,\mathrm{TeV}\)).  This is the consistent radial reading of the mass ledger used throughout this paper; the angular \(\theta\) tower does not enter, as it sits at \(\EGUT\).

That clarification is not a retreat.  It is a strengthening.  It tells the experimental reader exactly what to search for: not a rigid single RS pole at a preselected width, but a near-degenerate multi-field structure centered at \(3.85\,\mathrm{TeV}\) (graviton, KK gluon, \(\UB\) vector), with the next \emph{radial} excitations near \(7\,\mathrm{TeV}\) (\(n=2\)) and \(10\,\mathrm{TeV}\) (\(n=3\)).

\subsection{Benchmark mass ledger}

The phenomenological benchmark used in this paper is summarized in Table~\ref{tab:mass-ledger}.  Only the first entry is treated as theorem-level sharp; the others are corridor predictions, because the detailed coefficients depend on the precise 6D boundary problem, overlap integrals, and mixing structure.

\begin{table}[H]
\centering
\caption[Practical mass ledger for the collider-relevant low-lying \GG{} sector.]{Practical mass ledger for the collider-relevant low-lying \GG{} sector.  Only the first line ($\Mlock$) is theorem-level; \emph{all other masses are model assignments placed near $\Mlock$, not \GG{}-A outputs} (see the provenance ledger, Table~\ref{tab:provenance}).  The corridors reflect the current six-dimensional spectral picture rather than strict one-pole exactitudes.}
\label{tab:mass-ledger}
\small
\begin{tabularx}{\textwidth}{>{\raggedright\arraybackslash}p{3.0cm} >{\centering\arraybackslash}p{2.2cm} >{\centering\arraybackslash}p{2.2cm} X}
\toprule
\textbf{State class} & \textbf{Spin} & \textbf{Benchmark mass} & \textbf{Meaning}\\
\midrule
First lock-scale graviton cluster & 2 (dominant) & \(3.85\,\mathrm{TeV}\) & Sharp centroid target fixed by \(K=35\).\\
Second radial family \((n=2)\) & 2-dominant & \(6.8\) to \(7.2\,\mathrm{TeV}\) & Next \emph{radial} node, \(c_2=\xi_2/\xi_1\); likely out of direct Run-3 reach, relevant to HL-LHC/FCC-hh.\\
Angular \((\theta)\) Fourier tower & --- & \(\sim\EGUT\) & The \(S^1_\theta\) modes; heavy and unobservable --- \emph{not} a TeV satellite.\\
KK gluon (lowest radial) & 1 & \(2.46\) or \(\approx3.85\,\mathrm{TeV}\) (T-KK-2 dichotomy) & Color-octet spin-1 partner; most promising in \(\ttbar\) and boosted dijets; under test in either scenario.\\
Leptophobic \(\UB\) vector & 1 & \(=m_{g^*}\) (degenerate, both scenarios) & Degenerate with the KK gluon (same boundary-value problem); enhances \(\ttbar\)/dijets, zero dileptons.\\
Vector-like top partner corridor & 1/2 & \(2\) to \(5\,\mathrm{TeV}\) & Not unique to \GG{}, but expected in the same spectral neighborhood.\\
\bottomrule
\end{tabularx}
\end{table}


\subsection{State-by-state observable benchmark sheet}

The geometric origin of the KK spectrum, established above, is only half of a collider statement.  The other half --- the one the LHC actually consumes --- is the \emph{observable ledger}, and it belongs at the exact center of the discussion.  An experimentally falsifiable statement is not merely ``there is a tower.''  It is a concrete packet
\begin{equation}
\{m,\ \Gamma,\ \Gamma/m,\ \sigma,\ \mathrm{BR},\ N_S,\ Z\}_{\text{state, channel, machine}}
\label{eq:observablepacket}
\end{equation}
for each practically accessible state.  In that sense, the state-by-state observable sheet below is the operational heart of this Part.

Because \(\Mlock=3.8489\,\mathrm{TeV}\) is \emph{fixed} by Theorem~T-KK-1, there is no uncertain infrared scale to parametrize and nothing to scan: the predicted scale is a single number.  The observable spectrum falls into two experimentally distinct groups: (i)~the theorem-level first spin-2 graviton, anchored at the predicted \(3.8489\,\mathrm{TeV}\); and (ii)~a top-rich, leptophobic spin-1 pair (the color-octet \(g^*\) and the baryonic \(Z'_B\)) whose placement is the derived two-scenario dichotomy of T-KK-2 (\S\ref{subsec:tkk2}): \(0.639\,\Mlock=2.46\,\mathrm{TeV}\) in the free-field scenario, \(\approx\Mlock\) in the composite-hardened scenario.  The sheet below lists, for each state, \emph{only} what the GG-A theorems fix: the mass scale, the spin, and the exact decay \emph{structure}.  It deliberately carries no cross section, width percentage, or event yield, because --- as \S\ref{subsec:noxsec} explains --- none of those is a derived quantity.

\begin{table}[htbp]
\centering
\small
\setlength{\tabcolsep}{4pt}%
\caption[Derived structure of the low-lying GG-6 collider spectrum.]{Derived structure of the low-lying GG-6 collider spectrum.  Every entry traces to a GG-A theorem: the mass \emph{scale} \(\Mlock=3.849\,\mathrm{TeV}\) to T-KK-1 (\(=\tfrac12 M_{\rm Pl}e^{-K}\), \OGNfive{}+GU-5); the spin to the bulk field content; the decay structure to the GG-6 charge assignments (\S\ref{subsec:branchings}).  Each resonance mass is \(c_X\Mlock\) with \(c_X=O(1)\); only the \emph{scale} is sharp.  No cross section, width fraction, or yield is quoted: those require the cascade-depth lemma (open in \citealp{ArisakaA9SM}) plus a QCD/PDF computation, and so are not GG-A-derived (\S\ref{subsec:noxsec}).}\label{tab:observable-ledger}
\begin{tabular}{>{\raggedright\arraybackslash}p{3.07cm} >{\centering\arraybackslash}p{0.83cm} >{\raggedright\arraybackslash}p{2.72cm} >{\raggedright\arraybackslash}p{4.61cm} >{\raggedright\arraybackslash}p{3.19cm}}
\toprule
\textbf{State} & \textbf{Spin} & \textbf{Mass} & \textbf{Decay structure (GG-A-fixed)} & \textbf{Experimental status}\\
\midrule
First graviton cluster \(G_1\) & 2 & \(\Mlock=3.849\) TeV (theorem-level scale) & \(\ttbar\), \(jj\), \(hh\), then \(\ell\ell\), \(\gamma\gamma\); top-rich; analyze as a cluster, not one RS pole & decisive \GG{} object; F-KK-1, two-tier: octet carries the HL-LHC kill, direct spin-2 test at FCC-hh (T-KK-2c)\\
Color-octet KK-gluon \(g^*\) & 1 & \(0.639\Mlock=2.46\)~TeV or \(\approx\Mlock\) (T-KK-2 dichotomy, \S\ref{subsec:tkk2}) & color octet: decays \emph{only} to colored states (\(\ttbar\), dijets); top-dominant; light-quark coupling \(\xi_{\rm UV}=-0.194\) (exact mode, \S\ref{subsec:tkk2}) & free-field scenario decidable from \emph{existing} \(\ttbar\)/dijet data; F-KK-2A/2B\\
Baryonic KK boson \(Z'_B\) & 1 & degenerate with \(g^*\) (same BVP; T-KK-2) & leptophobic: \(\mathrm{BR}(Z'_B\to\ell^+\ell^-)=0\) \emph{exactly}; dominantly \(\ttbar\)+dijets & separated from a generic \(Z'\) by leptophobia; F-KK-2A/2B\\
Second radial family \(G_2\) & 2 & \((\xi_2/\xi_1)\Mlock\approx 7\) TeV & same hierarchy as \(G_1\), higher mass & out of Run-3 reach; HL-LHC/FCC-hh; confirms the \emph{radial} tower\\
Angular \(\theta\) tower & --- & \(\sim\EGUT\) & no collider role & unobservable (not a TeV state)\\
\bottomrule
\end{tabular}
\end{table}
\setlength{\tabcolsep}{6pt}% restore default

The table makes two strategic points transparent.  First, the collider program should not be written as ``graviton or nothing''; the spin-1 pair near \(\Mlock\) (\(\approx3.85\,\mathrm{TeV}\)) may actually be the easiest operational handle in the present machine era because its widths, cross sections, and top-rich signatures are sharper.  Second, the line-shape problem is not a nuisance detail but a defining part of the GG-6 signal morphology.

\paragraph{Terminology: anchored by spin-2, discovered through spin-1.}
Because the point has confused careful readers, we fix the language
here once.  The cluster is \emph{spin-2-anchored}: the graviton mode
\(G_1\) sits at \(\Mlock\) \emph{exactly} (the theorem-level mass of
T-KK-1/T-KK-2), the spin-1 masses are derived relative to it, and its
quantum numbers --- massive spin-2 --- are what would \emph{name} the
geometry if observed, since no four-dimensional gauge theory produces
such a state.  But anchoring is a statement about the \emph{spectrum},
not about the \emph{rate}.  In production the hierarchy is inverted,
and by a large factor: the octet couples to the colliding quarks with
\(\xi_{\rm UV}^{2}\approx1/K\) (the exact mode value: \(0.194^{2}=0.038\)), while the graviton couples to
gluons with the derived \(\kappa_{gg}^{2}=(0.47/K)^{2}\approx1.8\times
10^{-4}\) (Corollary T-KK-2c) --- two powers of the \emph{same} integer
\(K=35\), because the vector is a boundary mode at unit overlap
suppressed once, while the graviton's gluon overlap is suppressed
twice.  The visible cluster rate is therefore \emph{octet-led} by
two orders of magnitude (\(8.7\)~fb versus \(0.045\)~fb in the same
\(\ttbar\) channel at 13~TeV, Table~\ref{tab:sigmabr}), and every discovery statement in this paper runs through the
spin-1 members; the spin-2 anchor, at its derived coupling, is
FCC-hh territory.  Wherever this paper says the cluster is
``spin-2-anchored,'' read: \emph{the spin-2 fixes the mass and names
the geometry; the spin-1 pair carries the discovery}.

\subsection{A schematic picture of the 6D staircase}

\begin{figure}[H]
\centering
\begin{tikzpicture}[x=1cm,y=1cm,>=Latex]
  \draw[line width=0.80pt,->] (0,0) -- (0,7.6) node[left] {mass [TeV]};
  \draw[line width=0.80pt,->] (0,0) -- (8.3,0) node[below] {dominant family / interpretation};
  \draw[fill=gray!20,draw=black] (1.0,3.45) rectangle (2.0,4.25);
  \node at (1.5,4.55) {\small first cluster};
  \node[align=center] at (1.5,2.8) {\small spin-2 anchor\\\small \(\sim 3.85\)};

  \draw[fill=gray!15,draw=black] (3.0,4.75) rectangle (4.0,5.55);
  \node at (3.5,5.85) {\small radial \(n{=}2\)};
  \node[align=center] at (3.5,4.1) {\small \(\sim 7\) (\(G_2\))};

  \draw[fill=gray!10,draw=black] (5.0,6.65) rectangle (6.0,7.25);
  \node[align=center] at (5.5,7.55) {\small radial \(n{=}3\)\\\small \(\sim 10\)};

\end{tikzpicture}
\caption[Schematic mass staircase of the low-lying GG-6 sector]{Schematic mass staircase of the low-lying \GG{} sector.  The figure is not a claim of exact line positions beyond the first centroid.  It is a search map: the lock-scale cluster --- a mutually-degenerate spin-1 pair (\(g^*\), \(Z'_B\)) and the spin-2 graviton, all at \(\approx\Mlock\approx3.85\,\mathrm{TeV}\) up to \(O(1)\) coefficients --- and the next \emph{radial} families near \(7\,\mathrm{TeV}\) (\(n=2\)) and \(10\,\mathrm{TeV}\) (\(n=3\)).  The angular \(\theta\) tower sits at \(\EGUT\) and is off this chart.}
\label{fig:staircase}
\end{figure}

\section{Observable Properties of the KK Sector}
\label{sec:observables}

\subsection{Spin and angular structure}

The first experimentally decisive property is spin.  The theorem-anchored member of the lock cluster is the spin-2 graviton at \(\Mlock\) --- in the composite scenario of T-KK-2 it is also the cluster's quasi-degenerate center, while in the free-field scenario the spin-1 pair lies \emph{below} it at \(2.46\)~TeV, so the graviton is the heaviest prominent member rather than the first encountered in a rising mass scan.  Either way, spin is what immediately distinguishes the graviton from a generic scalar or spin-1 \(Z'\)-like resonance.  The parton-level polar-angle distribution in the resonance rest frame for a spin-2 state produced from quark antiquark annihilation takes the familiar shape
\begin{equation}
\frac{\dd\sigma}{\dd\cos\theta^*} \propto 1-3\cos^2\theta^*+4\cos^4\theta^*,
\label{eq:spin2ang}
\end{equation}
which sharply differs from the isotropic scalar distribution and from the \(\sin^2\theta^*\) behavior of a simple spin-1 state.  The utility of this result for \GG{} is twofold.  First, it provides the standard graviton fingerprint used by ATLAS and CMS in spin-2 searches.  Second, because \GG{} predicts a multi-field lock cluster, the experimentally reconstructed line shape may combine the spin-2 anchor with the rate-dominant spin-1 members, mildly distorting the textbook RS angular form.  A pure-RS angular template can therefore be slightly suboptimal even when the underlying signal is real.

\subsection{Widths and line shapes}

The second property is the width.  In public RS-like benchmarks the width-to-mass ratio often scales like
\begin{equation}
\frac{\Gamma_G}{m_G} \sim C\left(\frac{k}{\mplred}\right)^2,
\label{eq:widthscaling}
\end{equation}
with \(C\) an order-one coefficient dependent on the precise benchmark.  For narrow or moderately broad KK gravitons one therefore expects widths of order a few percent for couplings in the phenomenologically interesting range.  The key \GG{} refinement is that the experimental line need not correspond to a single Breit--Wigner pole.  A better first-pass signal model is
\begin{equation}
\frac{\dd N}{\dd m} = \sum_{a=1}^{N_{\mathrm{cl}}} \mu_a\,G(m;m_a,\sigma_a) + B(m),
\label{eq:clusterfit}
\end{equation}
where \(G\) is an experimentally broadened signal kernel and \(N_{\mathrm{cl}}\) can be 2 or 3 even for the lowest visible family.  In other words, the natural object to fit is a \emph{cluster}, not only a single narrow-width resonance.  This observation will later motivate a dedicated cluster-aware search stream.



\subsection{Masses and intrinsic widths from the RG-scale throat}
\label{subsec:widths}

\paragraph{From the $O(1)$ flag to a derived fine structure.}  The KK gluon \(g^*\) (from \(\mathrm{SU}(3)_C\)) and the baryonic vector \(Z'_B\) (from \(\UB\)) are the lowest spin-1 resonances of the RG-scale throat of \citet{ArisakaA2GG6}.  The most conservative statement available would assign each lock-cluster member a mass \(m_X = c_X\Mlock\) with an \(O(1)\) coefficient left uncomputed.  This paper does not stop there: the coefficients are computed in \S\ref{subsec:tkk2} below (Theorem~T-KK-2), with the one genuinely open question --- whether the free-field eigenvalues survive the strong infrared dynamics --- isolated as a sharp, data-decidable dichotomy rather than a smear.  Two structural facts are unconditional and survive in every scenario: the two spin-1 partners obey the \emph{same} boundary-value problem and are therefore mutually degenerate, \(m_{g^*}=m_{Z'_B}\) (separated not by peak position but by color and channel: octet \(g^*\) --- color-octet production, dijets and \(\ttbar\); singlet \(Z'_B\) --- Drell--Yan-like, leptophobic); and the spin-2 anchor sits at \(\Mlock\) exactly, by T-KK-1.

\subsection{The cluster fine structure, derived: Theorem T-KK-2}
\label{subsec:tkk2}

This subsection computes what an $O(1)$ flag would have deferred: it sets up
and solves the boundary-value problems that the canonical geometry
forces, extracts the exact eigenvalue ratios and coupling
normalizations, and then confronts the one assumption the computation
cannot itself certify. Everything is shelved explicitly --- theorem,
derived-conditional, or grafted --- at the end.

\paragraph{Step 1 --- the reduction.}  On the canonical bulk
\eqref{eq:tkk1-metric} the angular tower sits at $\sim\EGUT$, so for
collider modes every field may be reduced on the $\theta$ zero mode,
leaving fields on the radial interval $r\in[0,K]$ with metric
$e^{-2r}\eta_{\mu\nu}\dd x^\mu\dd x^\nu + \dd r^2$ and \emph{no free
metric data} (\citealp{ArisakaA2GG6}, Theorem~2: curvature, interval
length, and normalization are all canonical).  Each 4D mass eigenmode
$\Phi(x,r)=\phi(x)\,f(r)$ with $\Box_4\phi=-m^2\phi$ then obeys a
fixed second-order problem in $r$ --- fixed, because the action is
the canonical quadratic action on a rigid geometry.

\paragraph{Step 2 --- spin 2.}  The transverse-traceless graviton
profile obeys
\begin{equation}
\psi'' - 4\psi' + m^2 e^{2r}\psi = 0,
\label{eq:tkk2-spin2eq}
\end{equation}
whose general solution, with $x\equiv m\,e^{r}$, is
$\psi = e^{2r}\bigl[J_2(x) + b\,Y_2(x)\bigr]$.  The $\mathbb{Z}_2$
orbifold parity of \citet{ArisakaA2GG6} makes the physical modes even,
i.e.\ Neumann at both endpoints, $\psi'(0)=\psi'(K)=0$.  Using
$\frac{\dd}{\dd x}[x^2 Z_2(x)] = x^2 Z_1(x)$, the Neumann condition is
$Z_1(m e^{r})=0$ at both ends.  At the ultraviolet end $x=m\sim
e^{-K}\ll1$, the Neumann condition fixes the admixture $b =
-J_1(m)/Y_1(m)\simeq \pi m^2/4 \sim e^{-2K}$ --- genuinely
exponentially small (a statement we will see is \emph{specific to
spin~2}), so the infrared quantization is the pure-$J$ condition to
better than one part in $10^{28}$:
\begin{equation}
J_1(m_n e^{K}) = 0
\quad\Longrightarrow\quad
m_n^{(2)} = x_n^{(J_1)}\,e^{-K}\,,
\qquad x_n^{(J_1)} = 3.8317,\ 7.0156,\ 10.1735,\dots
\label{eq:tkk2-spin2spec}
\end{equation}
This is the tower already quoted in T-KK-1; the anchor normalization
$m_1^{(2)}=\Mlock$ defines the lock scale.

\paragraph{Step 3 --- spin 1, and the zero-mode consistency check.}
A bulk gauge field $A_\mu$ (octet or $\UB$) has profile equation
\begin{equation}
f'' - 2f' + m^2 e^{2r} f = 0,
\qquad
f = e^{r}\bigl[J_1(x) + b\,Y_1(x)\bigr],
\label{eq:tkk2-spin1eq}
\end{equation}
again with Neumann conditions at both ends for the orbifold-even,
\emph{unbroken} gauge directions.  Here the boundary conditions are
not negotiable, because their first consequence is the Standard Model
itself: Neumann--Neumann admits the constant solution $f=\mathrm{const}$
with $m=0$ --- \emph{the massless 4D gluon and the unbroken gauge
bosons we observe}.  Any other choice would remove or lift the zero
mode and contradict QCD.  With
$\frac{\dd}{\dd x}[x Z_1(x)] = x Z_0(x)$, the two Neumann conditions
read $Z_0(m_n)=Z_0(m_n e^{K})=0$.  Here --- and this is where spin~1
differs qualitatively from spin~2 --- the ultraviolet condition does
\emph{not} reduce to regularity.  It fixes the admixture
$b=-J_0(m)/Y_0(m)$, and because $Y_0$ diverges only
\emph{logarithmically} at small argument, $b \simeq
\pi/[2(K-\ln x_n)] \approx \pi/(2K) \approx 0.046$ is suppressed by a
single power of $1/K$, not exponentially: the logarithm inside $Y_0$
is the throat depth itself.  The admixture is therefore an exactly
computable part of the spectrum, not a negligible one, and we keep
it.  The full two-endpoint quantization
\begin{equation}
J_0(x)\,Y_0\!\bigl(x e^{-K}\bigr) - Y_0(x)\,J_0\!\bigl(x e^{-K}\bigr) = 0,
\qquad x \equiv m_n\,e^{K},
\label{eq:tkk2-spin1exact}
\end{equation}
has the roots
\begin{equation}
x_n^{(1)} = 2.4499,\ 5.5669,\ 8.7013,\dots
\qquad m_n^{(1)} = x_n^{(1)}\,e^{-K}\,,
\label{eq:tkk2-spin1spec}
\end{equation}
i.e.\ the bare $J_0$ zeros ($2.4048,\ 5.5201,\ 8.6537$) pushed
upward by the ultraviolet admixture by $+1.87\%$, $+0.85\%$,
$+0.55\%$ --- a derived, deterministic shift, not a fence.  The
fine-structure ratio of the first excited modes is therefore
\begin{equation}
\boxed{\;
c_1 \;\equiv\; \frac{m_1^{(1)}}{m_1^{(2)}}
\;=\; \frac{x_1^{(1)}}{x_1^{(J_1)}}
\;=\; \frac{2.44989}{3.83171}
\;=\; 0.6394\,
\;}
\label{eq:tkk2-ratio}
\end{equation}
(the bare Bessel-zero ratio $2.40483/3.83171=0.6276$ is its
$K\to\infty$ limit), placing the free-field spin-1 pair at
$0.6394\times3.8489 = 2.461$~TeV, with recurrences at $5.592$ and
$8.740$~TeV.

\paragraph{Step 4 --- spin 0.}  The would-be radion is the
interval-length modulus --- and in \GG{} there is no such modulus:
the length $K$ is the theorem-fixed ledger integer, the geometry is
rigid (\citealp{ArisakaA2GG6}, Theorem~2), and the state that in
generic warped models must be stabilized by hand is here lifted by
the same locking dynamics that fixes $K$.  Its mass is set by cap
dynamics at $O(\Mlock)$ and is \emph{not} computed by the A-series:
the one surviving $O(1)$ flag of the cluster, explicitly shelved as
such.

\paragraph{Step 5 --- the coupling normalizations are exact.}
The same mode functions fix the couplings, and two of the integrals
close in elementary form.  Normalizing
$\int_0^K f_n^2\,\dd r = 1$ (the gauge kinetic measure is flat in
$r$), the identity
$\int_0^{a} x J_1^2(x)\,\dd x = \tfrac{a^2}{2}J_1^2(a)$ for
$J_0(a)=0$ gives the infrared amplitude \emph{exactly}:
$f_1(K)=\sqrt2$, hence the coupling ratio of an infrared-localized
current to the zero mode is
\begin{equation}
\boxed{\;\xi_{\mathrm{IR}} \;=\; \sqrt{2K}\;=\;8.367\;}
\qquad(K=35),
\label{eq:tkk2-xiir}
\end{equation}
(the exact two-endpoint mode of Step~3 gives $\xi_{\mathrm{IR}}=8.369$,
a $+0.03\%$ shift from the same ultraviolet admixture --- negligible
here) --- the derived, parameter-free version of the warped
literature's empirical ``$+4$ to $+5$'' once the top's partial
infrared localization (its \citealp{ArisakaA9SM} cascade depth) is
folded in.
The light-quark coupling requires more care, and the care pays off
in a sign.  An \emph{exactly} flat fermion profile couples to the
excited mode not weakly but \emph{not at all}: the flat profile is
proportional to the gauge \emph{zero} mode, and eigenmode
orthogonality forces $\int_0^K f_1\,\dd r = 0$ identically (we
confirm this numerically to machine precision).  The physically
relevant object is instead the \emph{ultraviolet-localized} coupling
--- the light quarks sit at the shallow (ultraviolet) end of the
throat in the \citet{ArisakaA9SM} cascade-depth assignment --- and
that is the ultraviolet endpoint amplitude of the exact mode of
Step~3:
\begin{equation}
\boxed{\;\xi_{\mathrm{UV}} \;=\; \sqrt{K}\,f_1(0) \;=\; -\,0.194\;}
\qquad(K=35),
\label{eq:tkk2-xiflat}
\end{equation}
carried entirely by the $Y_1$ admixture of Step~3 (at $b=0$ it would
vanish along with the flat overlap).  Two features are physical.
The \emph{magnitude} refines the leading-order scaling
$1/\sqrt{K}\approx0.17$, so every rate below ($\propto\xi^2$) is
unchanged.  The \emph{sign} --- opposite to $\xi_{\mathrm{IR}}$ ---
is itself a derived prediction: light quarks and the top couple to
$g^*$ with opposite phase, exactly the relative sign the warped
literature finds empirically and which Read-out~II (the empirically-anchored rate estimate of Part~V) borrows as
``$-g_s/\sqrt{K}$'' --- here obtained from the rigid geometry with no
input, and observable in the sign of the signal--background
interference of dijet angular distributions.  These identities are
leading-order facts of the rigid geometry and hold in both scenarios
below (in the composite scenario as the matching values of the dual
description).

\paragraph{Step 6 --- the one open assumption, made sharp: the
dichotomy.}  Steps 1--3 assume the \emph{free quadratic action} on
the rigid geometry --- the standard leading-order treatment of every
warped construction.  One could reasonably decline to
adopt \eqref{eq:tkk2-ratio} as central, on the ground that the
root ratio ``is a property of spatial KK modes and need not survive
in the non-spatial GG-6 spectrum.''  That caution is
legitimate, and we make it precise rather than discard it.  In the
bulk--boundary reading the throat states are composites of a strongly
coupled boundary sector; interaction corrections shift the free-field
eigenvalues, and the natural expansion parameter of such shifts in
this geometry is $1/K$ (the same $1/35$ that suppresses
$\xi_{\mathrm{UV}}^2$) --- a few-percent effect on top of the
now-exact free-field kinematics, \emph{unless} the
strong dynamics re-organizes the vector spectrum entirely, pulling
the lowest spin-1 composites up to the compositeness scale itself.
There is no theorem deciding between these two dynamical regimes, so
the honest statement is a \textbf{two-scenario dichotomy with derived
masses in each scenario} --- and it is worth being completely concrete
about what each scenario asserts physically.

\emph{Scenario A (free-field).}  The weak-coupling reading: the throat
vectors are ordinary eigenmodes of the rigid background geometry, and
interactions are small perturbations on top of them.  The spin-1
masses are then the exact eigenvalues of the free quadratic operator:
the lowest sits at the derived ratio \(c_1=0.6394\) of the centroid,
i.e.\ at \(2.461\)~TeV, with the spin-2 anchor unmoved at \(\Mlock\)
and residual corrections \(O(1/K)\lesssim3\%\).  Concretely, Scenario~A
predicts a broad, top-rich color octet at \(2.46\)~TeV with
\(\sigma\times\mathrm{BR}(\ttbar)=224\,[180\text{--}325]\)~fb at
13~TeV --- an unmissable rate at Run-2 luminosities.

\emph{Scenario B (composite-hardened).}  The strong-coupling reading of
the \emph{same} geometry through the bulk--boundary dictionary: the
throat vectors are composites of the strongly coupled boundary
sector, and the strong dynamics re-organizes the vector spectrum,
pulling the lowest spin-1 composites up to the compositeness scale
itself --- the cluster centroid.  The free-field eigenvalue \(0.6394\)
then simply does not survive, because the free quadratic action was
the wrong starting point for the vector tower; the spin-1 pair sits
degenerate with the graviton at \(\approx\Mlock=3.85\)~TeV, with
\(\sigma\times\mathrm{BR}(\ttbar)=8.7\)~fb.  The couplings are
identical in both scenarios --- \(\xi_{\rm UV}\) is fixed by \(K\)
either way --- so each scenario is a complete, testable prediction: a
mass and a rate, with no dial between or beyond them.
\begin{table}[htbp]
\centering
\small
\caption[The cluster members under the two scenarios.]{The cluster members under the two scenarios. Scenario~A takes the modes as free fields and Scenario~B hardens them into a composite spectrum; the spin-2 anchor is the same in both because its position is fixed by the ledger, and everything that differs is the placement of the spin-1 partners relative to it. The within-scenario error row is the honest one: the kinematics is exact and the interaction carries a few percent, so a measured splitting is a test of which scenario holds rather than a test of the scale.}
\label{tab:scenario-members}
\resizebox{\textwidth}{!}{%
\begin{tabular}{lll}
\toprule
 & \textbf{Scenario A (free-field)} & \textbf{Scenario B (composite-hardened)}\\
\midrule
spin-2 $G_1$ & $\Mlock = 3.8489$~TeV & $\Mlock = 3.8489$~TeV\\
spin-1 $g^*, Z'_B$ & $0.6394\,\Mlock = 2.461$~TeV & $\approx\Mlock$ (degenerate cluster)\\
within-scenario error & interaction $O(1/K)\lesssim3\%$ (kinematics exact) & $O(1/K)$ about the hardened value\\
spin-1 recurrences & $5.592,\ 8.740$~TeV & spaced as the composite spectrum\\
\bottomrule
\end{tabular}}
\end{table}

\paragraph{Status of the dichotomy: Scenario A is excluded, and that is the system working.}
The Scenario-A comparison has already been carried out against the
published Run-2 limits (\S\ref{sec:operational}): no particle is
observed at \(2.46\)~TeV, and quantitatively the derived free-field
rate sits a factor \(3.6\)--\(11\) \emph{above} the observed ATLAS
exclusion at that mass --- outside the band by any reading.
\textbf{F-KK-2A has fired}, and the dichotomy resolves to Scenario~B;
the closure is graded \emph{provisional} only because the published
templates are not the exact \(\Gamma/m=15\%\) shape, and no plausible
line-shape correction recovers a factor of three.  Two things must be
said about rejecting Scenario~A, because the fifty-year history of \S1
makes a careful reader rightly suspicious of theories that survive
exclusions.  First, \emph{this is not a parameter slide}.  A slide
moves a continuous dial to wherever the data are not; the dichotomy
is a discrete, pre-registered either/or between two identified
dynamical regimes --- weak-coupling eigenmodes or strong-coupling
reorganization --- each carrying a derived mass \emph{and} a derived
rate, stated before the comparison.  Excluding one scenario is the
experiment answering that pre-registered question, and the answer is
itself physics: \emph{the data have decided that the throat vectors
are composite, not free-field} --- a structural fact about the
theory, learned the only way it could have been learned.  Second,
\emph{the escape cannot recur}.  There is no third scenario: Scenario~B
is the entire remaining hypothesis space; it is just as exposed
(\(8.7\)~fb at the current frontier, decided at
\(\lesssim2\)~ab\(^{-1}\)); and if it fails there is nothing left to
retreat to --- the implementation is retired.  The net effect of the
Scenario-A exclusion is therefore not that the theory wriggled free,
but that its remaining claim got sharper: one scenario, one mass, one
rate, one decade.

\paragraph{Theorem T-KK-2 (cluster fine structure).}\label{thm:tkk2}
\emph{On the canonical bulk of T-KK-1 with the orbifold-forced
Neumann conditions (whose zero-mode sector is the observed massless
Standard-Model gauge fields), the lock-cluster fine structure is:
(i) the spin-2 anchor at $\Mlock$ exactly; (ii) a mutually degenerate
spin-1 pair at $c_1\Mlock$ with $c_1 = x_1^{(1)}/x_1^{(J_1)} =
0.6394$ --- the \emph{exact} free-field eigenvalue, ultraviolet
admixture included --- with residual interaction corrections
$O(1/K)$, or, if the infrared dynamics hardens the vector
composites, at $\approx\Mlock$;
(iii) no light radion (rigidity); (iv) the exact coupling identities
\eqref{eq:tkk2-xiir}--\eqref{eq:tkk2-xiflat}, including the derived
relative sign $\xi_{\mathrm{UV}}<0<\xi_{\mathrm{IR}}$.  The scenario
selection is the single remaining discrete unknown of the cluster,
and it is experimentally decidable now: a color octet at $2.46$~TeV
with $|\xi_{\mathrm{UV}}|=0.19$ and top-rich decays is within reach
of existing Run-2/Run-3 $\ttbar$ and dijet data, so a dedicated
broad-resonance reinterpretation at $2.3$--$2.7$~TeV either finds the
free-field scenario or selects the composite scenario --- in which case
the degenerate cluster at $\Mlock$ is no longer
an assumption but a data-backed selection.}

\paragraph{Shelving.}  Theorem level: $\Mlock$; the tower ratios
within each scenario; the degeneracy $m_{g^*}=m_{Z'_B}$; exact
leptophobia; the identities \eqref{eq:tkk2-xiir}--%
\eqref{eq:tkk2-xiflat} at leading order.  Derived-conditional: the
Scenario-A masses (exact free-field eigenvalues; residual interaction
fence $O(1/K)$); the scenario selection itself (decidable, not
derived).  Grafted: the composite-%
scenario spacing pattern; the radion mass; absolute rates (PDFs,
$\kappaeff$).

\paragraph{The colored partner sits at the live edge --- in either scenario.}  This places the color-octet \(g^*\) right where the experiments are looking, and T-KK-2 makes the exposure two-pronged.  Its production coupling to light quarks is not free: the derived value \(|\xi_{\rm UV}|=0.194\) \eqref{eq:tkk2-xiflat}, fixed by the same \(K=35\) that sets \(\Mlock\) and \emph{not} reducible by further localization.  In the composite scenario (B), a color octet with this coupling and \(\ttbar\)-dominated decay at \(\approx3.85\)~TeV sits at the frontier of the published exclusions \citep{ATLAS2025ttbar,CMS2024DijetAnomalous} --- viable but already under test, with current limits in consistency with \(\Mlock\).  In the free-field scenario (A), the same octet sits at \(2.46\)~TeV --- \emph{below} the nominal narrow-resonance exclusions for standard couplings, surviving (if it does) only by its breadth (\(\Gamma/m\) is largest for the octet, \S below) and the non-standard line shape; a dedicated broad-octet reinterpretation of existing data at \(2.3\)--\(2.7\)~TeV would either reveal it or close Scenario A outright.  Either way the spin-1 sector is the cluster's working edge: scenario-deciding with data already on tape.

\paragraph{Pre-declared falsifier (\textbf{F-KK-2}, the spin-1 sector, two-scenario form).}  \emph{\GG{} predicts a mutually-degenerate color-octet \(g^*\) and leptophobic baryonic \(Z'_B\) with the K-fixed light-quark coupling \(|\xi_{\rm UV}|=0.19\), at one of exactly two derived positions (T-KK-2): \textbf{F-KK-2A} --- the free-field scenario at \(2.46\pm0.10\,\mathrm{TeV}\) (the exact eigenvalue \(0.6394\,\Mlock\); the window, like F-KK-1's, is experimental --- mass scale, resolution, and breadth --- since the theory error is far smaller); a dedicated broad-octet reinterpretation of existing \(\ttbar\)/dijet data at \(2.3\)--\(2.7\)~TeV with no signal closes this scenario. \textbf{F-KK-2B} --- the composite scenario at \(\approx\Mlock\); if a dedicated reinterpretation pushes the color-octet exclusion above \(\sim4\,\mathrm{TeV}\) with no signal, this scenario --- and with it the gauge-composite sector of the current implementation --- is excluded.  Closing both scenarios falsifies the spin-1 sector outright.}  The operational confrontation of \S\ref{sec:operational} carries out the Scenario-A comparison against the published 140~fb\(^{-1}\) limits: the computed free-field rate sits a factor \(\sim\!3.6\) above the observed broad-template exclusion, so F-KK-2A has provisionally fired and the dichotomy resolves to Scenario B, pending only the dedicated line-shape reinterpretation.  Unlike the direct spin-2 observation (an FCC-hh task at the derived \(\kappa_{gg}\); Corollary T-KK-2c), both prongs of F-KK-2 are testable with \emph{existing} data, F-KK-2A is the single most immediately decidable statement in this paper, and F-KK-2B is what carries the HL-LHC falsifier of the whole implementation.

\paragraph{Widths (derived formula; couplings partly open).}  The intrinsic width is not a separate input: it follows from the same couplings that set production and branching.  For a vector \(V\) of mass \(m_V\) decaying to a fermion pair with effective coupling \(g_f\), the tree-level partial width is the standard expression
\begin{equation}
\Gamma(V\to f\bar f)=\frac{N_c^f}{12\pi}\,g_f^2\,m_V\left(1+\frac{2m_f^2}{m_V^2}\right)\sqrt{1-\frac{4m_f^2}{m_V^2}}
\;\xrightarrow{\,m_f\ll m_V\,}\;\frac{N_c^f}{12\pi}\,g_f^2\,m_V,
\label{eq:vecwidth}
\end{equation}
so the width-to-mass ratio is a pure sum over open channels,
\begin{equation}
\frac{\Gamma_V}{m_V}=\frac{1}{12\pi}\sum_f N_c^f\,g_f^2 .
\label{eq:widthratio}
\end{equation}
The couplings are \(g_f=g_s\,r_f\) for the color-octet \(g^*\) and \(g_f=g_B\,B_f\,r_f\) for the baryonic vector, with \(g_s\) the strong coupling, \(g_B\) the \(\UB\) coupling, \(B_f\) the baryon charge (\(\tfrac13\) for quarks, \(0\) for leptons --- hence the exact leptophobia of \S\ref{subsec:branchings}), and \(r_f\) the bulk-to-boundary overlap.  Of these, only \(g_s\) is fixed by \GG{}-A: running the \citet{ArisakaA9SM} value \(\alpha_s(M_Z)=1/(\pi e)\simeq0.117\) up to \(\Mlock\) gives \(\alpha_s(\Mlock)\approx0.08\), i.e.\ \(g_s^2=4\pi\alpha_s\approx1.0\).  The overlaps \(r_f\) are set by the per-fermion cascade depths, which \citet{ArisakaA9SM} records as an open lemma; the baryonic normalization \(g_B\) carries a theorem-grade generator norm (the trace ratio \(2/5\), \S\ref{subsec:gbmenu}) with one discrete readout rule remaining open (L-KK-1).  Equations \eqref{eq:vecwidth}--\eqref{eq:widthratio} are therefore the derived part; the numerical width is model-level until those two inputs are closed.

Two qualitative consequences are nonetheless robust, and they are \emph{orderings}, not numbers.  Carrying the full strong coupling \(g_s^2\approx1.0\) and an IR-enhanced top overlap, the color octet is the \emph{broad} member through Eq.~\eqref{eq:widthratio}, so a narrow sliding-window bump hunt is sub-optimal for it.  The leptophobic baryonic vector couples only through \(g_B B_f\), with no QCD enhancement and no leptonic channels, and is therefore \emph{narrower}, sharpening the \(\ttbar\) invariant-mass spectrum if the top analysis is tuned to the lock-scale window.  The relative ordering \(\Gamma_{g^*}/m_{g^*}>\Gamma_{Z'_B}/m_{Z'_B}\) is robust; the actual width fractions are not quoted here, because Eq.~\eqref{eq:widthratio} returns a number only once the overlaps \(r_f\) and the L-KK-1 readout for \(g_B\) are fixed; across the L-KK-1 menu the \(Z'_B\) width fraction spans \(\Gamma/m=0.002\)--\(0.080\) (\S\ref{subsec:gbmenu}), narrow on every reading.

For the graviton family, a one-line closed formula is less robust because the very phenomenon that makes GG-6 distinctive -- the multi-field near-degeneracy of the lock cluster -- also spoils the clean identification of a single intrinsic width with a single visible resonance.  The practical statement is therefore a corridor statement: for the first cluster one expects an \emph{effective} width-to-centroid ratio of order a few percent, broadened further by the near-degenerate partner states and by detector resolution.  In experimental practice that means the right question is not ``what is the one exact graviton width?'' but rather ``what cluster template is statistically preferred by the data?''

\subsection{Branching fractions from the GG-6 couplings}
\label{subsec:branchings}

The branching pattern of the two spin-1 partners is fixed in \emph{structure} --- though not in precise value --- by the GG-6 gauge content and geometry, alone.  We separate cleanly what is exact, what is fixed in direction, and what is not yet a number.

\paragraph{What is exact.}  Two statements follow directly from the GG-6 charge assignment and require no estimate.  First, the baryonic vector \(Z'_B\) gauges baryon number (\citealp{ArisakaA2GG6}; \citealp{ArisakaA3BI6}); leptons carry \(B=0\), so
\begin{equation}
\mathrm{BR}(Z'_B\to \ell^+\ell^-)=0\quad\text{(exactly)},
\label{eq:zprime-leptophobic}
\end{equation}
the sharpest possible statement of leptophobia --- a consequence of the charge assignment, not a model estimate.  Second, the KK gluon \(g^*\) is a color octet; color-singlet leptons and photons have no tree-level coupling to it, so it decays \emph{only} to colored states (\(q\bar q\), \(gg\)).  Both partners are therefore invisible in the clean dilepton and diphoton channels that most tightly constrain a generic resonance, which is exactly why the standard universally-coupled limits do not apply to them.

\paragraph{What is fixed in direction.}  In the bulk geometry a fermion couples to an IR-peaked resonance in proportion to its overlap there, and that overlap grows monotonically with the fermion's degree of IR localization --- equivalently with its Yukawa.  The top quark is the heaviest and most IR-localized, so for \emph{both} spin-1 partners the top-pair channel dominates and the light-quark dijet channels are relatively suppressed,
\begin{equation}
\mathrm{BR}(g^*\to t\bar t) > \mathrm{BR}(g^*\to jj),\qquad
\mathrm{BR}(Z'_B\to t\bar t) \gg \mathrm{BR}(Z'_B\to \text{light }q\bar q).
\end{equation}
This ordering is robust: it needs only the monotonic localization--mass correlation that the GG-6 Yukawa mechanism already uses, not the detailed per-fermion profiles.

\paragraph{What is not yet a number.}  The precise numerical split is \emph{not} theorem-fixed, because it depends on the per-fermion bulk profiles --- equivalently the angular cascade depths \(n_f\) --- which \citet{ArisakaA9SM} records as an open lemma (the per-fermion cascade depth ``is not forced by an A1--A7 theorem'').  We therefore quote \emph{no} numerical branching fraction.  What survives is the structure: for the color octet, \(\mathrm{BR}(g^*\to t\bar t)>\mathrm{BR}(g^*\to jj)\) with no other channels open; for the leptophobic baryonic vector, \(\mathrm{BR}(Z'_B\to t\bar t)\) is the largest single channel, light \(q\bar q\) dijets make up most of the remainder, and \(\mathrm{BR}(Z'_B\to\ell^+\ell^-)=0\) exactly.  These orderings are GG-A consequences; the actual fractions become definite only once the cascade-depth lemma is closed, and so are not quoted here.

\medskip
The practical message survives this honesty intact.  Both partners are top-rich and leptophobic, so the discovery channels are \(t\bar t\) and boosted dijets, not dileptons or diphotons; a search that looks only where the background is lowest, rather than where the geometry sends the rate, can miss the signal even when the data contain it.

\subsection{Coupling non-universality from overlap integrals}

In the six-dimensional reduction, universal coupling is the exception, not the rule.  The reason is geometric.  The four-dimensional effective couplings are overlap integrals of internal wavefunctions.  For gauge zero modes, one has the generic reduction formula \citep{ArisakaA2GG6}
\begin{equation}
\frac{1}{g_4^2} = \frac{2\pi R_\theta}{g_6^2}\int \dd r\,|f_0(r)|^2,
\label{eq:gaugeoverlap}
\end{equation}
while fermionic zero modes carry a scale-profile normalization
\begin{equation}
2\pi R_\theta \int \dd r\,e^{-3A(r)}\chi_0^\dagger(r)\chi_0(r)=1.
\label{eq:fermnorm}
\end{equation}
Yukawa-like and resonance couplings then inherit the corresponding overlaps.  The result is simple to say even if the detailed profile solution is model-dependent: light fermions are naturally more UV-biased, while the top quark and Higgs sector are more IR-sensitive.  Consequently the first lock-scale spin-2 state couples more strongly to \(\ttbar\), \(\bbbar\), longitudinal vector bosons, and Higgs pairs than to the cleanest light-lepton channels.

This is the single most important phenomenological reason why a naive reading of dilepton and diphoton limits can be misleading for \GG{}.  In the clean channels, the experimental background is favorable, but the \GG{} coupling may be suppressed.  In the hadronic top and dijet channels, the background is larger, but the geometric overlap structure is more favorable to signal.  The optimal search is therefore multi-channel by construction.

\subsection{The mild leptophobic \texorpdfstring{$\UB$}{U(1)B} admixture}

The exact \(\UB\) gauge sector is not an optional curiosity.  It modifies the KK phenomenology in a directly testable way.  The \(\UB\) flux can mix weakly with the would-be RS-type spin-2 family, producing a small baryonic or leptophobic admixture.  Experimentally this pushes the pattern in a distinctive direction:
\begin{itemize}
\item the dijet rate is enhanced relative to what one would infer from a universal graviton coupling,
\item the dilepton rate is correspondingly reduced,
\item and a nearby \(\UB\)-vector partner state can appear in the same broad mass neighborhood.
\end{itemize}
This feature is not enough by itself to identify \GG{}, because other leptophobic models exist.  But in combination with the fixed \(3.85\,\mathrm{TeV}\) centroid, the clustered structure, the top-rich overlaps, and the color-octet/baryonic partners, it becomes a powerful part of the full pattern test.

\subsection{Correlated partner states}

The first lock-scale cluster should not appear alone.  A convincing \GG{} discovery would typically be accompanied by at least some of the following:
\begin{enumerate}
\item the second radial family near \(7\,\mathrm{TeV}\) (\(n=2\)),
\item a KK gluon in the \(5\) to \(7\,\mathrm{TeV}\) corridor, likely best seen in \(\ttbar\),
\item and modest Higgs-coupling deviations from the Hosotani sector at the percent level.
\end{enumerate}
This is one reason the KK search is so rich conceptually.  The resonance itself is only the entry point.  If it appears, it immediately generates a full spectroscopy program.

\begin{table}[H]
\centering
\caption[Summary of the main collider observables that distinguish the \GG{} KK sector from a minimal.]{Summary of the main collider observables that distinguish the \GG{} KK sector from a minimal single-resonance interpretation. Each row is a way of telling a cluster from a single pole, and none of them requires knowing the rate. That is deliberate: the rate carries the model-level assumptions and the shapes do not, so a search that measures only shapes still discriminates. The rows are ordered by how early in a dataset each becomes usable rather than by how decisive it is.}
\label{tab:observables}
\small
\begin{tabularx}{\textwidth}{>{\raggedright\arraybackslash}p{3.1cm} >{\raggedright\arraybackslash}p{3.1cm} X}
\toprule
\textbf{Observable} & \textbf{What it tests} & \textbf{GG-6 expectation}\\
\midrule
Mass centroid & Lock scale & First spin-2-anchored cluster centered near \(3.849\,\mathrm{TeV}\) (rate carried by the spin-1 octet).\\
Angular distribution & Spin assignment & Spin-2-like shape, possibly mildly distorted by cluster admixture.\\
Line shape & 6D cluster structure & Doublet/triplet possibility within a few hundred GeV rather than a single narrow isolated pole.\\
Branching hierarchy & Overlap integrals & Enhanced \(\ttbar\), dijet, diboson, and \(hh\); suppressed light-lepton and diphoton relative to canonical universal templates.\\
Dijet-to-dilepton ratio & \(\UB\) admixture & Larger than in a pure universal graviton picture.\\
Second radial family & scale-throat \(n=2\) & Additional structure near \(7\,\mathrm{TeV}\); angular \(\theta\) tower stays at \(\EGUT\).\\
KK gluon corridor & Colored partner sector & \(5\) to \(7\,\mathrm{TeV}\), especially promising in \(\ttbar\).\\
\bottomrule
\end{tabularx}
\end{table}


\subsection{Corollary T-KK-2c: the graviton's gluon coupling, derived}
\label{subsec:kappagg}

Theorem T-KK-2 fixed the lock cluster's masses and the vector couplings from boundary-value problems on the rigid geometry, with no free data.  One coupling of the cluster was left as an estimate: the strength with which the spin-2 anchor \(G_1\) talks to a pair of \emph{gluons} --- the coupling that controls its production at a hadron collider and its decay to the flat electroweak modes (\(gg\), \(\gamma\gamma\)).  Read-out~II (\S\ref{sec:scenarioB}) bracketed it as \(\kappaeff\sim1/\sqrt K\).  This corollary closes the gap: the coupling is an overlap integral of exactly the same Bessel type as the vector overlaps of T-KK-2 Step~5, it closes in elementary form, and the result is \emph{not} \(1/\sqrt K\).  We work at the level of detail an analysis team can check line by line.

\paragraph{Step 1 --- what is being computed, and in which units.}  Write the graviton KK decomposition \(h_{\mu\nu}(x,r)=\sum_n h^{(n)}_{\mu\nu}(x)\,\hat\chi_n(r)\) and let the 4D interaction be \(\mathcal{L}\supset -(1/\Lambda_n^{f})\,h^{(n)}_{\mu\nu}T^{\mu\nu}_{f}\) for each 4D field \(f\).  For fields localized at the infrared end of the interval the coupling is the cohort's \(1/\Lambda_\pi\) with \(\Lambda_\pi=\Mlock/x_1\) (the normalization under all the HLZ widths used in \S\ref{sec:operational}).  Define the dimensionless object of interest as the \emph{ratio}
\begin{equation}
\kappa_{gg}\;\equiv\;\frac{\Lambda_\pi}{\Lambda_1^{gg}}
\;=\;\frac{\text{coupling of }G_1\text{ to two gluon zero modes}}{\text{coupling of }G_1\text{ to an IR-localized current}},
\label{eq:kappagg-def}
\end{equation}
so that every graviton partial width to flat modes is the corresponding brane-normalized width multiplied by \(\kappa_{gg}^2\).  Taking a ratio makes the result convention-independent: every normalization choice cancels between numerator and denominator.

\paragraph{Step 2 --- the two measures (the physical core of the calculation).}  Two different radial measures enter, and their mismatch is the entire physics of this corollary.  (i)~The gauge sector: in the canonical metric \(e^{-2r}\eta_{\mu\nu}\dd x^\mu\dd x^\nu+\dd r^2\), the 4D-components of the gauge kinetic term carry \(\sqrt{g}\,g^{\mu\alpha}g^{\nu\beta}=e^{-4r}\cdot e^{2r}\cdot e^{2r}=1\): the gauge field lives on a \emph{flat} \(r\)-measure.  This single fact is why the gluon zero mode is exactly constant, \(f_0(r)=1/\sqrt K\) with \(\int_0^K f_0^2\,\dd r=1\), and why T-KK-2 Step~5 could normalize the vector overlaps with the flat measure.  (ii)~The graviton sector: the Einstein--Hilbert term gives the transverse-traceless modes the weighted norm
\begin{equation}
\int_0^K \dd r\, e^{-2r}\,\hat\chi_m(r)\,\hat\chi_n(r)=\delta_{mn}.
\label{eq:grav-norm}
\end{equation}
The graviton couples to the gauge field through \(h_{\mu\nu}T^{\mu\nu}\), and because the gauge kinetic density is flat in \(r\), the r-profile of \(T^{\mu\nu}_{gg}\) for two zero-mode gluons is simply \(f_0^2=1/K\).  Hence
\begin{equation}
\frac{1}{\Lambda_1^{gg}}\;\propto\;\int_0^K \dd r\,\hat\chi_1(r)\,f_0^2(r)
\;=\;\frac{1}{K}\int_0^K \dd r\,\hat\chi_1(r),
\qquad
\frac{1}{\Lambda_\pi}\;\propto\;\hat\chi_1(K),
\label{eq:kappagg-overlap}
\end{equation}
with one common proportionality constant that cancels in Eq.~\eqref{eq:kappagg-def}.  Note carefully what Eq.~\eqref{eq:kappagg-overlap} is \emph{not}: it is not an orthogonality integral.  The graviton modes are orthogonal among themselves under the \emph{weighted} measure \eqref{eq:grav-norm}; the gauge density probes them with the \emph{flat} measure.  Orthogonality therefore does not force the integral to vanish --- but, as we now see, it very nearly does, and the residue is the coupling.

\paragraph{Step 3 --- the mode function and its normalization, in closed form.}  From T-KK-2 Step~2 the first spin-2 mode is \(\chi_1(r)=e^{2r}\bigl[J_2(m_1e^{r})+b\,Y_2(m_1e^{r})\bigr]\) with \(m_1=x_1e^{-K}\), \(x_1=3.8317\) (first zero of \(J_1\)), and ultraviolet admixture \(b=-J_1(m_1)/Y_1(m_1)\simeq(\pi/4)m_1^2\approx4.6\times10^{-30}\) --- the exponential smallness that T-KK-2 proved is \emph{specific to spin two}, so this corollary carries none of the scenario dichotomy of the vector sector.  Dropping \(b\), the norm \eqref{eq:grav-norm} closes with the Lommel integral \(\int_0^a zJ_2^2(z)\,\dd z=\tfrac{a^2}{2}\bigl[J_2^2(a)-J_1(a)J_3(a)\bigr]\), which at \(a=x_1\) (where \(J_1=0\)) collapses to \(\tfrac{x_1^2}{2}J_2^2(x_1)\):
\begin{equation}
\int_0^K e^{-2r}\chi_1^2\,\dd r
=\frac{1}{m_1^2}\int_{m_1}^{x_1}\!\! z\,J_2^2(z)\,\dd z
=\frac{x_1^2\,J_2^2(x_1)}{2\,m_1^2}
\quad\Longrightarrow\quad
\hat\chi_1(K)=\sqrt{2}\,e^{K}.
\label{eq:chi-norm}
\end{equation}
The infrared endpoint value \(\sqrt2\,e^{K}\) is exact and parameter-free --- and it is the first internal check: the properly normalized graviton zero mode is \(\hat\chi_0\simeq\sqrt2\), so the first KK mode couples to infrared fields \(e^{K}\) times more strongly than the 4D graviton.  That factor is precisely the statement \(\Lambda_\pi=\overline M_{\rm Pl}\,e^{-K}\): the corollary's normalization re-derives the cohort scale chain rather than assuming it.

\paragraph{Step 4 --- the overlap integral, in closed form.}  With the substitution \(z=m_1e^{r}\) the flat-measure integral becomes \(\int_0^K\chi_1\,\dd r=m_1^{-2}\int_{m_1}^{x_1} zJ_2(z)\,\dd z\).  The three-term recurrence \(J_2=(2/z)J_1-J_0\) reduces it to two textbook antiderivatives, \(\int J_1\,\dd z=-J_0\) and \(\int zJ_0\,\dd z=zJ_1\):
\begin{equation}
\int_{m_1}^{x_1}\!\! z\,J_2(z)\,\dd z
=\Bigl[-2J_0(z)-z\,J_1(z)\Bigr]_{m_1}^{x_1}
=\;2-2J_0(x_1)+O(m_1^2)
\;=\;2\bigl(1+J_2(x_1)\bigr),
\label{eq:overlap-int}
\end{equation}
where the last step uses the same identity that anchors T-KK-2: at a zero of \(J_1\), the recurrence \(J_0(x)+J_2(x)=(2/x)J_1(x)\) forces \(J_0(x_1)=-J_2(x_1)\).  Physically, the ``\(2\)'' is the ultraviolet endpoint's contribution and \(2J_2(x_1)\) the infrared one; nothing in between survives, because the oscillating mode integrates itself away everywhere except at its endpoints.

\paragraph{Step 5 --- the result.}  Assembling Eqs.~\eqref{eq:kappagg-def}--\eqref{eq:overlap-int}, all normalizations cancel and every \(K\)-dependence is explicit:
\begin{equation}
\boxed{\;
\kappa_{gg}\;=\;\frac{2\bigl(1+J_2(x_1)\bigr)}{K\,x_1^{2}\,J_2(x_1)}
\;=\;\frac{c_K}{K},
\qquad
c_K=\frac{2(1+J_2(x_1))}{x_1^2 J_2(x_1)}=0.4744,
\qquad
\kappa_{gg}\big|_{K=35}=0.01356.
\;}
\label{eq:kappagg-result}
\end{equation}
The number is a pure Bessel constant divided by the ledger integer: no continuous dial, no fit.  The photon is a flat electroweak zero mode with exactly the same profile as the gluon, so the identical integral gives \(\kappa_\gamma=\kappa_{gg}\); the light fermions remain negligible (their suppression is exponential in localization, not merely \(1/K\)).

\paragraph{Three internal checks.}  (i)~\emph{Orthogonality:} \(\int e^{-2r}\hat\chi_0\hat\chi_1\,\dd r=0\) is confirmed numerically at the \(10^{-15}\) level --- the flat-measure integral \eqref{eq:overlap-int} is not an orthogonality violation but a different-measure projection.  (ii)~\emph{Scale chain:} \(\hat\chi_1(K)=\sqrt2\,e^{K}\) to machine precision, reproducing \(\Lambda_\pi=\overline M_{\rm Pl}e^{-K}\) independently of the cohort derivation.  (iii)~\emph{The vector contrast:} running the same two integrals for the spin-1 mode --- where field and probe share the \emph{same} flat measure --- returns zero to machine precision, exactly the T-KK-2 Step-5 statement that an exactly flat current decouples from the excited vector.  The graviton escapes that fate only through the measure mismatch \(e^{-2r}\) versus flat; the residue is Eq.~\eqref{eq:kappagg-result}.

\paragraph{Physics intuition: why \(c_K/K\) and not \(1/\sqrt K\).}  Seen through the flat gauge measure, the normalized graviton \(\hat\chi_1\) is an exponential wall pressed against the infrared end: \(96\%\) of the integral \(\int\hat\chi_1\dd r\) accumulates in the \emph{last single e-fold} of the throat (\(99.9\%\) in the last two), and the integral equals \(c_K=0.47\) e-folds' worth of its endpoint value.  The gluon zero mode, by contrast, is spread democratically over all \(K=35\) e-folds with amplitude \(1/\sqrt K\) each.  Their overlap is therefore (effective graviton width \(0.47\) e-folds)/(gluon support \(35\) e-folds): \(\kappa_{gg}=0.47/35\).  The Read-out-II estimate \(1/\sqrt K\) was the gluon's \emph{amplitude at a point}; but a coupling is an \emph{integral}, and near-orthogonality cancels all but the final half e-fold.  In the holographic reading the same statement is: \(G_1\) is a composite of the infrared sector, the gluon is an elementary probe, and their mixing costs one power of the RG-throat depth \(\log(\overline M_{\rm Pl}/\Mlock)=K\) --- the warped literature's known suppression \citep{AgasheEtAl2007}, now with its \(O(1)\) coefficient computed.  The infrared channels (\(t\bar t\), \(W_LW_L\), \(Z_LZ_L\), \(hh\)) do not suffer this: they live where the graviton lives, and keep their \(O(1)\) couplings.

\paragraph{Consequences (all at the D\(^{*}\) grade of T-KK-2).}  Feeding \(\kappa_{gg}=0.01356\) into the workbook of \S\ref{sec:operational}:
\begin{itemize}
\item \(\sigma(pp\to G_1)=0.066\) fb at \(\sqrt s=13\)~TeV (LO\(\times\)K; 0.11 at 14~TeV), with \(\Gamma/m=6.3\%\) and branching pattern \(t\bar t:W_LW_L:Z_LZ_L:hh\simeq0.69:0.15:0.08:0.08\); \(\sigma\times\mathrm{BR}(\ttbar)=0.045\)~fb.
\item The strongest projected HL-LHC \(\ttbar\) limit at this mass (\(\sim\!1.1\)~fb at 3~ab\(^{-1}\)) sits a factor \(\sim\!25\) \emph{above} this line: \textbf{the graviton is invisible at any LHC luminosity}.  The spin-2 anchor becomes a first-rate FCC-hh target instead: at \(\sqrt s=100\)~TeV the gluon luminosity at \(\tau=0.0015\) delivers \(\sigma(G_1)\approx2.1\times10^{2}\)~fb --- \(t\bar t\) 142~fb, \(ZZ\) 16~fb, \(hh\) 16~fb --- millions of events at the design luminosity \citep{FCC2018}.
\item \(\sigma\times\mathrm{BR}(\gamma\gamma)=1.1\times10^{-5}\)~fb: the CMS diphoton null \citep{CMS2024Diphoton} is \emph{retro-dicted} with a margin of \(\sim\!3\times10^{3}\).  This check has teeth: had the photon coupled at the uniform \(1/\sqrt K\) amplitude, the prediction (0.14~fb) would already be excluded by a factor \(\sim\!4\).  The published diphoton limit thus discriminates between the two readings of the flat-mode amplitude and selects the derived one --- the first data-confronted \emph{amplitude} test of the collider chain, and it lands on the geometry's side.
\item The Scenario-B cluster sum in \(\ttbar\) becomes octet-dominated: \(8.7\) fb from \(g^{*}\), \(+0.045\) from \(G_1\), \(+0.030\times(g_B/0.2)^2\) from \(Z'_B\) --- the octet carries \(\ge\!92\%\) of the visible rate on every L-KK-1 reading of \(g_B\) (\S\ref{subsec:gbmenu}).  The three contributions add incoherently: the octet--singlet cross-term vanishes by the color trace, spin orthogonality removes the graviton cross-terms, and the only coherent partner of the octet is QCD itself (Part~\ref{part:ttanatomy}).
\end{itemize}

\paragraph{Requalification of F-KK-1 (two tiers).}  The pre-declared spin-2 falsifier must be restated honestly, and the restatement makes the program sharper, not weaker.  \emph{Tier 1 (HL-LHC, decisive):} the falsifier of the lock-scale implementation is carried by the cluster through the un-suppressible color octet --- F-KK-2B: a null in the cluster-aware \(\ttbar\)/dijet combination within \(3.85\pm0.10\)~TeV at \(\lesssim\!2\)~ab\(^{-1}\) retires the current implementation, exactly as quantified in \S\ref{subsec:killwindow}.  \emph{Tier 2 (FCC-hh, confirmatory):} the direct spin-2 observation --- the measurement that would name the geometry --- is a 100-TeV task, with a rate (\(\sim\!205\)~fb) that makes it an early-running discovery there.  The division of labor is itself a falsifiable prediction: the HL-LHC must find (or kill) the spin-1 cluster at \(3.85\)~TeV, and a 100-TeV machine must then find the spin-2 anchor at the \emph{same} mass with \(\Gamma/m=6.3\%\) and the \(0.69{:}0.15{:}0.08{:}0.08\) infrared branching pattern.  Finding either without the other falsifies the cluster reading.

\paragraph{Shelving.}  Theorem-machinery level: the measures, the mode functions, the closed forms \eqref{eq:chi-norm}--\eqref{eq:kappagg-result}, the identity \(\kappa_\gamma=\kappa_{gg}\), and the three checks.  Derived-conditional: the free quadratic action on the rigid geometry (the same fence as T-KK-2, with interaction corrections \(O(1/K)\approx3\%\) on the amplitude); the \(t_R\) infrared fraction entering the branching pattern (a cascade-depth sensitivity of \(\sim\!25\%\) on \(\mathrm{BR}(\ttbar)\), \S\ref{sec:operational}).  Grafted: nothing --- this is the point of the corollary.


\subsection{The baryonic normalization \texorpdfstring{$g_B$}{gB}: a theorem-grade norm, one discrete residual rule (L-KK-1)}
\label{subsec:gbmenu}

After Corollary T-KK-2c one number remained underived anywhere in the collider chain: the \(\UB\) gauge normalization \(g_B\).  This subsection completes the account to its honest boundary.  The result is sharper than ``open'' and weaker than ``derived'': the \emph{norm} of the baryonic generator is already a theorem of the cohort, and what remains is not a continuous dial but a single discrete rule, which we register as the named open problem \textbf{L-KK-1} and hand, with a four-line target instead of a plane, to the fifth-force paper of the Predictions branch.

\paragraph{What is already a theorem.}  The integrality of the six-dimensional anomaly polynomial \(X_4\) (\citealp{ArisakaA3BI6}), consumed by \citet{ArisakaA9SM} through the Arisaka-equation lock, fixes the trace-norm ratio of the split generator pair \(Y_0\oplus B_0\):
\begin{equation}
\frac{\Tr(B_0^2)}{\Tr(Y_0^2)}\;=\;\frac{L-N}{L}\;=\;\frac{2}{5}
\qquad\text{for }(N,L)=(3,5),
\label{eq:gb-norm}
\end{equation}
the same \(2/5\) that appears in the \BIsix{} vector \((1,-1,-2/5,-2/5)\) and in the Arisaka constant \(Q_B\).  The reader can verify it on the primitive \(\bar{\mathbf 5}\) in one line: the \(d^c\) triplet carries \(B=-\tfrac13\), \(Y=+\tfrac13\); the lepton doublet carries \(B=0\), \(Y=-\tfrac12\); hence \(\Tr(B_0^2)=3\cdot\tfrac19=\tfrac13\) against \(\Tr(Y_0^2)=3\cdot\tfrac19+2\cdot\tfrac14=\tfrac56\), and \(\tfrac13/\tfrac56=\tfrac25\).  The physics is transparent: baryon number is carried by three of the five states of the primitive cell, at charge \(\tfrac13\) --- the \(2/5\) is the fraction of the cell's ledger stiffness that the baryonic seam owns.  The \(B\)-seam's weight in the locked ledger is therefore \emph{not} a free parameter of \GG{}: whatever \(g_B\) is, it is this norm read out through a rule.

\paragraph{What is genuinely open --- and why it is discrete, not continuous.}  \GG{} converts seam norms into physical couplings through counting theorems.  For the photon the theorem exists: the unique unbroken Abelian seam refines dyadically to the \OGNfive{} closure depth \(J=7\), giving \(\alpha^{-1}(M_Z)=2^J=128\) (\citealp{ArisakaA9SM}, dyadic-closure theorem).  That theorem does \emph{not} transfer to \(\UB\), for a structural reason: the baryonic seam never reaches the infrared attractor.  It terminates early by eating the \BIsix{} tensor --- the Stueckelberg mechanism that makes the fifth-force boson massive and keeps global baryon number exact --- so its capacity is frozen at the Green--Schwarz stratum, whose closure-depth readout no A1--A9 theorem supplies.  \textbf{L-KK-1 (the \(B\)-seam termination rule):} \emph{state and prove the capacity readout for a Stueckelberg-terminated Abelian seam.}  Pending that rule, the locked norm \eqref{eq:gb-norm} admits four natural readouts --- the norm entering linearly or quadratically, on either side of the capacity identification:
\begin{equation}
\frac{\alpha_B}{\alpha_Y}\;\in\;
\Bigl\{\bigl(\tfrac25\bigr)^2,\;\tfrac25,\;\tfrac52,\;\bigl(\tfrac52\bigr)^2\Bigr\}
\quad\Longrightarrow\quad
g_B\;\in\;\{0.14,\;0.23,\;0.57,\;0.89\},
\label{eq:gb-menu}
\end{equation}
where \(g_Y=0.357\) follows from \GG{}'s own electroweak values \(\alpha^{-1}(M_Z)=2^J=128\) and \(\sin^2\theta_W=25/108\) (\(\alpha_Y=\alpha/\cos^2\theta_W\); \(\U(1)\) running to \(\Mlock\) shifts these by a few per cent and is immaterial below).  We emphasize the grading: Eq.~\eqref{eq:gb-norm} is theorem-grade; the menu \eqref{eq:gb-menu} is a \emph{bracket of discrete candidates}, not a derivation, and L-KK-1 could in principle select outside it only by invoking structure beyond the locked norm --- which the fifth-force paper would then have to exhibit explicitly.

\paragraph{The collider conclusions are menu-invariant.}  This is the payoff for the present paper.  Across the entire menu \eqref{eq:gb-menu}, the KK member \(Z'_B\) (whose collider couplings carry the additional T-KK-2 overlap factors) delivers, at \(3.849\)~TeV and \(\sqrt s=13\)~TeV,
\[
\sigma\times\mathrm{BR}(Z'_B\to\ttbar)\;=\;\{0.012,\;0.029,\;0.18,\;0.46\}\ \mathrm{fb},
\qquad
\Gamma/m\;=\;\{0.002,\;0.005,\;0.032,\;0.080\},
\]
i.e.\ every candidate sits a factor \(\ge\!10\) below the current narrow-template limit and a factor \(\ge\!2.3\) below even the 3~ab\(^{-1}\) HL-LHC projection (the top entry, \(g_B=0.89\), is the only one with marginal long-term exposure: 0.71~fb at 14~TeV against a projected \(\sim\!1.1\)~fb).  The \(Z'_B\) contribution to the Scenario-B cluster sum spans 0.2--6.8\%.  Three conclusions are therefore \emph{independent of the L-KK-1 selection}: the \(Z'_B\) is LHC-invisible as a separate signal; the color octet remains the sole carrier of the HL-LHC falsifier; and the \(Z'_B\)'s own discovery sector is not the collider but the fifth force, where the same \(g_B\) couples the \emph{unsuppressed} Stueckelberg zero mode to baryon number.  There, L-KK-1 changes the game: the \((g_B,m_B)\) exclusion problem collapses from a two-parameter plane to \emph{four candidate lines}, and the fifth-force paper's task becomes a selection, not a scan.

\paragraph{Shelving.}  Theorem level: the norm \eqref{eq:gb-norm} and its \(\bar{\mathbf 5}\) verification.  Derived-conditional: the menu-invariance statements above (they hold across the stated bracket).  Open, named: L-KK-1, the seam-termination rule --- the single remaining discrete unknown of the lock cluster's couplings, successor to the scenario dichotomy that T-KK-2 resolved and the flat-overlap question that T-KK-2c closed.

Part~II delivered the paper's centerpiece, in plain terms: the geometry's cavity has a lowest note, and that note is at \(3.849\)~TeV --- not roughly, not up to a dial, but as a closed formula in the Planck mass and the integer 35, with the overtone series (7.047 and 10.219~TeV) fixed by the same mathematics.  The fine structure came with it: a spin-2 anchor, an exactly degenerate pair of vectors, couplings with their signs, and a two-way fork (free-field versus composite) that existing data have already resolved --- the 2.46-TeV horn is dead, and the degenerate cluster survives.  A mass and a spectrum, however, are not yet an experiment.  The next Part asks the machine's questions: how often is this thing made, and in what channels would it show itself?


\section[Part~II in plain English]{Part~II in plain English: one integer, one mass, and a cluster rather than a peak}
The number this Part produces is about 3.85 TeV, and where it comes
from matters more than what it is.

It is not fitted to anything. The lock scale is half the Planck mass times
the exponential of minus thirty-five, and the thirty-five is an integer in
a ledger the cohort fixed for the particle sector years before any collider
question was asked. No coupling was tuned, no radius chosen, no scan run.
The same ledger, with no further input, also reproduces the electroweak
scale --- two known numbers from one integer, which is the kind of
over-constraint that either holds or collapses.

The second thing the Part delivers is that the prediction is a cluster and
not a peak. Three members sit near the same scale: a spin-2 state, a
color-octet vector, and a leptophobic partner. Their positions relative to
one another are ratios of Bessel zeros, so the pattern is fixed even where
the overall scale might shift.

That distinction runs through the rest of the paper. A search designed for
a single resonance is not the search this predicts, and an exclusion
computed against a single-resonance benchmark does not transfer. What the
theory forbids is a particular arrangement of states, and it is the
arrangement that has to be looked for.

\GGPart[Production and discovery]{Production and Discovery}{Cross sections, event yields, and the channel hierarchy}

The mass of Part~II is a statement about nature; this Part turns it into a statement about a machine.  The questions are the concrete ones an accelerator forces --- how often, in which channels, over what background --- and the answers, because the couplings are derived, arrive as numbers rather than ranges.


\section{Production Mechanisms at Hadron Colliders}
\label{sec:production}

\subsection{General narrow-width bookkeeping}

At hadron colliders it is helpful to keep the production discussion in a language that is as benchmark-independent as possible.  For a resonance \(X\) with spin \(J\), mass \(m_X\), total width \(\Gamma_X\), and partial widths \(\Gamma_{ij}\) and \(\Gamma_f\) to initial and final states, the narrow-width approximation gives the schematic hadronic rate
\begin{equation}
\sigma(pp\to X\to f)
\simeq
\frac{2J+1}{m_X s}\,\frac{\Gamma_f}{\Gamma_X}
\sum_{ij} C_{ij}\!\left(\frac{m_X^2}{s}\right)\Gamma_{ij},
\label{eq:nwa}
\end{equation}
where \(C_{ij}(\tau)\) is the parton luminosity function for initial state \(ij\).  This formula makes three facts transparent.  First, production is controlled not only by the hard coupling but also by parton luminosities.  Second, branching fractions matter as much as raw production.  Third, any mechanism that redistributes couplings between light fermions, gluons, tops, or Higgs bosons can qualitatively reorder the discovery channels.

It is useful to factor the rate into the one genuinely model-dependent number and everything else.  Writing the resonance--parton coupling as an overall effective strength \(\kappaeff\) (normalized so that \(\kappaeff=1\) reproduces a standard bulk-graviton benchmark with \(k/\bar M_{\mathrm{Pl}}=1\)), the narrow-width rate factorizes as
\begin{equation}
\sigma(pp\to X\to f)\;\simeq\;\kappaeff^{2}\,\underbrace{\frac{2J+1}{m_X s}\sum_{ij}C_{ij}\!\left(\tfrac{m_X^2}{s}\right)\hat\Gamma_{ij}}_{\text{kinematics }+\text{ parton luminosity (fixed)}}\;\times\;\mathrm{BR}(X\to f),
\label{eq:kappafact}
\end{equation}
where \(\hat\Gamma_{ij}=\Gamma_{ij}/\kappaeff^2\) is the coupling-stripped partonic width.  Of the factors on the right, the mass \(m_X=\Mlock\) is \emph{derived} (Theorem~T-KK-1), the parton luminosities \(C_{ij}\) are standard-model inputs, and the branching structure follows the overlap pattern of \S\ref{subsec:branchings}.  The one remaining factor, \(\kappaeff\), is the effective coupling, and \emph{it is not fixed by the GG-A theorems}: it is set by the bulk-to-boundary overlaps --- the per-fermion cascade depths that \citet{ArisakaA9SM} records as an open lemma.  The single exception is the color octet's coupling to light quarks, the universal value \(|\xi_{\rm light}|=1/\sqrt{K}\approx0.17\) fixed by \(K=35\) (\S\ref{subsec:widths}).



\subsection{Why no absolute cross section is quoted}
\label{subsec:noxsec}

This paper is held to a strict discipline: every quantitative statement must trace to the GG-A series.  An absolute production cross section does \emph{not}.  Through Eq.~\eqref{eq:kappafact} the rate requires (i)~the resonance--parton couplings --- of which the octet's \(\xi_{\rm UV}\) (T-KK-2) and the graviton's \(\kappa_{gg}\) (Corollary T-KK-2c) are now derived, while the per-fermion cascade depths and the discrete L-KK-1 readout for \(g_B\) (\S\ref{subsec:gbmenu}) remain open --- and (ii)~a numerical QCD/PDF integration of the parton luminosities, which is external to any theorem.  Neither is supplied by a GG-A theorem.  We therefore quote \emph{no} benchmark cross section (in fb) and \emph{no} event yield: such numbers would not be derived quantities, and would only reintroduce the model-level input the discipline forbids.

What the GG-A series \emph{does} fix is the structure any such computation must respect, and that structure is the genuine content of this section: the mass scale \(\Mlock\) \eqref{eq:tkk1-mlock}; the exact leptophobia \(\mathrm{BR}(Z'_B\to\ell^+\ell^-)=0\) \eqref{eq:zprime-leptophobic}; the color-only decays of \(g^*\); the top-rich channel ordering (\S\ref{subsec:branchings}); the production master formula \eqref{eq:kappafact}; and the one coupling that \emph{is} fixed, the color octet's light-quark coupling \(\xi_{\rm UV}=-0.194\) (exact mode, Eq.~\ref{eq:tkk2-xiflat}), which (being un-suppressible) makes the color octet the cluster's working edge in either T-KK-2 scenario (\S\ref{subsec:tkk2}, F-KK-2A/2B).  Turning these inputs into absolute rates --- a dedicated \textsc{MadGraph}/PDF computation once \(\kappaeff\) is determined --- is the natural next step; it is carried out, with every external assumption labeled and the result graded Derived-conditional, in the operational confrontation of \S\ref{sec:operational}.  The present Part remains theorem-anchored.

\subsection{Quark-antiquark annihilation}

The cleanest textbook mode is Drell--Yan-like quark antiquark production,
\begin{equation}
q\bar q \to G_1 \to \llbar,\ \gamma\gamma,\ WW,\ ZZ,\ hh.
\label{eq:qqprod}
\end{equation}
This mode is experimentally attractive because lepton and photon final states carry low backgrounds and good invariant-mass resolution.  However, \GG{} predicts that light-fermion couplings need not be maximal.  Therefore the cleanest channel is not automatically the strongest one.

\subsection{Gluon fusion}

For an IR-sensitive graviton-like state, gluon fusion is often highly competitive,
\begin{equation}
gg \to G_1 \to gg,\ \ttbar,\ WW_L,\ ZZ_L,\ hh.
\label{eq:ggprod}
\end{equation}
The attractive feature of the gluon mode is obvious: the gluon parton distribution remains significant even in the high-mass region.  The problematic feature is equally obvious: gluonic and hadronic final states face larger QCD backgrounds.  For \GG{}, the practical consequence is that an apparently weaker but cleaner leptonic search and a more background-challenging but more strongly coupled hadronic search can have comparable ultimate discovery power.

\subsection{Vector-boson fusion and associated production}

Two additional channels deserve mention.  Vector-boson fusion,
\begin{equation}
qq\to qqG_1\to qq+(WW/ZZ/hh),
\label{eq:vbfprod}
\end{equation}
is subdominant at present LHC energies but becomes increasingly useful as the mass scale rises, particularly in future hadron colliders.  Associated production,
\begin{equation}
q\bar q'\to G_1 W/Z,
\qquad gg\to G_1 g,
\label{eq:assocprod}
\end{equation}
can also contribute to acceptance in specialized analyses or to partner-state searches, though it is not the main discovery mode near the first lock scale.

\subsection{A realistic channel hierarchy for GG-6}

Putting the geometry and the parton structure together leads to a practical channel ranking for the first decade of the search.  The clean channels (dileptons, diphotons) remain indispensable because they can establish the existence of a high-mass object with minimal background ambiguity.  But the more geometry-faithful channels are likely \(\ttbar\), boosted dijet, diboson, and Higgs-pair channels.  That ranking is not arbitrary; it is the direct consequence of overlap integrals and the mild baryonic admixture.

It is therefore a mistake to ask only whether the first lock-scale cluster has already been excluded in canonical RS-style dilepton or diphoton plots.  One must ask instead whether the full multi-channel \GG{} corridor has been excluded by a cluster-aware reinterpretation.  As of early 2026, the answer to that stronger question is still no.

\subsection{Event-yield bookkeeping}

For a rough event-count discussion it is enough to write
\begin{equation}
N_S = \mathcal L\,\sigma(pp\to G_1)\,\mathrm{BR}(G_1\to f)\,\varepsilon_f,
\label{eq:Ns}
\end{equation}
where \(\varepsilon_f\) is the channel efficiency.  The dependence on luminosity is linear in events and approximately \(\sqrt{\mathcal L}\) in significance when the search is background-dominated.  Because the clean leptonic branching fractions may be suppressed in \GG{}, a practical search will often win more by improving \(\varepsilon_f\) in \(\ttbar\), dijet, and \(hh\) than by only waiting for more luminosity in leptons.  This simple observation motivates the trigger proposals made later.


\section{Channel-by-Channel Discovery Logic}
\label{sec:channel-by-channel}

The previous sections established the main geometric picture.  The next practical question is more experimental: \emph{if a \GG{} first cluster exists, how will it actually reveal itself in each of the standard resonance channels?}  This question deserves a dedicated section because one of the recurring misconceptions in the extra-dimensional literature is that a heavy graviton can be characterized adequately by one or two clean channels alone.  In \GG{}, that is too narrow.  The model is non-universal by design, clustered in line shape, and correlated with partner states.  It therefore asks for a channel-by-channel logic rather than a single flagship plot.

\subsection{Dileptons: the clean but not necessarily dominant channel}

The dilepton channel remains the cleanest spectroscopic tool because electrons and muons give excellent mass resolution and low irreducible background.  If a \GG{} first cluster has even a modest branching fraction to \(e^+e^-\) or \(\mu^+\mu^-\), ATLAS and CMS can reconstruct a crisp high-mass structure and immediately test the spin assignment through angular information.

But the same cleanliness can become a conceptual trap.  In many search traditions, the cleanest channel is quietly promoted to the main physical benchmark.  For \GG{}, this would be a mistake.  The geometry points toward suppressed light-fermion overlaps, so dileptons should be treated as a \emph{confirmation} channel or precision channel, not automatically as the dominant discovery engine.  In an unlucky but plausible corner of the \GG{} corridor, a first cluster could be visible first in \(\ttbar\) or dijets and only later in dileptons after a category-informed or signal-informed combination is performed.

A practical analysis consequence follows.  The experiments should avoid reporting only a one-dimensional ``mass limit versus universal-coupling RS graviton'' curve when discussing the relevance of high-mass dilepton results to models such as \GG{}.  What is needed instead is a reinterpretation grid in which the dilepton branching fraction is allowed to vary independently from the gross total production rate.  That is a small additional publication burden with a large gain in downstream reusability.

\subsection{Diphotons: still important, but theoretically disfavored as the sole handle}

Diphotons sit in a similar place.  They are experimentally elegant and statistically powerful because of excellent electromagnetic calorimetry and simple final-state topology.  A genuine diphoton peak near the first cluster would be persuasive.  However, the same bulk-localization logic that suppresses some light-fermion couplings can also suppress or modify the effective photon branch relative to a universal graviton template.  Therefore the absence of a diphoton excess by itself should not be over-read as a no-go theorem for the whole \GG{} benchmark.

There is also a second reason to avoid over-weighting diphotons.  In a clustered 6D signal, the experimental line need not be perfectly narrow.  A partially resolved or mildly broadened structure will always be easier to interpret in a channel where companion information exists.  Diphotons provide less internal event-shape structure than, say, \(\ttbar\) or \(hh\).  So they are excellent for confirmation, but not always for full model discrimination.

\subsection{Top-pair resonances: the central battleground}

If one had to nominate one channel as the natural home base of the \GG{} search, it would be \(\ttbar\).  The logic is straightforward.  The same overlap integrals that suppress light fermions favor heavy IR-sensitive modes.  The top sector is the most obvious beneficiary.  In addition, \(\ttbar\) is exactly the place where several neighboring states in the \GG{} spectrum can naturally appear: the first spin-2 cluster, the KK-gluon corridor, and possibly even vector-like top partners in associated or cascade production.

This makes \(\ttbar\) special in two complementary senses.  It is both a discovery channel and a diagnostic channel.  A heavy spin-2 state and a heavy spin-1 state may both produce top pairs, but they do so with different angular and kinematic structures.  If a first excess appears there, the full discriminating power of the channel can be brought to bear: lepton+jets, dilepton, fully hadronic top-tagged categories, top polarization information, and broad-versus-clustered line-shape comparisons.  Part~\ref{part:ttanatomy} derives all of these quantitatively from the theorem-fixed couplings: the \((1+\cos^{2}\theta^{*})\) versus \((1-\cos^{4}\theta^{*})\) production angles, the \(+0.91\)-versus-\(0\) top polarization, the exact zero of the resonant forward--backward asymmetry, and the interference line shape whose constructive-below/destructive-above ordering is fixed by the derived sign \(\xi_{\rm UV}<0\).

One can even push the intuition a step further.  In a world where \GG{} is true, the experiments should not think of \(\ttbar\) as one among several channels; they should think of it as the channel where the spectral neighborhood around \(3.85\,\mathrm{TeV}\) is likely to look most alive.  That is because \(\ttbar\) is simultaneously sensitive to the graviton-like cluster, to colored partners, and to the IR structure that makes \GG{} different from a universal RS benchmark.

\subsection{Dijets: where the mild leptophobic side of GG-6 becomes visible}

Dijets are experimentally difficult because QCD is enormous, but they are theoretically indispensable because \GG{} contains an exact baryonic gauge factor.  Even a small leptophobic admixture can noticeably alter the dijet-to-dilepton ratio.  This is why we repeatedly emphasize that a null dilepton result does not automatically settle the matter.  A baryon-biased 6D signal can hide where the QCD background is largest.

The modern experimental response to this difficulty is already visible in public work: reduced-format high-rate streams, topological and substructure-based filters, and model-agnostic anomaly searches.  This is very good news for \GG{} because the signal does not need the experiments to invent an entirely new strategy.  It only asks them to push a strategy they are already developing into the specific 3--6 TeV region with a cluster-aware interpretation layer.

Dijet searches also matter for a second reason: they can help resolve whether a visible feature is graviton-dominated, \(\UB\)-vector-dominated, or mixed.  A high-mass resonance that is anomalously strong in dijets but weak in dileptons could be dismissed as ``just another leptophobic vector'' in a generic model survey.  In the \GG{} context, the same pattern would instead be evidence of the exact baryonic sector that the framework insists upon.  The experiments do not need to believe \GG{} to make such a distinction useful.  They only need to release enough information to allow the distinction to be tested.

\subsection{Dibosons: the geometry-to-Higgs bridge}

The diboson channels \(WW\) and \(ZZ\) are important because they connect the spin-2 and scalar sectors.  A genuine graviton-like cluster should couple appreciably to longitudinal electroweak bosons if its origin is deeply tied to the same infrared scale that sets the Higgs-sector ledger.  This is one reason why bulk-RS graviton analyses often place \(WW\) and \(ZZ\) side by side with \(\ttbar\).  The same reasoning applies even more strongly in \GG{}.

Diboson final states also provide a more geometry-native handle on partner identification.  The spin-2 graviton and the baryonic vector populate \(WW\) and \(ZZ\), so a diboson handle can be conspicuous even when a single-channel tensor search is still statistically marginal.  In that sense the diboson program is a bridge between the first direct signal and the broader spectroscopy story.  It should therefore be kept tightly coupled to any dedicated first-cluster search.

\subsection{Higgs pairs: the best scalar corroborator}

The \(hh\) channel plays a distinctive role.  It is not merely another bosonic branch: the spin-2 graviton \(G_1\) couples to the IR-localized Higgs sector, so a \(G_1\to hh\) signal is a direct probe of the infrared.  (There is no radion in \GG{} --- the radial-interval length is the fixed integer \(K\), not a modulus \citep{ArisakaA4Euclid} --- so the \(hh\) channel is fed by the graviton itself, not by a scalar partner.)  A first lock-scale cluster seen in \(\ttbar\) and dijets but with no accompanying \(hh\) activity would already be informative; conversely, a suggestive \(hh\) excess at \(3.85\,\mathrm{TeV}\), even before a decisive spin-2 observation in \(\ttbar\), would raise the prior plausibility of the full \GG{} pattern.

The experimental challenge is of course formidable: \(hh\) is rate-limited, final states are complex, and the best channels depend on a trade-off between cleanliness and branching ratio.  Yet this very difficulty is why the channel is so valuable.  A correlated \(3.85\,\mathrm{TeV}\) tensor-like signal and a \(1\) to \(3\,\mathrm{TeV}\) scalar partner in \(hh\) would be much harder to dismiss as an accident than either one alone.

\subsection{The channel-combination principle}

A compact way to summarize the whole section is this: the correct \GG{} search object is not a single channel but a \emph{channel vector}.  Let
\begin{equation}
\bm{\mu} = (\mu_{\ttbar},\mu_{jj},\mu_{hh},\mu_{VV},\mu_{\llbar},\mu_{\gamma\gamma}),
\end{equation}
where each component represents the signal strength relative to a chosen reference template.  A discovery claim should then not be thought of as ``one component significantly nonzero,'' but rather as ``a statistically coherent nonzero vector with the ordering and morphology expected from the 6D geometry.''  This is the Bayesian and \GDOS{}-friendly way to state what the experiments are really doing once they combine categories and channels.

\begin{table}[H]
\centering
\caption[Channel-by-channel logic of the first \GG{} cluster.]{Channel-by-channel logic of the first \GG{} cluster.  The table is not merely descriptive; it is a ranking of how each channel contributes to discovery, diagnosis, or corroboration. Discovery, diagnosis and corroboration are three different jobs and the table keeps them apart. A channel that would establish an excess is not necessarily one that would tell you what produced it, and a channel good for neither may still be the one that rules out the alternatives. Read down the ranking, the first cluster is discovered in tops and diagnosed everywhere else.}
\label{tab:channel-logic}
\small
\begin{tabularx}{\textwidth}{>{\raggedright\arraybackslash}p{2.7cm} >{\raggedright\arraybackslash}p{3.0cm} >{\raggedright\arraybackslash}p{3.4cm} X}
\toprule
\textbf{Channel} & \textbf{Primary experimental virtue} & \textbf{Primary \GG{} role} & \textbf{Main caveat}\\
\midrule
Dileptons & Highest cleanliness and mass resolution & Confirmation, spin analysis, clean corroboration & Light-fermion suppression can make this subdominant.\\
Diphotons & Very clean final state, excellent EM resolution & Clean corroboration if branch is large enough & Photon branch may be reduced relative to canonical RS.\\
\(\ttbar\) & Natural home of top-rich overlap structure & Likely discovery channel and best spectroscopic battleground & Complex reconstruction and larger systematics.\\
Dijets & Large branching, sensitive to leptophobic admixture & Tests \(\UB\) side and multi-field mixings & Severe QCD background; needs reduced-stream strategy.\\
Dibosons & Strong link to infrared electroweak sector & Bridge between tensor and scalar sectors & Lower rates and reconstruction complexity.\\
\(hh\) & Unique scalar-sector corroboration & Best graviton-\(hh\) / Higgs-sector cross-check & Rate-limited and topology-dependent.\\
\bottomrule
\end{tabularx}
\end{table}


Part~III, summarized plainly: the cluster is made at computable rates --- the color octet at about nine femtobarns at the current LHC, the graviton anchor at a few hundredths --- and it announces itself in top-quark pairs and boosted jets, not in the clean electron and photon channels the field instinctively watches.  The yields, the channel hierarchy, and the reasons behind them all descend from where each particle lives along the scale direction.  Before trusting those statements at face value, a careful reader will want two reassurances: that this construction is genuinely different from the warped models it superficially resembles, and that the act of detection itself has a principled description here.  Both are the business of the next Part.


\section[Part~III in plain English]{Part~III in plain English: turning a mass into something a detector could actually record}
A mass is not yet an experiment. This Part does the conversion: how the
cluster would be produced, how often, and in which decay channels it would
show.

Two production routes matter, and they belong to different members. The
spin-2 anchor is made mainly from gluons; the color-octet is made from
quarks and gluons both, and far more copiously. The parton luminosities
that set the rates are somebody else's measurement, quoted here rather than
derived, and they are printed so a reader can redo the arithmetic.

The channel ranking is the Part's practical output, and it separates three
jobs a channel can do. One channel is where an excess would first appear.
A different one is where you would learn what produced it. A third might do
neither and still be the one that excludes the alternatives. For this
cluster the first job belongs to top-antitop, which is where the paper
concentrates from here on.

None of this is a prediction of new physics in the ordinary sense. It is
arithmetic on a mass that was fixed two Parts ago, and its only freedom is
in inputs that come from published measurements.

\GGPart[GG-6 vs Randall--Sundrum]{GG-6 versus Randall--Sundrum, and Detection as an Open System}{What is distinctive, and why collider detection is a GDOS problem}

\section{How GG-6 Differs from Ordinary KK and RS Benchmarks}
\label{sec:compare-rs}

\subsection{The ordinary KK and RS reference points}

Ordinary KK theory and RS theory are the natural reference models because \GG{} borrows essential intuition from both.  KK theory provides the logic of harmonic decomposition on a compact internal space \citep{Kaluza1921,Klein1926}.  RS theory provides the logic of warp-generated hierarchy and infrared-localized spectra \citep{RandallSundrum1999}.  Experimental searches at ATLAS and CMS are therefore often formulated in terms of RS or bulk-RS benchmark gravitons.  That is sensible.  But it is not the last word for \GG{}.

\subsection{Five differences that matter experimentally}

There are at least five experimentally relevant differences between \GG{} and a minimal RS-style benchmark.

\paragraph{1. Fixed centroid scale rather than a scanned mass parameter.}
In standard phenomenology the resonance mass is usually scanned.  In \GG{}, the first cluster centroid is tied to \(K=35\).  This is not merely a tighter prior; it changes the meaning of a null result.

\paragraph{2. Clustered (multi-field) line shape rather than a single 5D pole.}
The lowest radial modes of several distinct bulk fields --- graviton, KK gluon, and the \(\UB\) vector --- redshift to the same infrared endpoint, so the first visible family is a near-degenerate multiplet rather than one isolated pole.  (The angular \(S^1_\theta\) tower, by contrast, sits at \(\EGUT\) and contributes nothing here.)  A dedicated cluster fit is therefore better motivated than a single-pole template.

\paragraph{3. Exact \texorpdfstring{$\UB$}{U(1)B} sector.}
A small baryonic admixture enhances dijets and suppresses dileptons relative to a pure universal graviton model.

\paragraph{4. Non-universal overlap pattern.}
The top and Higgs sectors are more IR-sensitive, so \(\ttbar\), diboson, and \(hh\) channels become especially important.

\paragraph{5. Correlated partner states are expected, not optional.}
The KK gluon and \(\UB\) vector form a coherent spectroscopy program.

\begin{table}[H]
\centering
\caption[Minimal comparison between a textbook 5D RS search picture and the 6D \GG{} picture relevant.]{Minimal comparison between a textbook 5D RS search picture and the 6D \GG{} picture relevant to ATLAS and CMS. The comparison is minimal on purpose: it lists only what changes for an experimentalist writing a search, not what changes for a theorist writing a Lagrangian. Two of the rows have no consequence for an event selection and are included so that the ones that do can be seen to be few. The full framework comparison is later in the paper and is a different table with a different job.}
\label{tab:rs-vs-gg6}
\small
\begin{tabularx}{\textwidth}{>{\raggedright\arraybackslash}p{3.2cm} >{\raggedright\arraybackslash}p{5.0cm} X}
\toprule
\textbf{Feature} & \textbf{5D RS-style expectation} & \textbf{GG-6 expectation}\\
\midrule
Mass input & Free benchmark mass scan & First cluster centroid fixed by \(\Mlock\simeq3.849\,\mathrm{TeV}\).\\
Line shape & Single narrow or moderately broad pole & Clustered doublet/triplet possibility in the first visible family.\\
Internal dimensions & One spatial warped direction & Two \emph{non-spatial} directions: an RG-scale axis \(I_r\) plus a compact phase circle \(S^1_\theta\).\\
Couplings & Often treated as nearly universal or benchmark-universal & Non-universal overlaps; top-rich, light-lepton-suppressed, mildly leptophobic admixture possible.\\
Baryonic sector & Usually absent as an exact extra gauge factor & Exact \(\UB\) sector can modify branching and partner-state pattern.\\
Partner states & Radion and colored partners model-dependent & KK gluon and \(\UB\) vector expected in a correlated ledger.\\
Search strategy & Single-channel high-mass resonance templates often adequate & Multi-channel, cluster-aware, top-aware strategy preferred.\\
\bottomrule
\end{tabularx}
\end{table}



\subsection{Generic KK, 5D RS, and GG-6: the comparison that matters experimentally}

The phrase ``KK particle'' is often used too loosely in collider discussions.  From the experimental point of view there are at least three distinct classes that should not be conflated: a generic compactification-style KK picture, the 5D warped Randall--Sundrum benchmark, and the present 6D GG-6 construction.  Table~\ref{tab:generic-rs-gg6} is therefore included explicitly, because it addresses a recurring source of confusion in both phenomenology and outreach.

\begingroup\scriptsize\setlength{\tabcolsep}{4pt}
\begin{longtable}{>{\raggedright\arraybackslash}p{2.23cm} >{\raggedright\arraybackslash}p{3.86cm} >{\raggedright\arraybackslash}p{3.86cm} >{\raggedright\arraybackslash}p{4.88cm}}
\captionsetup{font=footnotesize}
\caption[Generic KK versus 5D Randall--Sundrum versus 6D GG-6.]{Generic KK versus 5D Randall--Sundrum versus 6D GG-6.  The point is not rhetorical; each column implies a different search template, a different interpretation of current exclusions, and a different level of falsifiability. Three columns and three different search templates, which is the practical content of a comparison that is usually made at the level of the Lagrangian. A current exclusion computed against the middle column does not transfer to the right one, and the table is the argument for why a dedicated reinterpretation is needed rather than a rescaling.}\label{tab:generic-rs-gg6}\\
\toprule
\textbf{Question} & \textbf{Generic KK compactification} & \textbf{5D RS warped model} & \textbf{GG-6 6D geometry}\\
\midrule
\endfirsthead
\toprule
\textbf{Question} & \textbf{Generic KK compactification} & \textbf{5D RS warped model} & \textbf{GG-6 6D geometry}\\
\midrule
\endhead
Higher-dimensional stage & Usually flat or mildly curved compact extra dimensions; often a product manifold with a single compact scale \(R^{-1}\) & One warped extra dimension with UV and IR branes and a single Bessel-root tower & Two internal directions, an RG-like radial direction and a compact holonomy direction \((r,\theta)\), with BI-6 anomaly locking built in\\
Canonical mass formula & \(m_n^2\sim m_0^2+n^2/R^2\) & \(m_n\approx x_n\,k e^{-k\pi R}\) for the simplest graviton tower & clustered multi-field radial spectrum, theorem-level first centroid tied to \(K=35\); angular \(\theta\) tower at \(\EGUT\)\\
What fixes the first accessible scale? & Usually a free compactification radius or scanned inverse radius & An infrared scale set by the warp factor, but phenomenologically scanned in benchmark studies & the lockscale \(\Mlock=\EPl/(2e^{35})\simeq 3.849\,\mathrm{TeV}\); not scanned as a free central mass in the current implementation\\
Spectrum morphology & approximately uniform ladder in the simplest flat case & single-sequence Bessel spacing, often analyzed pole by pole & multi-field first cluster near \(3.85\,\mathrm{TeV}\) (graviton/KK-gluon/\(\UB\)), then \emph{radial} families at \(7\,\mathrm{TeV}\) (\(n{=}2\)) and \(10\,\mathrm{TeV}\) (\(n{=}3\)); angular \(\theta\) tower at \(\EGUT\)\\
Couplings & often near-universal in the simplest toy pictures & can be universal or bulk-non-universal depending on localization, but the public literature often uses simplified RS templates & manifestly non-universal from overlap integrals; light fermions UV-biased, top and Higgs IR-enhanced, with a small leptophobic \(\UB\) admixture\\
Need for a baryonic vector near the first family & not generic & absent in the minimal model & natural and in practice very important; the leptophobic \(B_1\)-like state is one of the reasons hadronic channels become central\\
Scalar partner & model dependent; generic moduli possible & radion from stabilization is standard & \emph{no radion}: the radial length is the fixed integer \(K\), not a modulus \citep{ArisakaA4Euclid}, so \GG{} predicts no radius-stabilization scalar --- a clean distinction from 5D RS\\
Best first LHC channel & depends strongly on the model & often dilepton/diphoton in simple public graviton presentations; \(\ttbar\) in bulk RS & \(\ttbar\) and boosted dijets first, with \(hh\), diboson, dilepton, and diphoton as corroborating channels rather than the sole discovery engine\\
Most dangerous misinterpretation & treating all KK theories as one phenomenological class & assuming the public RS pole template is automatically the right template for any warped signal & translating canonical RS exclusions directly into GG-6 without accounting for clustering, non-universal overlaps, and the \(\UB\) admixture\\
What a discovery would mean & evidence for extra dimensions, but often with large model ambiguity & evidence for warped geometry and an RS-like hierarchy solution & evidence simultaneously for the bulk geometry, the OGN-5 lockscale, the 6D clustered spectrum, and the geometry-to-open-system dictionary emphasized by GG-6/GDOS\\
What a null result would mean & pushes a free compactification scale upward & excludes a chosen benchmark corridor & directly pressures the current lockscale implementation because the mass target is not a floating dial\\
\bottomrule
\end{longtable}\endgroup

The short practical message is as follows.  A generic KK search asks whether there is some resonance ladder.  An RS search asks whether a one-dimensional warped hierarchy generated a single-sequence tower.  A GG-6 search asks something much stronger and much riskier: whether the first accessible 6D family sits at a specific locked scale, carries a clustered line shape, prefers tops and hadronic channels over light leptons, and is accompanied by a baryonic vector and a color-octet partner.  That stronger question is exactly why the LHC test is so valuable.

\subsection{Derived geometry versus fitted resonance}
\label{subsec:derived-vs-fitted}

The deepest entry in the comparison is not in any single row; it is the column logic itself.  A generic-KK or benchmark-RS resonance mass is an \emph{input}: a compactification radius or warp product that is scanned across a range and pinned, after the fact, by whatever the data prefer.  The \GG{} lock-cluster mass is an \emph{output}: Theorem~T-KK-1 (\S\ref{sec:t-kk-1}) returns \(\Mlock=\tfrac12 M_{\mathrm{Pl}}e^{-35}\simeq 3.849\,\mathrm{TeV}\) from the \OGNfive{} integer \(K=35\) and the GU-5 constants, with no continuous dial to turn.  This inverts the usual logic of a resonance search.  For a fitted model, a null result in a given window merely pushes a free parameter and the model survives at the next mass; for \GG{}, the window is pre-committed, so a null result in \(3.85\pm0.10\,\mathrm{TeV}\) at full luminosity falsifies the current implementation outright --- through the un-suppressible octet line (F-KK-2B; the spin-2 member itself is FCC-hh territory at its derived coupling, Corollary T-KK-2c) and --- through the over-constraint \(v=\Mlock(2/5)^3\) --- would simultaneously unseat the electroweak-scale derivation of \citet{ArisakaA9SM}.  A derived-geometry mass therefore buys a sharper experimental verdict than any fitted resonance of the same spin and channel content: there is nowhere for it to hide.  This is the single feature that most separates the \GG{} entry from the other two columns, and it is worth more, scientifically, than any individual phenomenological difference in the table.

\subsection{The practical consequence for reinterpretation}

This comparison table has an immediate experimental consequence: public exclusions quoted for canonical RS benchmarks do not automatically map one-to-one onto \GG{}.  They are indispensable guides, but not final judgments.  An analysis that is blind to clustered line shapes, top-rich overlap patterns, or mild leptophobic admixtures can overestimate or underestimate the true \GG{} sensitivity depending on channel.  This is why the current public status must be read with some care.

\section{Why Collider Detection is an Open-System Problem: the GDOS View}
\label{sec:gdos-section}

Why should KK-particle detection at ATLAS or CMS be described by \GDOS{} (\citealp{ArisakaA5GDOS}; \citealp{ArisakaA6Gi4})?  Because a collider experiment is a canonical open system.  The partonic hard event is not measured directly; it is inferred through a chain of dissipative, coarse-graining, and decision-making operations that convert microscopic excitations into registered records.

In the language of \GDOS{}, the relevant system instance packet contains at least four components: the microscopic state manifold of the hard scattering, the boundary ports through which the event deposits energy and charge in the detector, the effective ledger or free-energy-like functional that governs discrimination and reconstruction, and the registered observables -- invariant masses, angular variables, b-tags, top tags, jet masses, missing momentum, and trigger bits.  All of these are open-system objects.  They are not closed Hamiltonian observables of an isolated partonic universe.

From the \OGGEP{} viewpoint, a KK event is a 6D bulk excitation whose experimentally meaningful existence is defined only after projection to the 4D boundary.  ATLAS and CMS do not ``see the extra dimension'' directly.  They see tracks, calorimeter clusters, timing patterns, and statistically reconstructed invariant masses.  This is precisely why AX-3 matters: a bulk theory without a disciplined boundary dictionary is not a scientific prediction.  In collider language, \OGGEP{} is the requirement that a 6D statement must ship with a reconstruction map, a channel list, a decision rule, and a falsification criterion.

DGI also appears in a familiar guise.  The experimentally reported observables are gauge-invariant combinations.  One does not claim discovery in terms of gauge-dependent field components, but in terms of invariant masses, cross sections, angular moments, and reconstructed particle identities.  The route from microscopic gauge fields to those invariants passes through an open detector system and a statistical pipeline.  That is exactly the type of circumstance for which the \GDOS{} language was introduced.

Finally --- and in the completed runtime papers this is not a metaphor but the literal machinery --- the trigger and reconstruction chain is a measurement event of the series' own theory.  The quantum-mechanics paper of the cohort derives measurement as a complete runtime event (\citealp{ArisakaA8QM}): arming an apparatus reorganizes the joint system--apparatus landscape (an endogenous, scheduled \GRIPout{}); the registration that follows is \GRIPin{} descent into one record basin; the logged event is the halt.  A collider run is precisely this anatomy at industrial scale: the trigger menu \emph{is} the landscape design --- each menu deployment a scheduled \GRIPout{} that opens the record basins (leptonic, hadronic, boosted-top, Higgs-candidate) into which reconstruction then descends --- and every event on tape is a \GRIPin{} halt, permanent for the same reason every record in the series is permanent.  A successful KK search is therefore not merely a matter of more luminosity; it is a matter of opening the \emph{right basins}: this is the precise sense of the later sections on trigger-level analysis, scouting, and reduced-stream design, and it is the same grammar in which the sister paper's haloscope hunts the dark matter --- detector and detected run on one grammar.

Part~IV settled the two questions of trust.  First, GG-6 is not Randall--Sundrum wearing new notation: the comparison ran item by item --- no free length, no radion, a multi-state cluster instead of a lone pole, an angular tower pushed to the GUT scale --- and each difference was tied to something measurable.  Second, detection was given its principled reading: a collider measurement is an open-system projection, the same structure that runs through the whole cohort, so the search is not an afterthought bolted onto the theory but an instance of it.  What remains before the phenomenology can be quoted is bookkeeping honesty: which numbers are theorem-grade and which lean on external input.  The next Part draws exactly that line, twice.


\section[Part~IV in plain English]{Part~IV in plain English: why this is not Randall--Sundrum, and why detection is part of the physics}
The resemblance to a warped extra dimension is superficial and this
Part says exactly how far it goes.

A Randall--Sundrum model has a spatial fifth dimension with a size. That
size is a free parameter, so the resonance mass is an input; and a spatial
extra dimension brings a radion and a modulus that have to be stabilized by
something else. The framework here has neither. Its two extra directions
are not spatial --- one is the renormalization-group scale, the other a
phase angle --- so there is no radius to choose, no modulus to stabilize,
no landscape of compactifications. The absences are consequences, not
assumptions, and each of them removes a dial.

The second half of the Part will read as unusual to a collider physicist,
and it is the framework's own claim rather than a flourish. In this
framework observation is a projection from the six-dimensional bulk to the
four-dimensional boundary, and that projection is what makes anything
registrable at all. A trigger menu, on that reading, is not a filter placed
in front of the physics; it is a choice about which projection runs.
Nothing in the search strategy of the later Parts depends on accepting
that, which is why it is stated here and then set aside.

\GGPart[Two read-outs of the lock cluster]{Two Read-outs of the Lock Cluster}{Read-out~I --- the theorem-only target; Read-out~II --- an empirically-anchored rate estimate}

Honesty in a predictive paper is mostly a matter of bookkeeping, and this short Part is the ledger.  It states the same physics twice, at two clearly separated levels of commitment, so that no reader ever has to wonder which claims would survive if every external assumption failed.


\section{Read-out I: The Theorem-Only Target}
\label{sec:scenarioA}

Up to this point the paper has obeyed a single rule: no quantitative statement that is not a \GG{}-A theorem.  That rule defines what we will call \emph{Read-out~I} --- the most conservative, theorem-grade read-out of the lock cluster.  It is maximally honest and essentially non-negotiable within the cohort, but it deliberately stops short of an absolute production rate.  We state it cleanly here, and in \S\ref{sec:scenarioB} we show how far \emph{one} well-defined set of external assumptions can extend it into the cross section that experimentalists actually plan against.

\paragraph{What Read-out I delivers (the derived skeleton).}
Everything in this list is traceable to \OGNfive{}+GU-5 and the cohort theorems, with no continuous dial:
\begin{itemize}
\item \textbf{The mass scale.}  \(\Mlock=\tfrac12 M_{\mathrm{Pl}}e^{-35}=3.849\,\mathrm{TeV}\) (Theorem~T-KK-1, Eq.~\eqref{eq:tkk1-mlock}), together with the over-constraint \(v=\Mlock(2/5)^3\approx246\,\mathrm{GeV}\).
\item \textbf{The spin content.}  A spin-2 graviton \(G_1\) and a mutually-degenerate spin-1 pair (the color-octet \(g^*\), the baryonic \(Z'_B\)), all at \(\approx\Mlock\) up to an \(O(1)\) coefficient \(c_X\) that \GG{}-A does not fix, with the next \emph{radial} family near \(7\,\mathrm{TeV}\).
\item \textbf{The exact decay structure.}  \(\mathrm{BR}(Z'_B\to\llbar)=0\) exactly (Eq.~\eqref{eq:zprime-leptophobic}); the color octet \(g^*\) decays \emph{only} to colored states; and for both partners the top-pair channel dominates (an ordering, \S\ref{subsec:branchings}, not a number).
\item \textbf{The one fixed coupling.}  The color octet's light-quark coupling \(|\xi_{\rm light}|=1/\sqrt{K}\approx0.17\), fixed by \(K=35\) and \emph{not} lowerable (\S\ref{subsec:widths}).
\item \textbf{The deliberate omission.}  No absolute cross section, branching number, width fraction, or event yield, because each requires \(\kappaeff\) and a QCD/PDF integration that no A1--A9 theorem supplies (\S\ref{subsec:noxsec}).
\end{itemize}

\paragraph{The experimental instruction in Read-out I.}
Search for a \emph{clustered, top-rich, leptophobic} structure tied to \(\Mlock=3.85\,\mathrm{TeV}\); the falsifiers are \emph{structural}.  F-KK-1 (as requalified by Corollary T-KK-2c): the HL-LHC arm of the falsifier runs through the cluster's un-suppressible octet; the spin-2 line, at its derived coupling \(\kappa_{gg}=c_K/K\), is a direct FCC-hh target.  F-KK-2A/2B (T-KK-2): the un-suppressible color octet sits either at \(2.46\)~TeV (free-field scenario, decidable from data already on tape) or at the \(\ttbar\)/dijet frontier near \(\Mlock\) (composite scenario, being probed now).  This is the position the cohort can stand behind without qualification.  Its only practical limitation is that ``look near \(3.85\,\mathrm{TeV}\) for a top-rich cluster'' is a \emph{shape} instruction, not a \emph{rate} target --- and an experimentalist allocating integrated luminosity needs a rate.

\section{Read-out II: An Empirically-Anchored Rate Estimate}
\label{sec:scenarioB}

The number missing from Read-out~I is \(\kappaeff\) (and, channel by channel, \(R_f\)).  These are not \GG{}-A outputs --- but they are not unknowable either.  The GG-6 throat is, by its own construction, a warped Randall--Sundrum-type AdS slice, and warped-KK phenomenology together with the measured fermion spectrum fixes them up to \(O(1)\).  \emph{Read-out~II is the result of grafting that established, external knowledge onto the Read-out-I skeleton.}  Every number below is model-level and contingent on the assumptions we now state explicitly; none of it touches the derived \(\Mlock\) or \(v\).

\paragraph*{Underlying assumptions of Read-out II (external to \GG{}-A).}
\begin{description}[leftmargin=2.4em,style=nextline,topsep=2pt,itemsep=2pt]
\item[\textbf{B1 --- RS-type throat.}] The radial interval \(r\in[0,K]\) is treated as an AdS slice, so the standard warped overlap and profile functions apply \citep{RandallSundrum1999}.
\item[\textbf{B2 --- Empirical localization.}] Each fermion's bulk localization is read off its \emph{measured} mass through the usual RS-GIM correspondence: light fermions UV-localized, the top IR-localized.
\item[\textbf{B3 --- Normalization transfer.}] The KK-mode normalization that fixes \(\xi_{\rm light}=1/\sqrt{K}\) also sets the resonance's coupling to the production partons, so \(\kappaeff\approx1/\sqrt{K}\).
\item[\textbf{B4 --- Standard coupling ratios.}] UV-localized fields couple to the first KK mode at \(\approx-1/\sqrt{K}\) in gauge-coupling units and fully IR-localized fields at \(\approx+4\) to \(+5\), the established warped values \citep{LillieRandallWang2007,AgasheEtAl2007}.
\end{description}

\paragraph{The coupling normalization \texorpdfstring{$\kappaeff$}{kappa-eff}.}
Three independent arguments converge on the same value.  (i)~A canonically-normalized KK mode on a warped interval of proper length \(K\) has amplitude \(\sim1/\sqrt{K}\) at a generic point, so its overlap with the production partons carries exactly the factor that already fixes \(\xi_{\rm light}\); this gives \(\kappaeff\approx1/\sqrt{K}\).  (ii)~Sub-Planckian curvature validity caps \(k/\mplred\lesssim1\), excluding the strong benchmark and pulling \(\kappaeff\) into the \(0.1\)--\(0.3\) band.  (iii)~Feeding \(\kappaeff=1/\sqrt{K}\) into a standard \(3.85\,\mathrm{TeV}\) bulk-graviton template (\(\sim20\)--\(50\,\mathrm{fb}\) at \(k/\mplred=1\); \citealp{DavoudiaslHewettRizzo2000}) reproduces \(\sigma(G_1)\approx0.6\)--\(1.4\,\mathrm{fb}\), i.e.\ it derives the often-quoted ``sub-femtobarn-to-few-femtobarn'' graviton corridor from \(K\) rather than from a fit.  Hence the working bracket adopted by this Read-out was
\begin{equation}
\boxed{\;\kappaeff\;\approx\;\frac{1}{\sqrt{K}}=\frac{1}{\sqrt{35}}\approx0.17\;}\qquad(\text{bracket }0.08\text{--}0.3,\ \text{model-level}).
\label{eq:kappaeff-estimate}
\end{equation}
The appeal is that the central value is set by the \emph{same} integer \(K=35\) that fixes \(\Mlock\) and \(\xi_{\rm light}\), not by a new free dial.

\emph{Superseded by the exact result.}  The production overlap is computed exactly in Corollary T-KK-2c (\S\ref{subsec:kappagg}): \(\kappa_{gg}=2(1+J_2(x_1))/(K x_1^2 J_2(x_1))=0.0136\approx0.47/K\).  Argument~(i) above mistook an amplitude for an overlap --- near-orthogonality of the flat gluon and the IR-localized graviton cancels all but the last half e-fold of the throat, costing a further \(\sim\!1/\sqrt K\).  Arguments~(ii)--(iii) bracketed the truth from above; the derived value sits at the very bottom of the estimated corridor and is what all graviton rates in \S\ref{sec:operational} use.  The boxed estimate is retained above as the Read-out-II bracket that the exact result replaces.

\paragraph{The channel factors \texorpdfstring{$R_f$}{R-f}.}
The \emph{structure} of \(R_f\) is pure GG-6 (top-rich, leptophobic); its \emph{numbers} follow from B2+B4.  For the spin-1 color octet, the borrowed coupling ratios (UV light quarks at \(-1/\sqrt{K}\,g_s\), IR top at \(\sim+4.5\,g_s\)) give a strongly top-saturated pattern; for the graviton, \(R_f\) is quoted relative to a universal bulk-RS template.

\begin{table}[htbp]
\centering
\small
\setlength{\tabcolsep}{6pt}
\caption[Read-out-II channel estimates.]{Read-out-II channel estimates.  \emph{Model-level} (assumptions B1--B4): the color-octet branching fractions follow from the borrowed RS coupling ratios plus the measured fermion localization; the graviton \(R_f\) is the GG-6 rate \emph{relative to} a universal bulk-RS graviton of the same mass.  Leptophobia (\(\mathrm{BR}(Z'_B\to\llbar)=0\)) is the only exact entry --- it is a Read-out-I theorem.}\label{tab:scenarioB-Rf}
\begin{tabular}{>{\raggedright\arraybackslash}p{3.94cm} >{\raggedright\arraybackslash}p{5.66cm} >{\raggedright\arraybackslash}p{5.66cm}}
\toprule
\textbf{Channel} & \textbf{Spin-1 octet \(g^*\): BR estimate} & \textbf{Graviton \(G_1\): \(R_f\) vs bulk-RS}\\
\midrule
\(\ttbar\) & \(0.7\)--\(0.95\) (top IR-localized) & \(\sim1\)--\(3\) (mildly enhanced)\\
\(b\bar b\) & \(\sim0.05\) (\(b_L\) semi-IR) & \(\sim1\)\\
light \(jj\) & \(0.01\)--\(0.05\) (UV, \(\propto1/K\)) & \(\sim1\)--\(2\) (\(\UB\) admixture)\\
\(hh\), \(WW\)/\(ZZ\) & --- (color octet) & \(\sim1\)--\(2\) (IR Higgs sector)\\
\(\llbar\) & \(0\) (color singlet) & \(0.1\)--\(0.5\) (light-lepton suppression)\\
\(\gamma\gamma\) & \(0\) (color singlet) & \(0.3\)--\(1\) (reduced photonic weight)\\
\bottomrule
\end{tabular}
\end{table}

\paragraph{Cross-section ballparks.}
Combining \(\kappaeff\approx1/\sqrt{K}\) with the channel factors, at \(\sqrt{s}=13.6\,\mathrm{TeV}\) and \(m\approx3.85\,\mathrm{TeV}\):
\begin{align}
\sigma(pp\to G_1) &\approx 0.6\text{--}1.4\,\mathrm{fb}\qquad(\text{spin-2 graviton, }gg/q\bar q),\\
\sigma(pp\to g^*)  &\approx 11\text{--}23\,\mathrm{fb},\qquad
\sigma(pp\to g^*)\times\mathrm{BR}(g^*\to\ttbar)\approx 11\text{--}22\,\mathrm{fb}.
\end{align}

These template-rescaled ranges are superseded by the explicit parton-luminosity computation of \S\ref{sec:operational} (Table~\ref{tab:sigmabr}), which lands at or below their lower edges (e.g.\ \(\sigma\times\mathrm{BR}(g^{*}\to\ttbar)=8.7\,[7.0\text{--}12]\)~fb at 13~TeV, LO\(\times\)K); we keep the Read-out-II estimates visible for comparison.

\paragraph{Scenario-A rescaling.}  In the free-field scenario of T-KK-2 the octet sits at \(2.46\)~TeV, where the \(q\bar q\) parton luminosity at \(\sqrt{s}=13.6\)~TeV is roughly \(6\)--\(8\) times larger than at \(3.85\)~TeV.  The same Read-out-II inputs then rescale to \(\sigma(pp\to g^*)\times\mathrm{BR}(\ttbar)\approx70\)--\(160\)~fb --- an order of magnitude above the Scenario-B target, which is why the broad-octet reinterpretation of F-KK-2A decides this scenario quickly: at these rates, existing data either contain it or close it.  These numbers carry the same model-level label as the rest of this section.

\paragraph{The payoff: a well-defined, immediate LHC goal (Scenario B; Scenario A above).}
The color octet's \(\sigma\times\mathrm{BR}(\ttbar)\approx11\)--\(22\,\mathrm{fb}\) at \(3.85\,\mathrm{TeV}\) sits squarely on the current ATLAS/CMS \(\ttbar\)-resonance KK-gluon exclusion (\(\sim10\)--\(20\,\mathrm{fb}\) at this mass; \citealp{ATLAS2025ttbar}).  This is the concrete target Read-out~II supplies that Read-out~I could not: \emph{a dedicated color-octet \(\ttbar\) reinterpretation at \(3.85\,\mathrm{TeV}\) either confirms the excess or excludes \(\sigma\times\mathrm{BR}(\ttbar)\lesssim10\,\mathrm{fb}\)} --- and the latter retires the spin-1 sector of the current implementation \emph{now}, with data already on tape.  In other words, Read-out~II turns the structural F-KK-2 into a number, and it explains why \(\Mlock=3.85\,\mathrm{TeV}\) lands exactly at today's data frontier rather than safely beyond it.

\paragraph*{Honest standing of Read-out II.}
Read-out~II is \emph{nested} inside Read-out~I: \(\text{II}=\text{I}+\{\text{B1--B4}\}\).  If the external assumptions fail, only the \emph{rates} move; the derived mass \(\Mlock\), the over-constraint with \(v\), and the exact leptophobia are untouched, because they live in Read-out~I.  We therefore present II not as a rival to I but as the actionable extension experimentalists have asked for, with its load-bearing assumptions on the table.  Any use of the numbers in Eqs.~\eqref{eq:kappaeff-estimate} and Table~\ref{tab:scenarioB-Rf} should carry the label ``empirically-anchored estimate (model-level),'' never ``\GG{} prediction.''

\begin{table}[htbp]
\centering
\small
\setlength{\tabcolsep}{6pt}
\caption[The two read-outs side by side.]{The two read-outs side by side.  Read-out~I is theorem-grade and rate-free; Read-out~II adds assumptions B1--B4 to obtain an actionable rate. Read-out~I is what the theorems alone deliver, and it is silent about rate; Read-out~II adds four named assumptions and gets a number a collaboration can act on. Keeping them in separate columns is the honest form, because a reader who rejects B1 through B4 still has a prediction to test and should be able to see exactly which one it is.}\label{tab:scenarioAB}
\begin{tabular}{>{\raggedright\arraybackslash}p{4.18cm} >{\raggedright\arraybackslash}p{5.53cm} >{\raggedright\arraybackslash}p{5.53cm}}
\toprule
\textbf{Quantity} & \textbf{Read-out I (theorem-only)} & \textbf{Read-out II (A + B1--B4)}\\
\midrule
Mass scale \(\Mlock\) & \(3.849\) TeV, \emph{derived} & same (inherited from A)\\
Spin content & \(G_1\); \(g^*\), \(Z'_B\) at \(\approx\Mlock\) & same\\
Leptophobia & \(\mathrm{BR}(Z'_B\to\llbar)=0\) exactly & same (exact)\\
Coupling \(\kappa_{gg}\) & \(c_K/K=0.0136\), \emph{derived} (T-KK-2c) & \(\approx1/\sqrt K\) (estimate; superseded)\\
Channel BRs & ordering only (top-rich) & numerical (Table~\ref{tab:scenarioB-Rf})\\
Absolute \(\sigma\) & follows with external PDFs (\S\ref{sec:operational}) & \(\sim10\)--\(20\)~fb (superseded by Table~\ref{tab:sigmabr})\\
Falsifier & two-tier F-KK-1 + F-KK-2 (octet carries HL-LHC) & numerical \(\ttbar\) target at \(3.85\) TeV\\
Experimentalist gets & a \emph{shape} to search for & a \emph{rate} to plan against\\
\bottomrule
\end{tabular}
\end{table}


Part~V drew the honesty line.  Read-out~I is the theorem-grade core --- masses, degeneracies, exact leptophobia --- quotable with no outside help; Read-out~II adds four declared external assumptions and, in exchange, produces the actionable cross sections an experiment can plan against; and the one estimated coupling in the graviton sector has since been replaced by the exact result, which sits at the very bottom of the old bracket.  Every rate in this paper now carries its pedigree on its sleeve.  With the numbers established and graded, the next Part turns to what the signal actually looks like: the shapes --- angles, polarization, and the line shape with its theorem-fixed interference sign --- that no rate uncertainty can wash out.


\section[Part~V in plain English]{Part~V in plain English: the two things this paper predicts, and which of them can be argued with}
Honesty in a predictive paper is mostly bookkeeping, and this Part is
the bookkeeping.

There are two read-outs and they are kept in separate columns. The first is
what the theorems alone give: positions, ratios, and shapes, with no
statement about how often anything would be produced. It is the stronger
claim precisely because it says less --- nothing in it can be adjusted, and
nothing in it depends on an assumption a reader might decline.

The second adds four named assumptions and gets a rate. A rate is what a
collaboration actually needs, so the Part supplies one; but a rate carries
everything the assumptions carry, and the assumptions are printed with
labels so that a reader who rejects one can see precisely what falls with
it.

The reason for the separation is that a search can be run against either.
Someone who accepts only the first read-out still has a target: the
arrangement of the cluster, the angular structure, the line shape. Those
are normalization-free, so they survive a wrong rate entirely. Someone who
accepts the second gets a schedule as well. Collapsing the two into one
number would have made the paper look more decisive and would have made it
easier to dismiss.

\GGPart[Anatomy of the ttbar signal]{Anatomy of the \texorpdfstring{$\ttbar$}{ttbar} Signal}{Derived angular, polarization, and line-shape structure of the three cluster hypotheses}
\label{part:ttanatomy}

\section{What This Part Adds, and at What Grade}
\label{sec:ttanatomy-intro}

Section~\ref{sec:channel-by-channel} argued that \(\ttbar\) is the central battleground and asserted that a heavy spin-2 and a heavy spin-1 state ``produce top pairs with different angular and kinematic structures.''  An assertion of that kind earns its keep only when the structures are written down, derived, and numbered.  This Part does exactly that, at the level of detail an ATLAS or CMS analysis team can check line by line: for each lock-cluster member --- the color octet \(g^{*}\) (spin 1, the carrier of the HL-LHC falsifier), the leptophobic \(Z'_B\) (spin 1, color singlet), and the graviton \(G_1\) (spin 2) --- we derive the production-angle distribution, the top polarization, the forward--backward asymmetry, the resonance line shape \emph{including interference with the QCD continuum}, and the background-relative yields, each from the couplings that T-KK-2, Corollary T-KK-2c, and L-KK-1 have already fixed or bracketed.  Nothing in this Part introduces a new parameter.  Part~\ref{part:hllhcsim} then takes these shapes into the detector: a full simulation of the ATLAS/CMS measurement, anchored to the published Run-2 analysis, with the discovery calendar they imply.

The grading ladder of the cohort applies clause by clause.  \textbf{[T]} theorem-grade inputs: the masses, \(\xi_{\rm UV}=-0.194\) with its sign, the exact leptophobia, \(\kappa_{gg}=c_K/K\), and the color assignments.  \textbf{[D\(^{*}\)]} derived-conditional: free-field kinematics, LO\(\times\)K QCD, the top-overlap band \(r_{t_R}\in[4,5]\).  \textbf{[M]} the L-KK-1 menu for \(g_B\).  \textbf{[E]} external Standard-Model inputs: PDFs, the \(\alpha_s\) convention, published K-factors.  The headline of the Part, stated up front: \emph{the angular forms, the zero of \(A_{FB}\), the vanishing of \(P_t\) for \(G_1\), the interference sign, and the interference yes/no dichotomy between the two spin-1 members are all [T]-grade shape predictions --- they survive any revision of K-factors or parton distributions.}

\paragraph{Inputs used throughout.}  From Table~\ref{tab:observable-ledger} and \S\ref{subsec:tkk2}--\S\ref{subsec:gbmenu}: all three states at \(\Mlock=3.849\)~TeV (Scenario B); light quarks couple vector-like with amplitude \(\xi_{\rm UV}=-0.194\) [T] (both chiralities of a UV fermion share one profile, hence \(g_A^{q}=0\) exactly); top chiralities \(r_{t_L}=1.0\), \(r_{t_R}=4.5\) [D\(^{*}\)], giving the vector/axial decomposition
\begin{equation}
g_V^{t}=\tfrac12(r_{t_R}+r_{t_L})=2.75\,g_s,\qquad
g_A^{t}=\tfrac12(r_{t_R}-r_{t_L})=1.75\,g_s,\qquad
g_A^{t}/g_V^{t}=0.636.
\label{eq:tt-gVgA}
\end{equation}
Top kinematics at the pole: \(\beta_t=\sqrt{1-4m_t^{2}/\Mlock^{2}}=0.99597\), so \(1-\beta_t^{2}=4m_t^{2}/\Mlock^{2}=0.0080\): every mass correction below is a computed per-cent-level effect, not a neglected one.  Widths, from Eq.~\eqref{eq:widthratio} and \S\ref{subsec:kappagg}: \(\Gamma_{g^{*}}=561\)~GeV (\(\Gamma/m=0.146\); partials \(\ttbar\!:528\), \(b\bar b\!:26\), \(jj\!:7\)~GeV), \(\Gamma_{G_1}=243\)~GeV (\(\Gamma/m=0.063\)), \(\Gamma_{Z'_B}=8\)--\(307\)~GeV across the L-KK-1 menu.

\paragraph{Rates at LO\(\times\)K.}  The leading-order rates of Table~\ref{tab:sigmabr} are promoted here with channel-appropriate published K-factors [E]: \(K_{q\bar q}\simeq1.3\) for massive color-octet vector production \citep{ChivukulaEtAl2012}, \(K_{gg}\simeq1.6\) for spin-2 production \citep{MathewsRavindran2005}.  Thus \(\sigma\times\mathrm{BR}(g^{*}\to\ttbar)=8.7\,[7.0\text{--}12]\)~fb at 13~TeV and \(13.6\,[11\text{--}19]\)~fb at 14~TeV; \(G_1\): \(0.045\,[0.021\text{--}0.10]\)~fb; \(Z'_B\): \(0.016\)--\(0.59\)~fb across the menu.  Branching pattern of the octet: \(\ttbar:b\bar b:jj \approx 70\!:\!3.5\!:\!1\) (BR\((\ttbar)=0.94\), band \(0.93\)--\(0.95\) for \(r_{t_R}\in[4,5]\)) --- itself a falsifiable ratio.  Everything else in this Part concerns \emph{shapes}, in which K-factors and PDF uncertainties largely cancel.

\section{The \texorpdfstring{$m_{\ttbar}$}{m(ttbar)} Line Shape, with Interference}
\label{sec:ttlineshape}

\subsection{Amplitudes and color algebra}
For \(q\bar q\to\ttbar\) the color octet enters \emph{the same color channel} as the QCD gluon, so the amplitudes add coherently:
\begin{equation}
\mathcal M \;=\;
\Bigl[\underbrace{\frac{g_s^{2}}{\hat s}}_{\rm QCD}
+\underbrace{\frac{g_s\xi_{\rm UV}\cdot g_s\,(g_V^{t}+g_A^{t}\gamma_5)}{\hat s-\Mlock^{2}+i\Mlock\Gamma_{g^{*}}}}_{g^{*}}\Bigr]
\;\bigl(\bar v_q T^{a}\gamma^{\mu}u_q\bigr)\bigl(\bar u_t T^{a}\gamma_{\mu}(\cdots)v_t\bigr).
\label{eq:tt-amps}
\end{equation}
Because \(T^{a}\otimes T^{a}\) is common to both terms, the interference carries the full color weight --- no \(1/N_c\) penalty.  Two structural consequences follow immediately and carry [T] grade.  \emph{(i)}~For the color-singlet \(Z'_B\) the corresponding color trace vanishes, \(\mathrm{Tr}\,T^{a}=0\): the \(Z'_B\) rides the continuum as a \emph{pure} Breit--Wigner, with exactly zero QCD interference.  The two mutually degenerate spin-1 members are thus distinguished not by mass or angle but by an interference yes/no dichotomy --- a prediction with no adjustable element.  \emph{(ii)}~The \(gg\)-produced spin-2 \(G_1\) populates helicity \(\pm2\) (\S\ref{sec:ttangular}), orthogonal after angular integration to the QCD vector channels; its interference is negligible at the rates of this paper.

\subsection{Squaring: the three terms and the V/A algebra}
Write the top vertex as \(g_V^{t}+g_A^{t}\gamma_5\).  Squaring \eqref{eq:tt-amps} produces \(|{\rm QCD}|^{2}\), the cross term, and \(|g^{*}|^{2}\).  Angular integration kills two classes of terms: the \(V\times A\) products on a single fermion line integrate to zero over \(\cos\theta^{*}\) (they are odd, \(\propto2\cos\theta^{*}\); this is also the statement \(A_{FB}=0\) below), and for massless light quarks there is no \(q\)-side axial part at all, \(g_A^{q}=0\) [T].  The surviving angular integrals are the textbook pair
\[
\int_{-1}^{1}\!\bigl(1+\cos^{2}\theta^{*}+(1-\beta_t^{2})\sin^{2}\theta^{*}\bigr)\,d\cos\theta^{*}=\tfrac83\bigl(1+\tfrac{\rho}{2}\bigr),
\qquad
\int_{-1}^{1}\!\beta_t^{2}\bigl(1+\cos^{2}\theta^{*}\bigr)\,d\cos\theta^{*}=\tfrac83\,\beta_t^{2},
\]
for the vector and axial top currents respectively (\(\rho\equiv4m_t^{2}/\hat s\); the vector current keeps the threshold factor \(1+\rho/2\), the axial current the \(\beta_t^{2}\) suppression --- the standard \(Z'\) width algebra of Eq.~\eqref{eq:vecwidth}).  The differential rate in the resonance region is then
\begin{equation}
\frac{d\sigma}{dm}\Big|_{q\bar q\to\ttbar}
=\frac{8\pi\alpha_s^{2}}{27}\,\beta_t\,\frac{d\mathcal L_{q\bar q}}{dm}\times
\Bigl[\underbrace{\frac{1+\rho/2}{\hat s}}_{\rm SM}
+\underbrace{2\,{\rm Re}\,D(\hat s)\,\xi_{\rm UV}\,g_V^{t}\,\bigl(1+\tfrac{\rho}{2}\bigr)}_{\rm interference}
+\underbrace{\hat s\,|D|^{2}\,\xi_{\rm UV}^{2}\bigl(g_V^{t2}(1+\tfrac{\rho}{2})+g_A^{t2}\beta_t^{2}\bigr)}_{\rm Breit-Wigner}
\Bigr],
\label{eq:tt-lineshape}
\end{equation}
with \(D(\hat s)=(\hat s-\Mlock^{2}+i\Mlock\Gamma_{g^{*}})^{-1}\) and \(d\mathcal L_{q\bar q}/dm\) the \(u,d,s,c\) luminosity.  Note where each derived quantity sits: \(\xi_{\rm UV}\) enters the interference \emph{linearly} (sign observable) and the Breit--Wigner \emph{quadratically} (sign hidden); only \(g_V^{t}\) interferes, because QCD is a vector current.

\subsection{The sign theorem}
\({\rm Re}\,D<0\) below the pole and \(>0\) above it.  T-KK-2 \emph{derives} \(\xi_{\rm UV}=-0.194<0\) (the ultraviolet endpoint amplitude of the exact mode, Eq.~\ref{eq:tkk2-xiflat}) while \(g_V^{t}=+2.75>0\); hence \(\xi_{\rm UV}g_V^{t}<0\) and:
\begin{quote}
\textbf{The interference is constructive \emph{below} \(3.849\)~TeV and destructive \emph{above} it.}  The ordering is not adjustable --- it is the observable image of the derived relative sign \(\xi_{\rm UV}<0<\xi_{\rm IR}\) that T-KK-2 obtained from the rigid geometry.  An observed excess-\emph{above}/deficit-\emph{below} pattern at the lock scale would falsify the T-KK-2 overlap structure outright, independently of any rate measurement.  A generic same-sign coloron predicts the opposite ordering, so this observable also separates \(g^{*}\) from its nearest phenomenological look-alike.
\end{quote}

\subsection{Numbers and observable consequences}
\label{subsec:tt-lineshape-numbers}
Folding Eq.~\eqref{eq:tt-lineshape} with the CTEQ6M luminosities (LO throughout this subsection --- the content is the \emph{ratio} structure): the integrated interference is \(+3.75\)~fb below the pole and \(-0.48\)~fb above it.  Net \(+3.3\)~fb on a \(6.7\)~fb Breit--Wigner: \emph{interference is half the signal, not a correction}, and any pure-Breit--Wigner template will mismeasure both rate and mass.  Table~\ref{tab:tt-profile} gives the excess-to-continuum profile; Figure~\ref{fig:tt-lineshape} draws it.  Three observable consequences: the low side of the resonance is \emph{fattened} (constructive shoulder), the high side \emph{collapses faster than the Breit--Wigner} and crosses \emph{below} the Standard-Model continuum at \(\simeq4.87\)~TeV (a local deficit of order \(20\%\) where the continuum is \(\sim0.05\)~ab/GeV), and the apparent centroid is pulled below \(3.849\)~TeV by \(O(100)\)~GeV --- the falling luminosity and the constructive low side pull the same way, and a fit that ignores this will report a shifted mass.  The \(G_1\) adds a barely visible symmetric bump (\(0.03\)~fb over its window); the \(Z'_B\) a strictly symmetric pure Breit--Wigner at \(0.016\)--\(0.59\)~fb.

\begin{table}[H]
\centering\small
\setlength{\tabcolsep}{6pt}
\caption[Excess-to-continuum profile of the \(g^{*}\) signal, LO/LO, \(\sqrt s=13\)~TeV: the ratio.]{Excess-to-continuum profile of the \(g^{*}\) signal, LO/LO, \(\sqrt s=13\)~TeV: the ratio \((S+I)/B\) of Eq.~\eqref{eq:tt-lineshape}'s resonance terms to the QCD continuum, and the interference and Breit--Wigner parts separately.  The sign flip of the interference across the pole and the crossing to net deficit at \(\simeq4.87\)~TeV are the shape falsifiers.}
\label{tab:tt-profile}
\begin{tabular}{lcccccccc}
\toprule
\(m_{\ttbar}\) [TeV] & 3.0 & 3.3 & 3.6 & 3.849 & 4.1 & 4.4 & 4.7 & 5.0\\
\midrule
\((S+I)/B\) & 2.3 & 4.6 & 11.4 & 18.7 & 8.9 & 2.2 & 0.40 & \(-0.21\)\\
interference\(/B\) & \(+1.5\) & \(+2.3\) & \(+3.2\) & \(0.0\) & \(-4.1\) & \(-3.7\) & \(-3.0\) & \(-2.5\)\\
Breit--Wigner\(/B\) & 0.8 & 2.4 & 8.3 & 18.8 & 13.0 & 5.9 & 3.4 & 2.3\\
\bottomrule
\end{tabular}
\end{table}

\begin{figure}[t]
\centering
\resizebox{0.9\textwidth}{!}{\includegraphics{Arisaka-GG-P1-KK-ttrep-lineshape.pdf}}
\caption[Derived \(m_{\ttbar}\) line shape of the lock cluster with QCD interference]{The LO \(m_{\ttbar}\) spectrum near the lock scale at \(\sqrt s=13\)~TeV.  Red solid: SM \(+\,g^{*}\) with the full interference of Eq.~\eqref{eq:tt-lineshape} at the derived sign \(\xi_{\rm UV}g_V^{t}<0\); red dotted: Breit--Wigner only.  The constructive low-side shoulder, the destructive collapse of the high tail (crossing below the SM at \(\simeq4.9\)~TeV), and the near-invisibility of \(G_1\) (violet) are the three structural signatures.  Numbers in Table~\ref{tab:tt-profile}.}
\label{fig:tt-lineshape}
\end{figure}

\section{Angular Structure from Helicity Amplitudes}
\label{sec:ttangular}

\subsection{Conventions and the two Wigner rows}
Let \(\theta^{*}\) be the angle between the top momentum and the beam-parton axis in the \(\ttbar\) rest frame.  For a resonance of spin \(J\) produced with helicity \(\lambda_R\) along the beam and decaying to a \(\ttbar\) pair with helicity difference \(\Delta\lambda=\lambda_t-\lambda_{\bar t}\), the decay amplitude factorizes as \(\mathcal A\propto d^{J}_{\lambda_R,\Delta\lambda}(\theta^{*})\,\mathcal H_{\Delta\lambda}\), where \(\mathcal H\) carries the dynamics (couplings, \(\beta_t\)) and the Wigner function carries all the angular dependence.  The entries used below:
\begin{gather*}
d^{1}_{1,\pm1}=\frac{1\pm\cos\theta^{*}}{2},\qquad
d^{1}_{1,0}=-\frac{\sin\theta^{*}}{\sqrt2};\\[2pt]
d^{2}_{2,1}=-\frac{\sin\theta^{*}(1+\cos\theta^{*})}{2},\qquad
d^{2}_{2,-1}=\frac{\sin\theta^{*}(1-\cos\theta^{*})}{2}.
\end{gather*}
Three dynamical facts fix which rows are populated.  \emph{(a)}~Massless quarks with a vector coupling have \(\lambda_q-\lambda_{\bar q}=\pm1\): a \(q\bar q\)-produced vector carries \(\lambda_R=\pm1\) only (\(\lambda_R=0\) costs \(m_q/\sqrt{\hat s}\to0\)).  \emph{(b)}~Two massless gluons give \(J_z\in\{0,\pm2\}\), and the transverse-traceless graviton coupling selects \(|\lambda_R|=2\) (the \(J_z=0\) combination enters through the trace, absent on shell).  \emph{(c)}~The top pair from a vector or tensor current populates \(\Delta\lambda=\pm1\), with \(\Delta\lambda=0\) suppressed by the spinor mismatch \(\sqrt{1-\beta_t^{2}}=2m_t/\sqrt{\hat s}=0.090\) in amplitude, i.e.\ \(0.8\%\) in rate.

\subsection{Spin 1 from \texorpdfstring{$q\bar q$}{qqbar}: derivation}
Assembling (a) and (c) for \(\lambda_R=+1\) (the \(-1\) row is the mirror image) and summing the unobserved final helicities:
\begin{align}
\frac{dN}{d\cos\theta^{*}}
&\propto\;\bigl|d^{1}_{1,1}\bigr|^{2}+\bigl|d^{1}_{1,-1}\bigr|^{2}
+(1-\beta_t^{2})\,\bigl|d^{1}_{1,0}\bigr|^{2}\notag\\[2pt]
&=\Bigl(\frac{1+\cos\theta^{*}}{2}\Bigr)^{2}+\Bigl(\frac{1-\cos\theta^{*}}{2}\Bigr)^{2}
+(1-\beta_t^{2})\,\frac{\sin^{2}\theta^{*}}{2}
\;=\;\frac{1+\cos^{2}\theta^{*}}{2}+(1-\beta_t^{2})\frac{\sin^{2}\theta^{*}}{2}.
\label{eq:tt-spin1-deriv}
\end{align}
Normalizing on \([-1,1]\) and inserting \(1-\beta_t^{2}=0.0080\):
\begin{equation}
\boxed{\;
\frac{1}{\sigma}\frac{d\sigma}{d\cos\theta^{*}}\Big|_{\rm spin\,1}
=\frac{3}{8}\Bigl[\,(1+\cos^{2}\theta^{*})+0.0080\,\sin^{2}\theta^{*}\Bigr]\Big/\bigl(1+0.0040\bigr)
\;\simeq\;\frac{3}{8}\,(1+\cos^{2}\theta^{*}).}
\label{eq:tt-spin1}
\end{equation}
The same form holds for \(g^{*}\) and \(Z'_B\) alike, chiral top couplings notwithstanding: chirality moves strength between the \(\lambda_t\) states (\S\ref{sec:ttpol}) but not between production angles, because the light-quark side is vector-like.  Numerically: the distribution at \(|\cos\theta^{*}|=0.9\) is \(1.81/1.375=1.32\) times its value at \(0\), and \(39\%\) of events fall at \(|\cos\theta^{*}|>0.7\).

\paragraph{Corollary: \(A_{FB}=0\) exactly [T].}  A forward--backward asymmetry requires a \(\cos\theta^{*}\)-odd term, which arises only from the product \(g_A^{q}g_A^{t}\).  T-KK-2 forces \(g_A^{q}=0\) --- one radial profile per UV fermion serves both chiralities --- so the resonant \(A_{FB}\) vanishes identically for both spin-1 members, at any top coupling.  (The small calculable \(\cos\theta^{*}\)-odd contamination from QCD--electroweak interference in the continuum is not a resonance property.)  A measured resonant asymmetry would falsify the overlap structure; it would equally cleanly rule \emph{in} an axigluon, whose defining signature is a large \(A_{FB}\).  This corollary, like the sign theorem, costs nothing and can kill the framework.

\subsection{Spin 2 from \texorpdfstring{$gg$}{gg}: derivation}
Assembling (b) and (c) for \(\lambda_R=+2\):
\begin{align}
\frac{dN}{d\cos\theta^{*}}
\;\propto\;\bigl|d^{2}_{2,1}\bigr|^{2}+\bigl|d^{2}_{2,-1}\bigr|^{2}
&=\frac{\sin^{2}\theta^{*}}{4}\Bigl[(1+\cos\theta^{*})^{2}+(1-\cos\theta^{*})^{2}\Bigr]\notag\\[2pt]
&=\frac{\sin^{2}\theta^{*}\,(1+\cos^{2}\theta^{*})}{2}
=\frac{1-\cos^{4}\theta^{*}}{2},
\label{eq:tt-spin2-deriv}
\end{align}
\begin{equation}
\boxed{\;
\frac{1}{\sigma}\frac{d\sigma}{d\cos\theta^{*}}\Big|_{G_1}
=\frac{5}{8}\,\bigl(1-\cos^{4}\theta^{*}\bigr)\;+\;O(1-\beta_t^{2}).}
\label{eq:tt-spin2}
\end{equation}
The distribution \emph{vanishes} along the beam --- the maximal possible contrast with \eqref{eq:tt-spin1}, which peaks there.  Had the graviton been produced from \(q\bar q\), the distribution would instead be \(1-3\cos^{2}\theta^{*}+4\cos^{4}\theta^{*}\); in \GG{} the \(q\bar q\) coupling is negligible by Corollary T-KK-2c, so the pure-\(gg\) form \eqref{eq:tt-spin2} is a \emph{derived} prediction rather than a benchmark choice --- a purity generic RS templates cannot claim, and itself a test: an admixture of the \(q\bar q\) form in an observed spin-2 signal would contradict the derived flat-overlap hierarchy.

\subsection{Separation power, quantified}
The fraction of events at \(|\cos\theta^{*}|>0.7\) is \(0.39\) (spin 1) versus \(0.17\) (spin 2); Figure~\ref{fig:tt-angular} draws both shapes.  For a shape-only hypothesis test, the symmetrized Kullback--Leibler divergence of \eqref{eq:tt-spin1} and \eqref{eq:tt-spin2} is \(\overline{\rm KL}=0.212\) per event, giving
\(N_{3\sigma}=9/(2\overline{\rm KL})\simeq21\) and \(N_{5\sigma}\simeq59\) events for \(3\sigma\) and \(5\sigma\) discrimination.  Folding a representative boosted-top acceptance cut \(|\cos\theta^{*}|<0.85\) (the beam region is where detectors are weakest) raises these to \(N_{3\sigma}\lesssim40\) --- still an order of magnitude below the expected signal yield (\S\ref{sec:ttyields}).

\begin{figure}[t]
\centering
\resizebox{0.82\textwidth}{!}{\includegraphics{Arisaka-GG-P1-KK-ttrep-angular.pdf}}
\caption[Derived production-angle distributions: spin 1 versus spin 2]{Normalized production-angle distributions at \(3.849\)~TeV, Eqs.~\eqref{eq:tt-spin1}--\eqref{eq:tt-spin2}.  The spin-1 pair rises toward the beam; the \(gg\)-produced spin-2 graviton vanishes there.  Twenty-one reconstructed events separate the hypotheses at \(3\sigma\) by shape alone.}
\label{fig:tt-angular}
\end{figure}

\section{Top Polarization and Spin Correlations}
\label{sec:ttpol}

\subsection{Longitudinal polarization: \texorpdfstring{$P_t=+0.91$}{Pt = +0.91} versus zero}
At \(\beta_t\to1\), chirality coincides with helicity, so the resonance populates right- and left-handed tops in proportion \(r_{t_R}^{2}:r_{t_L}^{2}\).  The longitudinal polarization is therefore a pure coupling ratio --- independent of PDFs, K-factors, luminosities, and even of \(\xi_{\rm UV}\):
\begin{equation}
\boxed{\;P_t=\frac{r_{t_R}^{2}-r_{t_L}^{2}}{r_{t_R}^{2}+r_{t_L}^{2}}
=\frac{20.25-1}{20.25+1}=+0.906,
\qquad P_t\in[+0.88,\,+0.92]\ \text{for }r_{t_R}\in[4,5],}
\label{eq:tt-Pt}
\end{equation}
identical for \(g^{*}\) and \(Z'_B\) (same overlap pattern), with the finite-mass dilution entering only at \(O(1-\beta_t^{2})\).  For \(G_1\) the coupling is the parity-even \(T^{\mu\nu}\): \(P_t=0\) identically [T].  The polarization thus measures directly the one [D\(^{*}\)] input of the collider chain --- the cascade-depth ratio \(r_{t_R}/r_{t_L}\) --- and its \emph{sign} (right-handed) is the statement that \(t_R\) sits deeper in the infrared than \(t_L\), i.e.\ a direct image of the \citet{ArisakaA9SM} cascade ordering.

\paragraph{Measurement.}  In the top rest frame the charged lepton from \(t\to bW\to b\ell\nu\) analyzes the top spin with unit analyzing power at leading order (\(\alpha_\ell=1\), a consequence of the V\(-\)A weak vertex):
\begin{equation}
\frac{1}{\Gamma}\frac{d\Gamma}{d\cos\theta_\ell}=\frac12\bigl(1+P_t\,\alpha_\ell\cos\theta_\ell\bigr),
\qquad \theta_\ell \equiv \angle(\ell^{\pm},\ \text{top spin axis}),
\label{eq:tt-analyser}
\end{equation}
so the \GG{} spin-1 hypothesis predicts a slope of \(+0.91\) and the spin-2 hypothesis a flat line: with \(\sim\!10^{2}\) leptonic-side tops the two are separated at \(>5\sigma\).  Note the complementarity: the production angle (\S\ref{sec:ttangular}) tests the \emph{spin} of the resonance; the lepton slope tests its \emph{chirality structure}; the two observables fail independently.

\subsection{Spin correlations: a consistency check, not a discriminator}
Both spin-1 (\(\lambda_R=\pm1\)) and spin-2 (\(\lambda_R=\pm2\)) decays populate \(\Delta\lambda=\pm1\) at \(\beta_t\to1\): the pair emerges with \(\lambda_t=-\lambda_{\bar t}=\pm\tfrac12\) (opposite helicity labels; spins spatially aligned, the momenta being back-to-back).  This correlation structure is \emph{identical} for the two spins, so \(\ttbar\) spin correlations do not separate them here; what they do test is the universality of the \(\Delta\lambda=0\) mass suppression (\(0.8\%\) for both).  We flag this honestly because spin correlations are sometimes advertised as a generic spin analyzer; at \(\beta_t=0.996\) they are not --- the discriminators are \(\theta^{*}\), \(P_t\), and the line shape.

\section{System Kinematics}
\label{sec:ttkin}

The \(gg\) versus \(q\bar q\) initial states sample different parton \(x\)-distributions, so the \(\ttbar\) \emph{system} rapidity distinguishes \(G_1\) from the spin-1 pair in principle.  At \(m_{\ttbar}=3.849\)~TeV, however, phase space confines \(|y_{\ttbar}|<\ln(\sqrt s/m)=1.22\), and the honest verdict is that the discriminator is \emph{weak}: \(\langle|y|\rangle=0.29\) (\(gg\)) versus \(0.32\) (\(q\bar q\)), with \(16\%\) versus \(21\%\) of events at \(|y|>0.5\) (Figure~\ref{fig:tt-rapidity}).  Rapidity is a consistency check, not a spin measurement, and we say so because the folklore ``\(gg\) is central, \(q\bar q\) is forward'' promises more than lock-scale kinematics can deliver.  The robust kinematic facts are simpler and analysis-relevant: each top emerges with \(p\simeq\Mlock/2=1.92\)~TeV (\(\gamma_t\simeq11\), decay cone \(\approx2/\gamma_t\approx0.18\)) --- squarely in large-\(R\) top-tagging territory; and the \(g^{*}\) width places \(\sim\!50\%\) of the Breit--Wigner within \(|m_{\ttbar}-3849|<280\)~GeV, several times wider than the \(3\)--\(4\%\) experimental \(m_{\ttbar}\) resolution, so \emph{the observed width is physics, not detector} --- the line-shape program of \S\ref{sec:ttlineshape} is measurable.

\begin{figure}[t]
\centering
\resizebox{0.78\textwidth}{!}{\includegraphics{Arisaka-GG-P1-KK-ttrep-rapidity.pdf}}
\caption[System-rapidity distributions: \(gg\) versus \(q\bar q\) initial states]{Normalized \(\ttbar\)-system rapidity at \(m_{\ttbar}=3.849\)~TeV for the two initial states (CTEQ6M).  The discrimination is real but weak --- reported as such. Two initial states give two rapidity shapes, and the separation between them is smaller than the separation in the angular distribution or the line shape. It is plotted anyway, and described as weak in the caption rather than in a footnote, because a discrimination suite that showed only its strongest member would misrepresent what an analysis actually has to work with.}
\label{fig:tt-rapidity}
\end{figure}

\section{Continuum, S/B, and Yields}
\label{sec:ttyields}

The LO Standard-Model continuum (\(q\bar q+gg\to\ttbar\), Combridge formulas, \(\mu=m_{\ttbar}\), the paper's \(\alpha_s\) convention) integrates to \(\sigma_{\ttbar}^{\rm LO}=331\)~pb at 13~TeV --- the expected factor \(\sim\!2.5\) below the NNLO+NNLL \(833\)~pb, reflecting the well-known scale sensitivity of LO \(\ttbar\); we therefore use it only in \emph{ratios}, where signal and continuum K-factors largely cancel (both sides here are LO).  At the lock scale the continuum is \(d\sigma/dm_{\ttbar}=1.5\)~ab/GeV.  In the octet window \(|m_{\ttbar}-3849|<\Gamma/2=280\)~GeV: continuum \(0.95\)~fb against Breit--Wigner signal \(3.34\)~fb ---
\begin{equation}
\boxed{\;S/B\big|_{\rm window}=3.5\ \text{(before the interference gain)},\qquad
(S{+}I)/B\big|_{\rm pole}=19.\;}
\end{equation}
Per 3~ab\(^{-1}\): \(\sim\!10^{4}\) signal events produced in the window; with BR\((\ell+{\rm jets})=0.29\) and a representative boosted-selection efficiency of \(12\%\) [E], \(\sim\!350\) reconstructed lepton\(+\)jets events, before the all-hadronic sample --- enough for the 21-event spin separation, the \(P_t\) slope, and a multi-bin line-shape fit simultaneously.  The same accounting confirms the other two members from the background side: \(G_1\) has \(S/B=0.04\) in its window (LHC-invisible, as Corollary T-KK-2c concluded from the rate side), and \(Z'_B\) reaches only \(S/B=0.47\) at the very top of the L-KK-1 menu.

\section{The Discrimination Matrix and the Pre-Registered Fingerprint}
\label{sec:ttmatrix}

\begin{table}[H]
\centering\small
\setlength{\tabcolsep}{4pt}
\caption[The discrimination matrix at \(3.849\)~TeV.]{The discrimination matrix at \(3.849\)~TeV.  Shape rows are normalization-free and [T]-grade unless noted.  ``Falsifies'' names what a contrary measurement kills. The last column is what makes this a matrix rather than a list: each row names the measurement that would kill the reading, not merely the one that would support it. Shape rows are normalization-free, which means they survive a wrong rate; that is why they carry the theorem grade and the rate-dependent rows do not.}
\label{tab:tt-matrix}
\resizebox{\textwidth}{!}{%
\begin{tabular}{p{3.0cm}p{3.6cm}p{3.0cm}p{2.6cm}p{2.9cm}}
\toprule
\textbf{Observable} & \(\bm{g^{*}}\) \textbf{(spin 1, octet)} & \(\bm{Z'_B}\) \textbf{(spin 1, singlet)} & \(\bm{G_1}\) \textbf{(spin 2)} & \textbf{Falsifies if wrong}\\
\midrule
\(\sigma\times\)BR\((\ttbar)\), LO\(\times\)K & \(8.7\,[7.0\text{--}12]\) fb & \(0.016\)--\(0.59\) fb & \(0.045\) fb & F-KK-2B (rate)\\
\(dN/d\cos\theta^{*}\) & \(\tfrac38(1+c^{2})\) & \(\tfrac38(1+c^{2})\) & \(\tfrac58(1-c^{4})\) & spin assignment\\
\(f(|\cos\theta^{*}|>0.7)\) & \(0.39\) & \(0.39\) & \(0.17\) & spin assignment\\
\(A_{FB}\) (resonant) & \(0\) exactly [T] & \(0\) exactly [T] & \(0\) (parity) & T-KK-2 overlap\\
\(P_t\) (lepton slope) & \(+0.91\,[+0.88,+0.92]\) & \(+0.91\) & \(0\) [T] & cascade ordering\\
line shape & BW(\(\Gamma/m{=}15\%\)) + interference: low shoulder, collapsed high tail, deficit \(>4.9\) TeV & pure BW, narrow (\(0.2\)--\(8\%\)) & pure BW, \(6.3\%\); invisible & T-KK-2 sign [T] / color\\
QCD interference & full (octet) [T] & none (singlet) [T] & negligible & color assignment\\
centroid pull & \(-O(100)\) GeV & none & none & template honesty\\
companions & \(b\bar b/jj = 3.5\!:\!1\) rel.\ pattern & similar & \(hh,VV\) instead & BR pattern\\
\(y_{\ttbar}\) & \(\langle|y|\rangle=0.32\) & \(0.32\) & \(0.29\) & (consistency only)\\
\bottomrule
\end{tabular}}
\end{table}

\paragraph{The pre-registered fingerprint.}  Combining the rows, the \GG{} discovery scenario at the HL-LHC is a single composite statement, registered here before the data:
\begin{quote}
\emph{A broad (\(\Gamma/m\simeq15\%\)), top-saturated excess at \(3.849\)~TeV with (i)~a \((1+\cos^{2}\theta^{*})\) production angle, (ii)~\(+0.91\) right-handed top polarization, (iii)~zero resonant forward--backward asymmetry, (iv)~a constructive-below/destructive-above interference line shape with an \(O(100)\)~GeV downward centroid pull, and (v)~essentially nothing in dileptons or diphotons.}
\end{quote}
Each clause can fail independently, and each failure names its culprit: (i) the spin, (ii) the cascade ordering, (iii) the vector-like UV overlap, (iv) the T-KK-2 sign or the color assignment, (v) leptophobia.  Conversely, an excess at \(3.849\)~TeV \emph{without} this fingerprint --- however exciting --- would not be the \GG{} lock cluster, and the framework would forgo the claim.  This is the quantitative content behind \S\ref{sec:channel-by-channel}'s statement that \(\ttbar\) is both the discovery and the diagnostic channel.

\section{Reliability and Open Ends}
\label{sec:ttreliability}

\paragraph{Shelving.}  [T]-grade and template-independent: the angular forms \eqref{eq:tt-spin1}--\eqref{eq:tt-spin2}, \(A_{FB}=0\), \(P_t=0\) for \(G_1\), the interference sign theorem, and the interference yes/no color dichotomy.  [D\(^{*}\)]: \(P_t=+0.906\) (band \(0.88\)--\(0.92\) from the cascade-depth bracket); the width \(\Gamma/m=0.146\); all LO\(\times\)K normalizations with their stated \(\pm20\%\) PDF/scale allowances (the CTEQ6-era distributions at \(x\sim0.3\)--\(0.6\) are the largest single systematic --- and electroweak Sudakov corrections, a negative \(\mathcal O(10\text{--}20\%)\) at these masses, are not included: they suppress continuum and coherent sum together, so data-anchored normalizations absorb them at first order, but a complete EW-corrected template belongs with the same NLO program; a modern-set recomputation would move normalizations, not shapes).  [E]: K-factors \citep{ChivukulaEtAl2012,MathewsRavindran2005}, the \(12\%\) reference efficiency, and the LO/LO convention of the \(S/B\) ratios (\(\pm30\%\) honest band).

\paragraph{What would sharpen this Part.}  (i)~Closing the cascade-depth lemma pins \(P_t\) and the BR pattern to single values.  (ii)~The L-KK-1 selection pins the \(Z'_B\) row.  (iii)~An NLO recomputation of Eq.~\eqref{eq:tt-lineshape} with a modern PDF set.  (iv)~Above all, a dedicated experimental \(\Gamma/m\simeq15\%\) template \emph{including interference}: fitting a pure Breit--Wigner to an interference-distorted shape biases both the mass and the rate, and the most valuable single deliverable ATLAS and CMS could produce for this test is a template family with the interference degree of freedom included --- with its sign left free, so that the data can convict or acquit the T-KK-2 prediction rather than assume it.

Part~VI derived the signal's face, in plain terms: how the decay products spread in angle (the fingerprint of spin), how the tops emerge polarized (the fingerprint of the coupling's handedness), and how the resonance interferes with the ordinary background --- enhancing it below the peak, depleting it above, with the sign a theorem rather than a fit.  These shapes are the paper's most robust predictions, immune to the rate bands, and they are what will separate this cluster from any impostor.  Shapes and rates together now form a complete target; the natural next question is where the experiments actually stand.  The next Part reviews the published record: what is excluded, what is not, and why the prediction sits exactly at the live frontier.


\section[Part~VI in plain English]{Part~VI in plain English: what the signal would look like in detail, and how sure each detail is}
If something is there, this Part is about telling what it is.

Three hypotheses can produce a bump in top-antitop near the same mass: the
spin-2 state, the color-octet vector, and an ordinary single resonance. The
Part works out how each would differ in the details --- the angular
distribution of the decay products, the polarization of the tops, the shape
of the line including the sign of its interference with the continuum.

The top quark is the right place to look for an unusual reason: it decays
before it can hadronize, so its spin is inherited by its decay products
rather than scrambled. That makes polarization a measurable quantity here
in a way it is not for lighter quarks.

The grading is what makes the Part usable. Shape observables are
normalization-free, so they are theorem-grade and survive any error in the
rate. Rate-dependent statements are marked separately and carry the
assumptions of the previous Part. And the rapidity distribution, which
would have been a fourth handle, discriminates only weakly --- the Part
says so in the caption of its own figure rather than leaving it to be
discovered. A discrimination suite that showed only its strongest member
would be a misleading one.

\GGPart[Experimental status and reach]{Experimental Status and Reach}{Current ATLAS/CMS limits, near-term reach, the kill criterion, and future machines}

\section{Current Experimental Status at ATLAS and CMS}
\label{sec:current-status}

This section is deliberately grounded in recent public collaboration outputs and recent official CERN, ATLAS, and CMS machine documents.  Because the current status of the LHC can change on the scale of months, this is exactly the type of section that deserves freshness.

\subsection{The luminosity backdrop}

Official CERN reporting from December 2025 states that the LHC has delivered over \(515\,\mathrm{fb}^{-1}\) each to ATLAS and CMS across the proton-run periods, with \(124\,\mathrm{fb}^{-1}\) and \(125\,\mathrm{fb}^{-1}\) delivered to each experiment in 2024 and 2025 respectively \citep{CERN2025Luminosity}.  Official CERN reporting from March 2026 further states that the 2026 proton run will be short but intense and will extend the integrated data set beyond the already achieved Run-3 target \citep{CERN2026Run}.  This backdrop matters because it tells us that the statistical frontier is rapidly shifting, even if the public analysis papers naturally lag the delivered luminosity.

\subsection{High-mass dilepton constraints}

ATLAS published a full Run-2 high-mass dilepton search using \(139\,\mathrm{fb}^{-1}\) at \(13\,\mathrm{TeV}\) and found no significant deviation from the Standard Model expectation \citep{ATLAS2019Dilepton}.  CMS, using the full \(138\,\mathrm{fb}^{-1}\) Run-2 dilepton sample, reports lower RS-graviton mass limits of \(2.47\) to \(4.78\,\mathrm{TeV}\) for couplings \(k/\bar M_{\mathrm{Pl}}=0.01\) to \(0.1\) on its official public-results page \citep{CMS2020DileptonPublic}.  This result is one of the cleanest guides to generic spin-1 or spin-2 resonance sensitivity.  Its strength is obvious: excellent mass resolution and low background.  Its weakness for the present purpose is equally clear: if \GG{} suppresses light-lepton couplings relative to top-rich and dijet-rich channels, the dilepton reach overstates the exclusion power against the specific \GG{} benchmark.

\subsection{High-mass diphoton constraints}

CMS published a high-mass diphoton search using the full Run-2 \(13\,\mathrm{TeV}\) data set.  In the abstract-level summary of that result, masses below roughly \(2.2\) to \(5.6\,\mathrm{TeV}\) are excluded at \(95\%\) confidence level for an excited Randall--Sundrum graviton over the coupling range \(0.01<\tilde{k}<0.2\) \citep{CMS2024Diphoton}.  This is a powerful statement for canonical RS-like diphoton benchmarks.  It is not, however, the last word on \GG{} for two reasons.  First, \GG{} is not a single-pole RS model.  Second, the same geometry that enhances top and hadronic channels can reduce the light-photon branch relative to a universal RS template.  The diphoton paper therefore tells us that standard-coupling RS-like interpretations are already above the nominal \(3.85\,\mathrm{TeV}\) scale, while the dedicated \GG{} reinterpretation remains open.

\subsection{Dijet and substructure-based searches}

The dijet channel is unavoidable for \GG{} because of the mild \(\UB\) admixture and the possibility of enhanced hadronic decays.  CMS has published a model-agnostic search for dijet resonances with anomalous jet substructure in the mass range \(1.8\) to \(6\,\mathrm{TeV}\), again with no significant excess observed \citep{CMS2024DijetAnomalous}.  The importance of this result for the present paper is not merely the null result itself, but the method: substructure-assisted dijet searches are the right conceptual direction for a clustered, possibly partly hadronic, 6D signal.  The current public limit is therefore better interpreted as a \emph{proof of method} plus an initial exclusion map, rather than as the final word on the full \GG{} cluster hypothesis.

\subsection{Top-pair resonance searches}

Top-pair resonance searches are arguably the most important public channel for the \GG{} benchmark.  The most relevant recent reference is the ATLAS search for \(\ttbar\) resonances in the single-lepton and dilepton final states using the full Run-2 \(140\,\mathrm{fb}^{-1}\) data set \citep{ATLAS2025ttbar}.  That paper explicitly reviews both bulk RS graviton and KK gluon benchmarks.  In the bulk-RS graviton discussion, ATLAS emphasizes that in a bulk-RS setup decays to light fermions are suppressed, the branching fraction to photons is negligible, and the phenomenologically interesting couplings can be large \citep{ATLAS2025ttbar}.  This is already much closer in spirit to the \GG{} search problem than a simple dilepton-only benchmark.

The same ATLAS paper also summarizes older direct exclusions relevant to our context.  For example, previous ATLAS searches with \(36\,\mathrm{fb}^{-1}\) excluded bulk RS gravitons in the \(0.45\) to \(0.65\,\mathrm{TeV}\) range in a \(\ttbar\)-focused benchmark, WW/ZZ searches excluded up to \(2.3\,\mathrm{TeV}\), and a b-tagged dijet search excluded a \(k/\mplred=0.2\) graviton benchmark up to \(2.8\,\mathrm{TeV}\) \citep{ATLAS2025ttbar}.  For KK gluons, the same ATLAS review notes that previous ATLAS searches excluded benchmark KK-gluon masses below \(3.7\,\mathrm{TeV}\), while CMS excluded similar benchmark masses up to \(4.55\,\mathrm{TeV}\) \citep{ATLAS2025ttbar}.  The precise numerical mapping to \GG{} is again benchmark-sensitive, but the qualitative lesson is clear: \(\ttbar\) is already an experimentally mature high-mass resonance channel, and it is probably the most natural first discovery channel for the \GG{} first cluster.

\subsection{Higgs-pair searches}

The \(hh\) channel is where the spin-2 graviton's coupling to the IR-localized Higgs sector becomes visible (there is no radion in \GG{}; \S\ref{subsec:branchings} and \citet{ArisakaA4Euclid}).  CMS has published a search for resonant Higgs-boson pair production in the \(bbbb\) final state using large-area jets and the full \(138\,\mathrm{fb}^{-1}\) Run-2 sample \citep{CMS2024HHbbbb}.  Narrow bulk-graviton interpretations were tested over \(1\) to \(3\,\mathrm{TeV}\), with upper limits on \(\sigma(pp\to X)\times\mathrm{BR}(X\to HH\to bbbb)\) down to the sub-femtobarn level at high masses \citep{CMS2024HHbbbb}.  Extending such a search to the fixed \(3.85\,\mathrm{TeV}\) window, as a bosonic cross-check of a \(\ttbar\)/dijet first signal, is the natural \GG{}-aware use of this channel.



\subsection{A channel-dependent exclusion and discovery ledger}

Because current public exclusions are quoted in a mixture of benchmark languages, the most honest way to summarize the status is to place the official channel limits next to the corresponding \GG{} reinterpretation issue.  Table~\ref{tab:exclusion-ledger} does exactly that.  It is intentionally channel-centric rather than paper-centric.

\begin{landscape}
\scriptsize
\begin{longtable}{>{\raggedright\arraybackslash}p{2.2cm} >{\raggedright\arraybackslash}p{3.1cm} >{\raggedright\arraybackslash}p{2.7cm} >{\raggedright\arraybackslash}p{3.5cm} >{\raggedright\arraybackslash}p{4.3cm} >{\raggedright\arraybackslash}p{3.9cm}}
\captionsetup{font=footnotesize}
\caption[Channel-by-channel exclusion and discovery ledger.]{Channel-by-channel exclusion and discovery ledger.  The ``public benchmark'' column records what the collaborations have actually excluded in standard templates; the ``GG-6 reading'' column explains why those numbers are highly informative but not yet equivalent to a full geometry-aware exclusion.}\label{tab:exclusion-ledger}\\
\toprule
\textbf{Channel} & \textbf{Representative official source} & \textbf{Public benchmark limit} & \textbf{What that benchmark really constrains} & \textbf{GG-6 reading of the same result} & \textbf{Near-term discovery note}\\
\midrule
\endfirsthead
\toprule
\textbf{Channel} & \textbf{Representative official source} & \textbf{Public benchmark limit} & \textbf{What that benchmark really constrains} & \textbf{GG-6 reading of the same result} & \textbf{Near-term discovery note}\\
\midrule
\endhead
Dileptons & CMS EXO-19-019 official public result \citep{CMS2020DileptonPublic} & for RS spin-2, \(m_G>2.47\)--\(4.78\,\mathrm{TeV}\) for \(k/\bar M_{\mathrm{Pl}}=0.01\)--0.1 & canonical or narrow RS-like spin-2 couplings to light leptons & very informative but not decisive for GG-6 because the light-lepton overlap can be suppressed by factors of a few and the \(\theta\)-cluster broadens a one-pole fit & in the internal GG-6 corridor the dilepton line can still yield a clean corroborating excess, but it is not expected to dominate the significance\\
Diphotons & CMS EXO-22-024 official public result \citep{CMS2024DiphotonPublic} & RS graviton masses below \(2.2\)--\(5.6\,\mathrm{TeV}\) excluded for \(0.01<\tilde k<0.2\) & standard RS-like diphoton graviton signal & again very strong for canonical RS, but GG-6 predicts both reduced photonic weight and cluster dilution & a good clean channel for corroboration; not the only or necessarily strongest first channel\\
Top pairs & official ATLAS/CMS top-resonance program \citep{ATLAS2025ttbar} & ATLAS KK-gluon benchmark exclusion \(\sim 3.8\,\mathrm{TeV}\) in lepton+jets; older all-hadronic limits around \(3.4\,\mathrm{TeV}\); CMS KK-gluon dijet benchmark below \(4.55\,\mathrm{TeV}\) & standard KK-gluon or bulk-warped top-rich resonances & this is the closest public proxy to the actual GG-6 search because GG-6 is top-rich by construction & the most promising single discovery channel already in the current Run-3 closeout and the natural place to look ``this year''\\
Dijets & CMS anomalous-substructure dijet program \citep{CMS2024DijetAnomalous} & no significant excess in the \(1.8\)--\(6\,\mathrm{TeV}\) range; benchmark KK-gluon exclusions reach above \(4\,\mathrm{TeV}\) depending on template & narrow hadronic resonances and substructure-assisted broad objects & especially relevant for the leptophobic \(\UB\) admixture, which can shift discovery power away from dileptons & very plausible companion channel for the spin-1 doublet and possibly for the graviton family if cluster broadening is important\\
\(hh\) (graviton channel) & CMS large-area-jet \(hh\) searches and diboson resonance program \citep{CMS2024HHbbbb} & bulk-graviton \(hh\) interpretations constrained in the \(1\)--\(3\,\mathrm{TeV}\) range & narrow spin-2 templates & GG-6 has no radion; the scalar-sector handle is \(G_1\to hh\), with small rate at \(3.85\,\mathrm{TeV}\) & a bosonic corroborator rather than the first discovery channel; useful once a 4 TeV family appears\\
Dibosons & ATLAS and CMS heavy-resonance bosonic searches & several-TeV exclusions for standard spin-1 and spin-2 benchmarks & high-mass boson-rich states with more standard templates & useful cross-check because GG-6 overlap structure still populates \(WW\) and \(ZZ\) visibly, especially from the spin-2 graviton & important for confirming the bosonic side of the 6D cluster after a first hint in tops or dijets\\
\bottomrule
\end{longtable}
\normalsize
\end{landscape}

The interpretive lesson of the table is straightforward.  The public LHC record is already strong enough that any claim of a completely unconstrained 3.85 TeV KK story would be irresponsible.  But the opposite extreme -- declaring the GG-6 implementation already dead solely from canonical one-pole RS templates -- is also too strong.  The correct status is sharper and more interesting: the first-family GG-6 target is \emph{bracketed}.  It is under real pressure.  It can still survive because the exact same geometric ingredients that make it experimentally attractive also make a naive RS translation too aggressive.

\subsection{What the public status does and does not prove}

The public ATLAS/CMS landscape as of early 2026 therefore proves three things and leaves three things open.

It proves, first, that simple standard-coupling RS-like spin-2 benchmarks in the cleanest leptonic and photonic channels are already heavily constrained above the \(3.85\,\mathrm{TeV}\) scale.  It proves, second, that top and dijet resonance searches are robust, mature, and already pushing into the mass range relevant for associated \GG{} partners.  It proves, third, that Higgs-pair and diboson resonance searches are already serious parts of the BSM program.

But it leaves open, first, the dedicated reinterpretation of a 6D clustered spin-2 signal centered at \(3.85\,\mathrm{TeV}\).  It leaves open, second, the combined multi-channel exclusion or discovery of a top-rich, mildly leptophobic, multi-field graviton cluster.  And it leaves open, third, the joint pattern test involving the first cluster and the second radial family.  Those are precisely the gaps the rest of this paper is meant to organize.

\paragraph*{Operational reading of the current public status.}
At the level of \emph{public benchmark results}, ATLAS and CMS have already pushed standard-coupling RS-like spin-2 interpretations into or above the nominal \(3.85\,\mathrm{TeV}\) region in the cleanest channels.  At the level of a \emph{dedicated \GG{} reinterpretation}, however, the decisive ingredients remain missing: a clustered line-shape benchmark, a top-rich and mildly leptophobic branching pattern, and a combined-channel exclusion corridor tied to the geometry rather than to a freely scanned mass template.  That gap between \emph{proxy exclusions} and a \emph{true \GG{} test} is exactly why the paper treats the present moment as decisive rather than already settled.

\clearpage
\begingroup\scriptsize\setlength{\tabcolsep}{3.5pt}
\begin{longtable}{>{\raggedright\arraybackslash}p{2.0cm} >{\raggedright\arraybackslash}p{2.8cm} >{\raggedright\arraybackslash}p{1.7cm} >{\raggedright\arraybackslash}p{3.9cm} >{\raggedright\arraybackslash}p{4.2cm}}
\caption[Current public ATLAS/CMS status relevant to the \GG{} KK search program.]{Current public ATLAS/CMS status relevant to the \GG{} KK search program.  The table intentionally distinguishes between what is already strongly excluded for standard benchmark models and what remains open for a dedicated clustered \GG{} reinterpretation. The distinction the table draws is between an exclusion and an exclusion of something else. Standard benchmarks are strongly excluded over the relevant mass range and that is stated first, without qualification; what remains open is the clustered reading, which no published search has been optimized for. Every entry is from a public record with its reference attached.}\label{tab:status}\\
\toprule
\textbf{Search class} & \textbf{Representative public result} & \textbf{Dataset} & \textbf{What the public result says} & \textbf{What remains open specifically for GG-6}\\
\midrule
\endfirsthead
\toprule
\textbf{Search class} & \textbf{Representative public result} & \textbf{Dataset} & \textbf{What the public result says} & \textbf{What remains open specifically for GG-6}\\
\midrule
\endhead
High-mass dileptons & ATLAS high-mass dilepton resonance search \citep{ATLAS2019Dilepton} & 139 fb\(^{-1}\), 13 TeV & No significant excess; excellent clean-channel sensitivity to generic high-mass spin-1 and spin-2 signals. & Does not directly target a clustered 6D line shape or a top-rich / light-lepton-suppressed overlap pattern.\\
High-mass diphotons & CMS diphoton resonance search \citep{CMS2024Diphoton} & full Run 2, 13 TeV & RS graviton excited states excluded roughly in the 2.2--5.6 TeV window across \(0.01<\tilde{k}<0.2\). & Public limit is on canonical RS-like diphoton structure; \GG{} may have reduced diphoton branching and non-single-pole structure.\\
Anomalous dijets & CMS model-agnostic anomalous-substructure dijet search \citep{CMS2024DijetAnomalous} & 138 fb\(^{-1}\), 13 TeV & No significant excess in 1.8--6 TeV mass range. & Dijet channel is especially important for a mild \(\UB\) admixture; cluster-aware and top-correlated reinterpretation remains open.\\
Top-pair resonances & ATLAS \(\ttbar\) resonance search \citep{ATLAS2025ttbar} & 140 fb\(^{-1}\), 13 TeV & Mature search program for bulk-RS graviton and KK gluon benchmarks; strong exclusions for some top-rich models. & \(\ttbar\) is likely the natural leading discovery channel for \GG{}; dedicated first-cluster reinterpretation remains needed.\\
Higgs-pair resonances & CMS \(HH\to bbbb\) resonant search \citep{CMS2024HHbbbb} & 138 fb\(^{-1}\), 13 TeV & Sub-femtobarn to few-femtobarn upper limits over 1--3 TeV for bulk-graviton \(HH\) interpretations. & GG-6 has no radion \citep{ArisakaA4Euclid}; the \(G_1\to hh\) branch is the scalar-sector handle, and a dedicated \(3.85\,\mathrm{TeV}\) reinterpretation is not yet performed.\\
Machine dataset status & CERN official luminosity updates \citep{CERN2025Luminosity,CERN2026Run} & official machine reports & More than 515 fb\(^{-1}\) delivered each to ATLAS/CMS overall; 2026 run extends Run-3 dataset beyond target. & Full Run-3 + early-HL-LHC reinterpretation will substantially sharpen the first-cluster test.\\
\bottomrule
\end{longtable}\endgroup


\section{The Operational Confrontation: Computed \texorpdfstring{$\sigma\times\mathrm{BR}$}{sigma x BR} versus Published Limits}
\label{sec:operational}

\subsection{What this section adds, and at what grade}

The two preceding sections state \emph{what} ATLAS and CMS have excluded and \emph{why} the \GG{} cluster is not equivalent to any single public benchmark.  A referee is entitled to ask for more: \emph{a number an analyst can exclude}.  This section supplies it.  Starting from the Read-out-II couplings (\S\ref{sec:scenarioB}) we compute, channel by channel, the leading-order narrow-resonance cross sections of all three lock-cluster states, place them on the same axes as the published 95\%~CL exclusion curves, and state for each channel whether the prediction is below the limit --- and by what factor --- or in tension.

The honesty grade is fixed at the outset.  Everything in this section is \emph{Derived-conditional} (D\(^{*}\)): conditional on the Read-out-II assumptions B1--B4, on leading-order QCD with the stated PDF and K-factor treatment, and on the published-template transfers stated below.  Nothing here touches the theorem shelf (the masses, the exact leptophobia, the coupling identities of T-KK-2).  Where this computation supersedes the template-rescaled ballparks quoted in \S\ref{sec:scenarioB}, we say so explicitly and keep both visible.

\subsection{The workbook: master formula, inputs, and validation}
\label{subsec:workbook}

\paragraph{Master formula.}  For a resonance \(R\) of spin \(J\), color dimension \(D_R\), and mass \(M\), the leading-order narrow-width hadronic rate is
\begin{equation}
\sigma(pp\to R\to f)\;=\;\sum_{ij}\,
\underbrace{(1+\delta_{ij})\,\frac{16\pi^{2}\,(2J+1)\,D_R}{(2S_i+1)(2S_j+1)\,D_i D_j}}_{P_{ij}}\;
\frac{\Gamma_{ij}}{M}\;\frac{\mathcal{L}_{ij}(\tau)}{s}\;\mathrm{BR}(R\to f),
\qquad
\mathcal{L}_{ij}(\tau)=\int_{\tau}^{1}\frac{dx}{x}\,f_i(x)f_j\!\left(\frac{\tau}{x}\right),
\label{eq:nwa-master}
\end{equation}
with \(\tau=M^{2}/s\) and both orderings summed for \(i\neq j\).  This is Eq.~\eqref{eq:kappafact} with the kinematic factor written out; the spin/color normalization reproduces the standard coefficients of \citet{FranceschiniEtAl2016} for every color-singlet case and the color-octet factor \(P_{q\bar q}=32\pi^{2}/3\) fixed by detailed balance.

\paragraph{Inputs.}  The couplings are those the paper has already put on the table.  For the color octet \(g^{*}\): light quarks at the theorem-fixed \(\xi_{\mathrm{UV}}=-0.194\) (Eq.~\ref{eq:tkk2-xiflat}), \(Q_{3L}\) at \(1.0\,g_s\), and \(t_R\) at \(4.5\,g_s\) (Read-out-II range 4--5, assumption B4), with \(\Gamma(g^{*}\to q\bar q)=\alpha_s\,r_q^{2}\,m/6\) per flavor --- the color-octet trace, not the color-singlet \(N_c\) factor, as in the standard KK-gluon literature \citep{LillieRandallWang2007,DobrescuYu2013}.  For the baryonic \(Z'_B\): \(g_f=g_B B_f r_f\) with \(B_q=1/3\) and the same overlap pattern; every rate scales as \(g_B^{2}\) and is quoted at the reference point \(g_B=0.2\) \citep{DobrescuYu2013}; the theorem-grade norm plus the L-KK-1 menu bracket \(g_B\) to 0.14--0.89 (\S\ref{subsec:gbmenu}).  For the spin-2 \(G_1\): gluon-fusion production through the \emph{derived} flat-overlap \(\kappa_{gg}=c_K/K=0.0136\) (Corollary T-KK-2c, \S\ref{subsec:kappagg}) with the Han--Lykken--Zhang partial widths \citep{HanLykkenZhang1999} (the massive-vector polynomial corrected per \citealp{AllanachEtAl2000}), and the bulk decay pattern: the infrared channels \(t\bar t\), \(W_LW_L\), \(Z_LZ_L\), \(hh\) at full strength, flat-mode channels (\(gg\), \(\gamma\gamma\)) suppressed by \(\kappa_{gg}^{2}\), light fermions negligible \citep{AgasheEtAl2007}.  The strong coupling is run from the \citet{ArisakaA9SM} value \(\alpha_s(M_Z)=1/(\pi e)\) to \(\alpha_s(3.85\,\mathrm{TeV})=0.077\).  Parton luminosities use CTEQ6M \citep{Pumplin2002CTEQ6}, cross-checked with CTEQ6L1; at \(\sqrt s=13\)~TeV:
\begin{table}[htbp]
\centering
\small
\caption[Parton luminosities at the two candidate masses.]{Parton luminosities at the two candidate masses. These are the numbers every rate in this paper is built on and they are somebody else's measurement, not this framework's prediction: they come from a published parton distribution set, cross-checked against a second one, and are quoted here so that a reader recomputing a cross section starts from the same inputs. The gluon-gluon and quark-antiquark columns are given separately because the two cluster members are produced through different initial states.}
\label{tab:parton-lumi}
\small
\begin{tabular}{lccccc}
\toprule
\(M\) & \(\tau=M^{2}/s\) & \(\mathcal{L}_{gg}\) & \(\mathcal{L}_{u\bar u}\) & \(\mathcal{L}_{d\bar d}\) & \(\sum_q \mathcal{L}_{q\bar q}\)\\
\midrule
3.849 TeV & 0.0877 & 0.0337 & 0.0452 & 0.0139 & 0.0596\\
2.461 TeV & 0.0358 & 1.440  & 0.974  & 0.462  & 1.462\\
\bottomrule
\end{tabular}
\end{table}
Any analyst can redo the arithmetic of this section from Eq.~\eqref{eq:nwa-master} and this table alone; that is the point.

\paragraph{Validation.}  The pipeline was checked against three independent published anchors before any \GG{} number was produced: (i)~the reference parton-luminosity coefficients of \citet{FranceschiniEtAl2016} at \(M=750\)~GeV are reproduced to 2--5\%; (ii)~the Randall--Sundrum graviton \(\sigma\times\mathrm{BR}(e^+e^-)\) table of \citet{AllanachEtAl2000} is reproduced to 15--25\% (the expected LO/PDF-era spread), with the total widths matching to 2--3\%; (iii)~the warped KK-gluon branching pattern of \citet{LillieRandallWang2007} (\(\mathrm{BR}(t\bar t)=92.5\%\)) is reproduced exactly, and the \citet{AgasheEtAl2007} production benchmark at 3~TeV to \(\sim\!25\%\).

\paragraph{Uncertainties.}  The quoted bands fold together the NLO K-factor (1.0--1.3 for \(q\bar q\) resonances, stated rather than silently applied) and a \(\pm20\%\) PDF-and-scale allowance for \(q\bar q\)-initiated rates (the CTEQ6M/CTEQ6L1 spread is 3--9\%; the allowance covers modern-set drift at large \(x\)); the gluon-gluon luminosity at \(x\sim0.3\) carries a factor \(\sim\!2\) spread, absorbed into the \(\kappaeff\) corridor it multiplies.  These are LO estimates with stated errors --- not precision QCD, and not dressed as such.

\subsection{Results}
\label{subsec:sigmabr-results}

\begin{table}[H]
\centering
\small
\setlength{\tabcolsep}{5pt}
\caption[Computed \(\sigma\times\mathrm{BR}\) for the lock-cluster states at \(\sqrt{s}=13\)~TeV.]{Computed \(\sigma\times\mathrm{BR}\) for the lock-cluster states at \(\sqrt{s}=13\)~TeV (LO\(\times\)K, CTEQ6M: \(K_{q\bar q}=1.3\) on the vector rows, \(K_{gg}=1.6\) on the \(G_1\) rows; bands fold the PDF/scale spread).  All entries Derived-conditional on B1--B4.  The \(G_1\) rows use the \emph{derived} \(\kappa_{gg}=c_K/K=0.0136\) (Corollary T-KK-2c); their band is the gluon-luminosity spread (factor \(\sim\!2\)) with the \(O(1/K)\) amplitude fence, and \(\mathrm{BR}(\ttbar)\) carries a \(\sim\!25\%\) cascade-depth sensitivity.  The \(Z'_B\) rows scale as \((g_B/0.2)^{2}\); the L-KK-1 menu spans \(g_B=0.14\)--\(0.89\), i.e.\ \(\times0.5\)--\(20\) on the quoted reference (\S\ref{subsec:gbmenu}).}
\label{tab:sigmabr}
\begin{tabular}{llp{3.8cm}p{2.6cm}c}
\toprule
\textbf{State (mass)} & \textbf{Channel} & \(\bm{\sigma\times\mathrm{BR}}\) \textbf{central [band], fb} & \textbf{Scaling} & \(\bm{\Gamma/m}\)\\
\midrule
\(g^{*}\), Scenario B (3.849 TeV) & \(\ttbar\) & 8.7 [7.0--12] & fixed (\(\xi_{\rm UV}\) theorem) & 0.15\\
 & \(jj+b\bar b\) & 0.55 [0.44--0.78] & fixed & \\
\(g^{*}\), Scenario A (2.461 TeV) & \(\ttbar\) & 224 [180--325] & fixed & 0.15\\
 & \(jj+b\bar b\) & 14 [11--21] & fixed & \\
\(G_1\) (3.849 TeV) & \(\ttbar\) & 0.045 [0.021--0.10] & derived (T-KK-2c) & 0.063\\
 & \(W_LW_L\) & 0.010 & derived & \\
 & \(Z_LZ_L\) & 0.0051 & derived & \\
 & \(hh\) & 0.0050 & derived & \\
 & \(jj\;(gg)\) & \(9.0\times10^{-5}\) & derived & \\
 & \(\gamma\gamma\) & \(1.1\times10^{-5}\) (retro-diction) & derived & \\
\(Z'_B\) (3.849 TeV) & \(\ttbar\) & \(0.030\times(g_B/0.2)^{2}\) & \(\propto g_B^{2}\) & \(4\times10^{-3}\)\\
 & \(jj+b\bar b\) & \(0.0018\times(g_B/0.2)^{2}\) & \(\propto g_B^{2}\) & \\
\midrule
cluster sum, Scenario B & \(\ttbar\) & 8.7--9.3 (octet-dominated; \S\ref{subsec:gbmenu}) & --- & 0.06--0.15\\
\bottomrule
\end{tabular}
\end{table}

Two structural facts deserve emphasis.  First, the color octet's \(\ttbar\) rate carries \emph{no} continuous freedom: its production coupling is the theorem-fixed \(\xi_{\mathrm{UV}}\) and its branching pattern is saturated by the top.  It is the one line in Table~\ref{tab:sigmabr} that \GG{} cannot lower.  Second, the three states are not interchangeable targets: the octet is broad (\(\Gamma/m\simeq15\%\)), the graviton moderately narrow (\(\Gamma/m=6.3\%\) at the derived coupling, inside the width family of the published spin-2 templates), and the \(Z'_B\) very narrow --- the cluster line shape is a \emph{sum} of these, which is exactly why \S\ref{sec:channel-by-channel} asks for a cluster-aware fit.

\begin{figure}[p]
\centering
\resizebox{0.810\textwidth}{!}{\includegraphics{Arisaka-GG-P1-KK-overlay-sigmaBR.pdf}}
\caption[Computed lock-cluster $\sigma\times$BR versus published 95\% CL limits]{The operational confrontation at \(\sqrt s=13\)~TeV, all couplings at their derived values (T-KK-2, T-KK-2c).  \emph{Panel (a):} observed 95\%~CL limits from the ATLAS 140~fb\(^{-1}\) \(\ttbar\)-resonance search \citep{ATLAS2025ttbar} (narrow-\(Z'\), broad-KK-gluon \(\Gamma/m=30\%\), and spin-2 templates; HEPData ins3094414) against the predictions of Table~\ref{tab:sigmabr}: the Scenario-B octet at 8.7~fb (star) --- the un-suppressible line that carries the HL-LHC falsifier; the Scenario-A octet at 2.46~TeV (red cross), a factor \(\sim\!3.6\) \emph{above} the observed broad-template limit (F-KK-2A provisionally fired); and \(G_1\) at the derived \(\kappa_{gg}=c_K/K\) (violet triangle, 0.045~fb) --- a factor \(\sim\!25\) below even the 3~ab\(^{-1}\) HL-LHC projection, hence LHC-invisible.  \emph{Panel (b):} the CMS narrow-dijet \(\sigma B A\) limit \citep{CMS2020Dijet}, diphoton spin-2 limit \citep{CMS2024Diphoton}, and dilepton limit over its retrieved extent \citep{CMS2020DileptonPublic}, against the octet dijet rate, the \(g_B^{2}\)-scaled \(Z'_B\), and \(G_1\to\gamma\gamma\) at the derived coupling (\(7\times10^{-6}\)~fb): the published diphoton null is \emph{retro-dicted} with a margin of \(\sim\!3\times10^{3}\), whereas a uniform-\(1/\sqrt K\) photon amplitude would already be excluded by a factor \(\sim\!4\).  All predictions LO with stated bands.}
\label{fig:overlay}
\end{figure}

\subsection{The survival statement, channel by channel}
\label{subsec:survival}

\paragraph{Scenario A (free-field): excluded at the stated uncertainties.}  The free-field octet at \(2.461\)~TeV predicts \(\sigma\times\mathrm{BR}(\ttbar)=224\,[180\text{--}325]\)~fb.  The observed ATLAS limit at that mass is 62~fb in the \emph{most conservative} (broad, \(\Gamma/m=30\%\)) template and 20~fb in the narrow template; the predicted \(\Gamma/m\simeq15\%\) lies between the two, so the applicable limit is if anything \emph{stronger} than 62~fb.  The prediction therefore sits a factor 3.6--11 above the observed exclusion, outside the band by any reading.  Within the stated uncertainties, \textbf{F-KK-2A has fired}: the free-field scenario of T-KK-2 is closed by existing data, and the dichotomy resolves to the composite-hardened scenario~B --- the degenerate cluster at \(\Mlock\).  We register the one formal caveat: this transfer uses ATLAS's published templates rather than a dedicated \(\Gamma/m=15\%\) reinterpretation, so we grade the closure \emph{provisional} until that reinterpretation exists; but the factor is large, and no plausible line-shape correction recovers it.

\paragraph{Scenario B octet: alive, and decidable soon.}  At \(3.849\)~TeV the octet predicts \(8.7\,[7.0\text{--}12]\)~fb in \(\ttbar\).  The observed broad-template limit is 25.5~fb --- a factor \(\sim\!2.9\) of headroom --- while the narrow-template limit, 4.9~fb, is already \emph{below} the prediction.  The state survives today precisely because of its breadth: a broad, top-saturated object at the lock scale is not yet excluded, but the window between the two templates is exactly where a dedicated \(\Gamma/m\simeq15\%\) reinterpretation has discriminating power \emph{now} --- and Part~\ref{part:ttanatomy} supplies the full differential anatomy (angle, polarization, interference line shape with its derived sign) that such a reinterpretation should implement.  In dijets the octet is safe by a factor \(\sim\!35\) (0.23~fb against the CMS \(\sigma B A\) limit of 8.0~fb): the dijet null is consistent, not constraining.

\paragraph{The graviton: derived, and out of LHC reach.}  With the coupling now derived (Corollary T-KK-2c), the \(G_1\) line is a point, not a corridor: \(\sigma\times\mathrm{BR}(\ttbar)=0.045\,[0.021\text{--}0.10]\)~fb.  Even the 3~ab\(^{-1}\) HL-LHC projection of the strongest template (\(\sim\!1.1\)~fb) sits a factor \(\sim\!25\) above it: \textbf{no LHC dataset will see the graviton}, and its absence from every LHC search is henceforth a \emph{prediction} of the framework, not a loose end.  The diphoton channel supplies the retro-diction that gives this claim teeth: had the photon coupled at the uniform \(1/\sqrt K\) amplitude, \(\sigma\times\mathrm{BR}(\gamma\gamma)=0.14\)~fb would already be excluded by a factor \(\sim\!4.2\) by the observed CMS limit of 0.033~fb; at the derived \(\kappa_\gamma=\kappa_{gg}\) the prediction is \(7\times10^{-6}\)~fb, a factor \(\sim\!3\times10^{3}\) below the limit.  The published null therefore discriminates between the two readings of the flat-mode amplitude and lands on the derived one --- the first amplitude-level confrontation of the collider chain, resolved in the geometry's favor.  The direct spin-2 test moves to FCC-hh, where \(\sigma(G_1)\approx205\)~fb at 100~TeV makes the anchor an early-running discovery target (\S\ref{subsec:killwindow}).

\paragraph{The \(Z'_B\): bracketed by the derived norm, tested elsewhere.}  Scaling the \(Z'_B\) rate to the current \(\ttbar\) limit bounds \(g_B\lesssim2.9\) --- far above the entire derived-norm menu \(g_B\in[0.14,\,0.89]\) of \S\ref{subsec:gbmenu}.  Across that menu \(\sigma\times\mathrm{BR}(\ttbar)=0.012\)--\(0.46\)~fb: a factor \(\ge\!10\) below the current narrow-template limit and \(\ge\!2.3\) below the 3~ab\(^{-1}\) projection, so the state is LHC-invisible as a separate signal on every L-KK-1 reading and enters only through the cluster sum (0.2--6.8\%).  Its decisive test remains the fifth-force \((g_B,m_B)\) window of the Predictions branch \citep{ArisakaA9SM} --- now four candidate lines rather than a plane.  Its exact leptophobia keeps the dilepton null consistent by construction.

\paragraph{The cluster, summed.}  The Scenario-B cluster delivers \(\simeq8.8\)--\(9.3\)~fb (central, spanning the L-KK-1 menu) of \(\ttbar\) signal spread over the near-degenerate line shape --- a factor \(\sim\!2.8\) below the broad-template limit.  The correct summary of the public record is therefore neither ``already excluded'' nor ``comfortably safe'': the lock cluster is \emph{bracketed}, with the un-suppressible octet line carrying the exposure.

\subsection{The kill window, quantified}
\label{subsec:killwindow}

With background-dominated \(\sqrt{\mathcal L}\) scaling (Eq.~\ref{eq:sqrtL}) from the 140~fb\(^{-1}\) baseline, the observed broad-template \(\ttbar\) limit at \(3.85\)~TeV projects to \(\sim\!17\)~fb at 300~fb\(^{-1}\), \(\sim\!9.6\)~fb at 1~ab\(^{-1}\), and \(\sim\!5.5\)~fb at 3~ab\(^{-1}\) --- while the octet prediction rises to \(13.6\,[11\text{--}19]\)~fb at \(\sqrt s=14\)~TeV.  \textbf{The projected exclusion crosses the un-suppressible octet band between roughly 0.5 and 2~ab\(^{-1}\)}: F-KK-2B is decided --- discovery or exclusion --- within the first half of the HL-LHC program, on the collaborations' own published sensitivity, with no \GG{}-side freedom left to move the target.  The graviton, at its derived \(\kappa_{gg}\), contributes 0.045~fb in \(\ttbar\) --- no LHC dataset reaches it, and the spin-2 test moves cleanly to FCC-hh, where \(\sigma(G_1)\approx205\)~fb at 100~TeV (\(t\bar t\) 142~fb, \(ZZ\) 16~fb, \(hh\) 16~fb) makes the anchor an early discovery target \citep{FCC2018}.  The division of labor is itself a falsifiable pattern: the HL-LHC must find or kill the spin-1 cluster; a 100-TeV machine must then find the spin-2 anchor at the same mass.  This, together with the Scenario-A closure above, is the operational content of the 3.85-TeV falsifier: one scenario already decided by data on tape, the other pre-registered against a dated luminosity milestone.  Figure~\ref{fig:timeline} (\S\ref{subsec:timeline}) recasts the same statement in calendar years.

\section{Near-Term Discovery Potential}
\label{sec:near-term}

\subsection{The Run-3 to early-HL-LHC bridge}

The next experimentally decisive interval is not the distant future; it is the bridge from full Run~3 to early HL-LHC analysis.  Official CERN planning documents describe HL-LHC as a \(14\,\mathrm{TeV}\) program targeting about \(3000\,\mathrm{fb}^{-1}\) per experiment, with luminosity gains of roughly a factor of \(5\) to \(7.5\) over the nominal LHC design \citep{CERN2023HLLHC}.  That long-term target matters, but the shorter-term issue is equally important: the completion of Run~3 and the appearance of public analyses that incorporate the full Run-2 + Run-3 sample in high-mass resonance channels.

For a background-dominated search, one expects the simplest significance scaling to follow
\begin{equation}
\mathcal Z(\mathcal L_2) \approx \mathcal Z(\mathcal L_1)\sqrt{\frac{\mathcal L_2}{\mathcal L_1}}.
\label{eq:sqrtL}
\end{equation}
This familiar relation does not decide the discovery by itself, but it tells us something important: a signal that is presently marginal or template-diluted can become clearly visible once a cluster-aware multi-channel analysis is applied to a data set larger by a factor of two to five.  Part~\ref{part:hllhcsim} replaces this back-of-envelope scaling with the full anchored simulation --- and shows that \eqref{eq:sqrtL} fails in precisely the regime that matters: beyond \(\sim\!0.5\)~ab\(^{-1}\) the search is systematics-limited and significance no longer grows as \(\sqrt{\mathcal L}\) at all.

\subsection{Why \texorpdfstring{$\ttbar$}{ttbar} is the priority channel}

If the light-lepton and photon overlaps are even moderately suppressed, then the first robust excess will most likely emerge in \(\ttbar\) rather than in \(\llbar\) or \(\gamma\gamma\).  There are three reasons.
\begin{enumerate}
\item The geometry naturally favors the top sector via infrared-localized overlaps.
\item The bulk-RS literature already demonstrates that high-mass spin-2 and KK-gluon searches in \(\ttbar\) are mature and technically feasible.
\item Top final states are also the natural meeting point of the graviton, KK-gluon, and vector-like-top partner stories, making them ideal for a pattern test.
\end{enumerate}
This does not mean that dilepton or diphoton channels are unimportant.  On the contrary, they can still be crucial for a clean corroborating line if the signal is sufficiently strong.  But a \GG{}-aware discovery program should not treat hadronic top channels as merely complementary.  They are central.

\subsection{The role of boosted dijets and Higgs pairs}

Dijet and \(hh\) channels serve different but complementary purposes.  Dijet channels test the mild leptophobic or baryonic admixture and may carry substantial branching power.  Higgs-pair channels test the scalar or Hosotani-related side of the theory.  A combined \(\ttbar\)+dijet+\(hh\) excess pattern near the first cluster would be much more diagnostic of \GG{} than any single excess in isolation.

\subsection{A benchmark significance corridor}

Because the exact first-cluster line shape is model-dependent, a completely rigid cross-section prediction is premature.  A more honest intermediate product is a \emph{discovery corridor}.  Let the effective signal strength for a given channel \(f\) be parameterized by
\begin{equation}
\mu_f = \left(\frac{\kappaeff}{0.1}\right)^2 R_f,
\label{eq:muf}
\end{equation}
where \(\kappaeff\) is the effective lock-induced coupling normalization and \(R_f\) packages branching and acceptance relative to a reference bulk-RS template.  In this language, the decisive question becomes whether the combined ATLAS+CMS program can exclude or discover \(\mu_f\) across the full geometry-allowed corridor rather than at a single arbitrary benchmark point.

Within this Part's theorem-anchored discipline \(\kappaeff\) carries no number --- it is not a GG-A output: it is set by the bulk-to-boundary overlaps that \citet{ArisakaA9SM} leaves as an open lemma.  The conceptual point survives without a number: \(\kappaeff\) is not free in the spirit of arbitrary model scans; it is constrained by the same geometry that fixes the mass scale.  This is another reason the eventual collider test is unusually meaningful --- and it is why the decisive deliverable is the corridor \(\mu_f\), not a single benchmark cross section.  Read-out~II (\S\ref{sec:scenarioB}) estimates \(\kappaeff\) from the throat geometry, and \S\ref{sec:operational} confronts the resulting rates, corridor and all, with the published limits.



\subsection{Could ATLAS and CMS already get the first events in the current 2026 run?}

Yes -- in the limited but important sense of \emph{hints, small excesses, or channel-local anomalies}.  No -- if by ``get'' one means an indisputable five-sigma combined discovery with a model-complete pattern test before LS3.  The reason is simple arithmetic.

Official CERN reporting states that the 2025 proton run alone delivered \(125.4\,\mathrm{fb}^{-1}\) to both ATLAS and CMS, bringing the total delivered since LHC startup above \(500\,\mathrm{fb}^{-1}\) \citep{CERN2025Luminosity}.  Official CERN reporting from March 2026 further states that stable beams for physics were declared on 7 March 2026, that the 2026 run is ``short but densely packed,'' and that the proton program includes a low-pileup phase followed by a high-pileup phase that will extend the integrated dataset beyond the already achieved Run-3 target before LS3 begins at the start of July \citep{CERN2026Run}.  In practical language, that means there is still genuine discovery leverage \emph{this year}, but it sits in a compressed window.

We deliberately do \emph{not} convert this dataset into a predicted event count.  Through Eq.~\eqref{eq:Ns} an event yield requires the absolute cross section, which (\S\ref{subsec:noxsec}) is not a GG-A-derived quantity: it depends on \(\kappaeff\) and a QCD/PDF integration that lie outside the theorems.  Quoting ``\(N\) tagged \(\ttbar\) events this year'' would therefore present a model-level number as if it were a prediction, which the discipline of this paper forbids.  (The clearly-labeled Read-out-II rates of \S\ref{sec:operational}, Table~\ref{tab:sigmabr}, convert to yields via Eq.~\eqref{eq:Ns} for any reader who wants them at that grade.)  What \emph{is} robust, and is the only thing we assert here, is structural: the color octet carries the un-suppressible light-quark coupling \(1/\sqrt{K}\approx0.17\), so it sits at the current \(\ttbar\)/dijet exclusion frontier and is being probed by existing data (F-KK-2); and the lock scale is fixed at \(3.849\,\mathrm{TeV}\), squarely inside the corridor where the full Run-2\,+\,Run-3 top-pair and dijet samples already have leverage.

The operational consequence stands without any yield number: a cluster-aware, top-rich, leptophobic reinterpretation of the existing high-mass \(\ttbar\) and dijet data is the test that matters now, and the absence of any emerging structure in that reinterpretation would itself begin to constrain the current implementation.

The right way to say it is therefore this.  ATLAS and CMS \emph{could} see the first real GG-6-motivated hints already in the current 2026 closeout analyses, especially in \(\ttbar\) and boosted hadronic channels.  A completely decisive verdict is more plausibly a Run-3-plus-early-HL-LHC story.  But the theory has already entered the regime in which ``wait until 2040'' is no longer the correct experimental attitude.

\subsection{Near-term reach in practical language}

It is prudent not to overstate significance projections.  The cleanest public machine-based statements are these.  Official CERN documents define HL-LHC as the facility that will deliver the data volume required for decisive tests in the TeV range \citep{CERN2023HLLHC}.  Official ATLAS and CMS upgrade pages describe detector and trigger improvements specifically aimed at maintaining performance under \(\mathcal O(200)\) pile-up conditions \citep{ATLAS2024HLUpgrade,CMS2024UpgradeWeek}.  Public high-mass resonance analyses already reach to several-TeV masses in the clean channels.  Therefore the cautious but strong statement is: \emph{a dedicated cluster-aware combined-channel analysis of the full Run-2 + Run-3 sample can plausibly bring the \(3.85\,\mathrm{TeV}\) \GG{} target into the first genuinely discriminating sensitivity regime, and the HL-LHC should be able either to confirm it or to exclude it across the geometry-allowed corridor.}

\begin{table}[H]
\centering
\caption[Near-term discovery logic for the first \GG{} cluster.]{Near-term discovery logic for the first \GG{} cluster.  The entries are intentionally qualitative where public collaboration reach projections do not yet map one-to-one onto the clustered \GG{} benchmark. The qualitative entries are qualitative because the public projections they would have to be read against were computed for single-resonance benchmarks, and this paper's target is a cluster. Forcing a number onto those rows would give a precision the underlying projections do not have. Where a number can be stated honestly it is stated.}
\label{tab:nearterm}
\small
\begin{tabularx}{\textwidth}{>{\raggedright\arraybackslash}p{3.3cm} >{\raggedright\arraybackslash}p{3.6cm} X}
\toprule
\textbf{Program stage} & \textbf{What changes} & \textbf{Implication for the \GG{} first cluster}\\
\midrule
Run 2 only & Mature high-mass searches exist, but mostly in benchmark-specific one-pole templates & Important bracketing information exists, but not a decisive dedicated \GG{} test.\\
Run 2 + full Run 3 & Larger data set, improved detector performance, first public analyses with full combined sample & First genuinely meaningful cluster-aware reinterpretation should become possible.\\
Early HL-LHC & Substantially larger \(\mathcal L\), improved timing, tracking, and reduced-stream flexibility & Best stage for first 3--5\(\sigma\) evidence if the \GG{} signal sits near the center of the expected corridor.\\
Full HL-LHC & \(3000\,\mathrm{fb}^{-1}\) per experiment, mature Phase-2 detector and trigger ecosystem & Should provide either discovery-level combined sensitivity or a strong module-level falsification of the current \GG{} implementation.\\
\bottomrule
\end{tabularx}
\end{table}


\subsection{The falsification timeline in calendar years}
\label{subsec:timeline}

Every sensitivity statement in this Part has so far been phrased against integrated luminosity.  This closing subsection converts the whole picture into the coordinate an experimental program actually runs on: the calendar.  The conversion is natural for \GG{} in a way it is not for scanned-parameter models.  Because the theory predicts a \emph{single} derived mass, the standard sensitivity-versus-mass presentation (Fig.~\ref{fig:overlay}) collapses to one vertical slice at \(3.849\)~TeV; Figure~\ref{fig:timeline} re-plots that slice against time, with the derived predictions appearing as \emph{horizontal bands} --- they do not move, because there is nothing in the theory that can move them --- and the projected experimental sensitivities as descending curves.  A theory with a mass dial cannot draw this figure; its bands would slide.  The entire numerical content of the figure is collected in Table~\ref{tab:timeline-inputs}.

\paragraph{Construction.}  The bands are the derived \(\sqrt s=13\)~TeV rates of Table~\ref{tab:sigmabr}, carried unchanged.  The sensitivity curves take the observed full-Run-2 95\%~CL limits at \(3.849\)~TeV (Table~\ref{tab:timeline-inputs}, middle block) and scale them as \(\sqrt{140\,\mathrm{fb}^{-1}/L(Y)}\), where \(L(Y)\) is the integrated luminosity on tape per experiment in year \(Y\) under the CERN schedule as of July 2026 (bottom block): Run~3 ended on 14~June~2026 with approximately \(540\)~fb\(^{-1}\) delivered since startup, of which \(\approx466\)~fb\(^{-1}\) is analysis-quality \citep{CERN2026LS3}; LS3 runs to 2030; the HL-LHC takes physics data from mid-2030 at \(250\)--\(350\)~fb\(^{-1}\) per year, accumulating \(\approx3.4\)~ab\(^{-1}\) by the end of the current baseline in 2041 \citep{CERN2023HLLHC,CERN2026LS3}.  Beyond the LHC the calendar is now fixed by the December-2025 European Strategy recommendation of FCC-ee as the next flagship \citep{ESPP2025}: FCC-ee (physics from \(\sim\!2047\)) probes the cluster only indirectly, and the first machine to see the lock scale directly again is FCC-hh, with physics starting around 2074 in the staged Feasibility-Study scenario \citep{FCC2025FSR}.

\paragraph{Honesty conventions.}  Three conservatisms are built in, all pushing the drawn crossings \emph{later} than reality.  The curves are single-experiment (an ATLAS+CMS combination gains a further \(\approx\sqrt2\) in effective luminosity); the bands are the 13-TeV rates, whereas at 14~TeV every signal rises by 35--55\% while the backgrounds rise less steeply; and the octet is read against the broad (\(\Gamma/m=30\%\)) template although its predicted \(\Gamma/m\simeq15\%\) width would be better matched --- and more strongly constrained --- by a dedicated intermediate-width reinterpretation.  One convention runs the other way and must be stated just as plainly: the curves are drawn against the year the \emph{data are on tape}, not the year the analysis appears.  Historically, flagship full-dataset resonance results lag their data by two to seven years (the full-Run-2 \(\ttbar\) search \citealp{ATLAS2025ttbar} appeared seven years after the end of Run~2), so published verdicts trail these curves by that much.

\paragraph{Reading.}  The figure compresses the collider fate of \GG{} into one visual sentence.  The broad-template \(\ttbar\) curve --- the only curve that matters, because after F-KK-2A's closure and T-KK-2c's requalification the octet carries the entire LHC falsifier --- crosses the top of the \(g^{*}\) band with data on tape around 2032 and exits its bottom around 2040: \textbf{F-KK-2B is decided, discovery or exclusion, within Run~4--5 of the HL-LHC}.  Every other band lies below every curve everywhere in the window: the \(Z'_B\) across its entire L-KK-1 menu and the graviton in every channel remain LHC-invisible through 2050, so their absence from LHC searches is a standing prediction of the framework, not a deficit.  After the HL-LHC baseline ends in 2041 the curves go flat --- no approved machine probes \(3.85\)~TeV directly for a generation --- which gives the window its weight: the LHC's verdict on the \GG{} collider sector will be substantially in hand by \(\sim\!2040\), final (in published form) by \(\sim\!2047\), and nothing else can speak to it until FCC-hh, where \(\sigma(G_1)\approx205\)~fb would turn the surviving cluster into early-running spectroscopy.

\begin{table}[H]
\centering
\small
\setlength{\tabcolsep}{4pt}
\caption[Numerical inputs to Figure~\ref{fig:timeline}.]{Numerical inputs to Figure~\ref{fig:timeline}.  Top: the \GG{} bands (derived LO\(\times\)K rates at \(\sqrt s=13\)~TeV, from Table~\ref{tab:sigmabr}; bands fold \(\pm20\%\) PDF/scale).  Middle: the observed 95\%~CL anchors at \(3.849\)~TeV from which the curves are \(\sqrt L\)-scaled (\(\ttbar\) rows: HEPData record ins3094414).  Bottom: the luminosity-on-tape model per experiment (CERN schedule, July 2026).  Sensitivity in year \(Y\): \(\mathrm{limit}_{140}\times\sqrt{140/L(Y)}\).}
\label{tab:timeline-inputs}
\resizebox{\textwidth}{!}{%
\begin{tabular}{llll}
\toprule
\multicolumn{4}{l}{\textbf{GG bands at \(M=3.849\)~TeV} (13 TeV, LO\(\times\)K)}\\
\textbf{Signal} & \(\bm{\sigma\times}\)\textbf{BR [band], fb} & \(\bm{\Gamma/m}\) & \textbf{Coupling provenance}\\
\midrule
\(g^{*}\to\ttbar\) & 8.7 [7.0--12] & 0.15 & \(\xi_{\rm UV}=-0.194\), theorem (T-KK-2)\\
\(g^{*}\to jj{+}b\bar b\) (\(\sigma BA\), \(A{=}0.55\)) & 0.30 [0.23--0.43] & 0.15 & same\\
\(Z'_B\to\ttbar\) & 0.016--0.60 & 0.002--0.080 & L-KK-1 menu, \(g_B=0.14\)--0.89 (\S\ref{subsec:gbmenu})\\
\(G_1\to\ttbar\) & 0.045 [0.021--0.10] & 0.063 & \(\kappa_{gg}=c_K/K=0.0136\), derived (T-KK-2c)\\
\midrule
\multicolumn{4}{l}{\textbf{Observed 95\% CL anchors at \(3.849\)~TeV} (full Run 2, single experiment)}\\
\textbf{Channel / template} & \textbf{Limit [fb]} & \textbf{Dataset} & \textbf{Source}\\
\midrule
\(\ttbar\), broad \(\Gamma/m=30\%\) & 25.5 & 140 fb\(^{-1}\) & \citet{ATLAS2025ttbar}\\
\(\ttbar\), narrow & 4.9 & 140 fb\(^{-1}\) & \citet{ATLAS2025ttbar}\\
dijet, narrow \(qq\) (\(\sigma BA\)) & 8.0 & 137 fb\(^{-1}\) & \citet{CMS2020Dijet}\\
\midrule
\multicolumn{4}{l}{\textbf{Luminosity on tape per experiment} \(L(Y)\) (13-TeV-equivalent)}\\
\textbf{Year / era} & \(\bm{L}\) \textbf{[fb\(^{-1}\)]} & \multicolumn{2}{l}{\textbf{Basis}}\\
\midrule
2025 (Run 3) & 442 & \multicolumn{2}{l}{Run-2 140 + Run-3 on tape \citep{CERN2025Luminosity}}\\
2026--29 (LS3 from 29 Jun 2026) & 466 & \multicolumn{2}{l}{Run 3 ended 14 Jun 2026; \(\sim\!540\) delivered \citep{CERN2026LS3}}\\
2030 (HL-LHC from mid-year) & \(\approx\)526 & \multicolumn{2}{l}{ramp half-year \citep{CERN2026LS3}}\\
2031--41 (HL-LHC) & \(+200\)--275/yr & \multicolumn{2}{l}{leveled \(250\)--\(350\)/yr \citep{CERN2023HLLHC}; \(\approx\!3450\) by end-2041}\\
2042--50 (post-LHC gap) & 3451 & \multicolumn{2}{l}{baseline ends 2041; next direct probe FCC-hh \(\sim\!2074\) \citep{FCC2025FSR}}\\
\bottomrule
\end{tabular}}
\end{table}

\begin{figure}[p]
\centering
\resizebox{0.94\textwidth}{!}{\includegraphics{Arisaka-GG-P1-KK-timeline-sigmaBR.pdf}}
\caption[The lock-cluster falsification timeline in calendar years, 2025--2050]{The falsification timeline of the \GG{} lock cluster at \(M_{\rm lock}=3.849\)~TeV, 2025--2050.  Horizontal bands: the derived \GG{} predictions of Table~\ref{tab:timeline-inputs} --- flat in time because the theory has no dial to move them.  Descending curves: projected single-experiment 95\%~CL sensitivities at \(3.849\)~TeV, \(\sqrt L\)-scaled from the observed full-Run-2 anchors on the July-2026 CERN schedule (shaded eras: Run 3, LS3, HL-LHC, post-LHC gap).  The broad-template \(\ttbar\) curve crosses the \(g^{*}\) band with data on tape between \(\sim\!2032\) and \(\sim\!2040\): F-KK-2B --- the sole LHC arm of the falsifier after F-KK-2A's closure and T-KK-2c's requalification of F-KK-1 --- is decided within HL-LHC Run~4--5.  All other bands remain below every curve through 2050 (their LHC absence is a prediction); the \(G_1\to\gamma\gamma\) line (\(7\times10^{-6}\)~fb) lies far below the plotted window.  Conservatisms and the publication-lag convention are stated in the text; all inputs in Table~\ref{tab:timeline-inputs}.}
\label{fig:timeline}
\end{figure}

\section{Why This is the Best Collider Killer Experiment}
\label{sec:killer}

\subsection{Sharpness compared with other energy-frontier claims}

The phrase ``killer experiment'' should be used carefully.  In the broadest sense, \GG{} has other important tests: axion-sector tests, dark-energy tests, Higgs-coupling tests, and possible quantum-open-system tests.  But at the energy frontier, the KK search is categorically the cleanest falsifier because it combines three features that rarely coexist in BSM physics.

First, the target mass is unusually sharp.  There is no scanned supersymmetric spectrum, no landscape of vacua, and no flexible mediator mass.  The first cluster is tied to \(K=35\).

Second, the same integer that fixes the lock scale also participates in other successful internal ledger relations, such as the weak scale.  So the mass prediction is not isolated; it is part of a coherent triad.  The interlocking extends across the predictions branch in both directions: the dark-matter companion stakes a haloscope line at \(1.95\pm0.12\)~GHz, twice split by exactly \(1/35\) --- the same ledger integer that sets \(\Mlock\) --- and the black-hole companion reads gravitational-wave ringdowns as curvatures of the same free-energy functional.  Kill the lock cluster and those accounts inherit the wound; conversely, a confirmed \(1/35\) doublet would make the \(3.85\)~TeV search a confirmation hunt rather than an exploration.  No other energy-frontier proposal carries hostages in neighboring experiments of this kind.

Third, the experimental statement is binary in a practically useful way.  Either the spin-2-anchored, octet-led signal cluster appears in the neighborhood of \(3.85\,\mathrm{TeV}\), or increasingly powerful data exclude it across the geometry-allowed coupling corridor.  That is much more decisive than many BSM searches in which a null result simply pushes a free scale upward.

\subsection{What discovery would establish}

A convincing discovery of the spin-2-anchored cluster near \(3.85\,\mathrm{TeV}\), together with a top-rich branching pattern and at least one of the expected partner signatures, would establish several things at once.  It would establish that an extra-dimensional description is realized in nature.  It would establish that the first accessible scale is not generic but locked to a highly specific infrared value.  It would give strong support to the non-universal overlap picture.  And it would make a direct future search for the second radial family and the KK-gluon corridor mandatory.

\subsection{What absence would establish}

Conversely, a sufficiently strong combined-channel exclusion would do more than say ``one benchmark failed.''  It would pressure the \GG{} constitutional chain itself.  If the full Run-2 + Run-3 + early-HL-LHC program excludes the first-cluster corridor across \(\ttbar\), dijet, \(hh\), dilepton, and diphoton channels within a cluster-aware window around \(3.85\,\mathrm{TeV}\), then at least one of the following must be false:
\begin{enumerate}
\item the OGN-5 lock index \(K=35\) does not control the collider infrared scale as claimed,
\item the 6D spectral interpretation of the first lock family is wrong,
\item or the overlap/coupling structure is radically different from the presently derived corridor.
\end{enumerate}
Any of these would count as a deep revision of the current \GG{} implementation.

\subsection{A pre-declared kill criterion}

Following the GG-Constitution's insistence on measurement and falsifiability, the collider-level kill criterion can be written explicitly:
\begin{quote}
\emph{If, after analysis of the combined Run-2 + full Run-3 + early-HL-LHC data set, ATLAS and CMS observe no excess above background in any cluster-aware combination of \(\ttbar\), boosted dijet, \(hh\), \(\llbar\), and \(\gamma\gamma\) searches within a window of order \(\pm 0.6\,\mathrm{TeV}\) around \(3.849\,\mathrm{TeV}\), and if the corresponding \(95\%\) CL upper limits on the combined signal strength fall below the full geometry-allowed \GG{} corridor, then the present lock-scale implementation of \GG{} is ruled out at the collider level.}
\end{quote}
This wording is intentionally stronger than a single-channel exclusion and intentionally weaker than an impossible demand for a fully exact one-pole template.  It is the right balance between honesty and decisiveness.  \S\ref{subsec:killwindow} converts it into dated numbers: the un-suppressible color-octet line is crossed between roughly 0.5 and 2~ab\(^{-1}\) of HL-LHC data.  (With Corollary T-KK-2c the spin-2 member sits a factor \(\sim\!25\) below any HL-LHC sensitivity, so the criterion's teeth at the LHC are entirely the spin-1 sector's; the graviton line is FCC-hh's to test.)

\begin{table}[H]
\centering
\caption[Why the KK search is the sharpest \emph{collider} falsifier of the current \GG{} implementation.]{Why the KK search is the sharpest \emph{collider} falsifier of the current \GG{} implementation. Sharpest is a claim about the falsifier and not about the physics: the collider test is the one whose negative result is hardest to argue with, because the mass is fixed by an integer and the machine already exists. Other tests in the cohort probe deeper structure and take longer to settle; this one is decidable inside a doctoral career.}
\label{tab:whybest}
\small
\begin{tabularx}{\textwidth}{>{\raggedright\arraybackslash}p{3.6cm} X}
\toprule
\textbf{Criterion} & \textbf{Why the KK search is unusually strong}\\
\midrule
Specificity of mass target & The first cluster is centered at \(3.849\,\mathrm{TeV}\), not ``somewhere in the few-TeV region.''\\
Connection to other theory outputs & The same lock ledger also enters electroweak-scale relations and the broader hierarchy story.\\
Binary character & Discovery or combined-channel exclusion both have immediate meaning; a null result does not merely move a free parameter.\\
Energy-frontier accessibility & The relevant mass scale lies in the LHC/HL-LHC domain rather than only in far-future or purely cosmological observables.\\
Pattern richness & The search probes not only one state but a correlated pattern: cluster shape, top-rich branching, and higher partners.\\
\bottomrule
\end{tabularx}
\end{table}

\section{Long-Term Future Prospects}
\label{sec:longterm}

\subsection{HL-LHC as the decisive machine}

Official CERN documentation frames the HL-LHC as a \(14\,\mathrm{TeV}\) machine delivering more than a factor-of-five luminosity gain over the nominal LHC design, with a target integrated luminosity of about \(3000\,\mathrm{fb}^{-1}\) per experiment \citep{CERN2023HLLHC}.  That statement alone already implies why HL-LHC is decisive for \GG{}.  The first cluster is not at some inaccessible tens-of-TeV frontier; it is precisely in the corridor where HL-LHC will have maximal leverage.

\subsection{ATLAS Phase-2 upgrades}

ATLAS describes a Phase-2 detector centered on a fully new all-silicon inner tracker (ITk), forward tracking extended to \(|\eta|<4\), a new level-0 trigger architecture, and the High-Granularity Timing Detector (HGTD).  ATLAS states that the ITk extends pseudorapidity coverage from \(|\eta|<2.5\) to \(|\eta|<4\) and that the HGTD will provide \(30\) to \(50\,\mathrm{ps}\) timing in the forward region \citep{ATLAS2024HLUpgrade}.  These are not generic improvements; they are directly relevant to a \GG{} search because a clustered signal at \(3.85\,\mathrm{TeV}\) is especially sensitive to pile-up rejection, forward acceptance, and improved boosted-object tagging.

\subsection{CMS Phase-2 upgrades}

CMS similarly describes the HL-LHC era as a major transformation with a new inner tracker, the high-granularity endcap calorimeter (HGCAL), and the MIP Timing Detector (MTD).  Official CMS descriptions emphasize that the MTD aims at about \(35\,\mathrm{ps}\) time resolution for charged particles, precisely to untangle \(140\) to \(200\) pile-up interactions per bunch crossing \citep{CMS2019MTD,CMS2024UpgradeWeek}.  Again, for \GG{} this is directly useful: top-rich and dijet-rich high-mass searches benefit strongly from pile-up mitigation and improved object assignment.

\subsection{FCC-hh, muon collider, and CLIC}

If \GG{} is confirmed, the future collider program becomes a spectroscopy enterprise.  The official FCC-hh concept is a \(100\,\mathrm{TeV}\) hadron collider with integrated luminosity goals at least five times larger than HL-LHC and a mass reach extended by almost an order of magnitude \citep{FCC2018}.  Such a machine would not merely confirm the first cluster.  It would map the entire low-lying \GG{} staircase.

Official CERN materials also note that a \(10\,\mathrm{TeV}\) muon collider could be competitive with a \(100\,\mathrm{TeV}\) proton collider for some high-energy-frontier questions \citep{CERN2024Muon}.  For a \(3.85\,\mathrm{TeV}\) spin-2 state, a muon collider would offer a uniquely clean environment and exquisite angular analysis.  CLIC, on the other hand, is officially staged at \(380\,\mathrm{GeV}\), \(1.5\,\mathrm{TeV}\), and \(3\,\mathrm{TeV}\) \citep{CERN2024CLIC}.  That makes direct production of a \(3.85\,\mathrm{TeV}\) state inaccessible at the baseline stages, but virtual effects and Higgs-coupling precision would still provide important complementary information.

\subsection{Indirect probes}

Even if the first signal comes directly, indirect probes remain important.  In the current \GG{} picture one expects modest Higgs-coupling deviations from Hosotani-related mixing, likely at the percent level, and electroweak precision shifts of order \(10^{-3}\) in some observables.  Those effects are not by themselves the decisive falsifier, but they are valuable pattern checks.  If a \(3.85\,\mathrm{TeV}\) cluster appears with no companion distortions anywhere in Higgs or electroweak observables, the interpretation would need care.

\section{A Trigger and Analysis Roadmap for ATLAS and CMS}
\label{sec:trigger-roadmap}

This section proposes concrete improvements to ATLAS/CMS triggers and related search strategy.  The proposal is intentionally pragmatic.  The goal is not to ask the experiments to rebuild their trigger systems for one speculative model.  It is to point out modifications that are already aligned with broader new-physics goals and that would be especially useful for a clustered \GG{} signal.

\subsection{Use the high-rate frontier aggressively}

ATLAS has already demonstrated the power of trigger-level analysis (TLA), in which only essential reconstructed high-level quantities are stored, reducing event size dramatically and permitting much higher rates \citep{ATLAS2025HighRate}.  CMS has similarly introduced Level-1 Scouting, which stores reduced information from all collisions in a triggerless or trigger-reduced format \citep{CMS2025Scouting}.  These methods are already changing what the experiments can do in high-rate regimes.

For the \GG{} program the lesson is immediate: a dedicated reduced-format stream for the \(3\) to \(6\,\mathrm{TeV}\) region should be treated as a priority.  The relevant reduced quantities are not exotic.  They include high-mass dijet kinematics, boosted top substructure variables, Higgs-tagging observables, b-tagging summary information, coarse timing variables, and enough leptonic information to retain a clean cross-check in the \(\llbar\) channels.

\subsection{Make the search cluster-aware at trigger level}

Most online and semi-online resonance thinking is one-pole thinking.  A clustered \GG{} signal changes the optimization.  Instead of only centering windows on one putative pole mass, the trigger and scouting logic should support a broad \(3.2\) to \(5.2\,\mathrm{TeV}\) search region in which multiple nearby structures are explicitly allowed.  At analysis level this means storing enough resolution and calibration information to fit Eq.~\eqref{eq:clusterfit}.  At trigger level it means avoiding overly narrow single-mass assumptions that could dilute sensitivity to a split or shoulder-like feature.

\subsection{Promote top and Higgs boosted objects to first-class citizens}

Because \GG{} is likely top-rich, the trigger grammar should treat boosted top and boosted Higgs categories as first-class citizens in the high-mass new-physics program.  In practice this means:
\begin{itemize}
\item preserving reduced top-tag and Higgs-tag discriminants in scouting/TLA streams,
\item keeping high-\(p_T\) b-tag summary information available online or quasi-online,
\item and designing combined \(\ttbar\)+\(hh\)+dijet anomaly streams rather than siloed single-object streams.
\end{itemize}
These are useful changes even beyond \GG{}.  That is a virtue, not a problem.

\subsection{Exploit precision timing as a resonance-quality variable}

Both ATLAS and CMS emphasize precision timing for pile-up control at HL-LHC \citep{ATLAS2024HLUpgrade,CMS2019MTD}.  In the \GG{} context, timing should be treated not just as a vertexing aid but as a resonance-quality variable.  High-mass boosted hadronic final states are especially sensitive to pile-up contamination in substructure observables.  Timing-assisted cleaning should therefore be included explicitly in any dedicated \GG{} reinterpretation.

\subsection{A joint benchmark release would help enormously}

The experiments are understandably conservative about model proliferation.  The most useful single step for the community would therefore be modest: define a public benchmark family capturing the essential \GG{} phenomenology without overloading the analysis with theory details.  Such a benchmark could include
\begin{enumerate}
\item a first cluster centered at \(3.85\,\mathrm{TeV}\),
\item an optional split of order a few hundred GeV,
\item non-universal branching ratios favoring \(\ttbar\), dijets, and \(hh\),
\item a mild suppression of \(\llbar\) and \(\gamma\gamma\),
\item and the second radial family near \(7\,\mathrm{TeV}\) (\(n=2\)).
\end{enumerate}
That alone would transform the current status from a set of partial proxies into something much closer to a real test.

\begingroup\scriptsize\setlength{\tabcolsep}{3.5pt}
\begin{longtable}{>{\raggedright\arraybackslash}p{2.0cm} >{\raggedright\arraybackslash}p{2.6cm} >{\raggedright\arraybackslash}p{2.6cm} >{\raggedright\arraybackslash}p{3.6cm} >{\raggedright\arraybackslash}p{3.8cm}}
\caption[Proposed trigger and reduced-stream improvements for the \GG{} KK search.]{Proposed trigger and reduced-stream improvements for the \GG{} KK search.  All proposals are framed so that they also benefit a wider class of high-mass BSM programs. Every proposal here is framed to serve a wider class of high-mass searches, and that is a condition on the proposal rather than a courtesy. A trigger change justified by one theory's target is a change no collaboration should make; a change that improves the reach of a whole category and happens to enable this search as well is a change worth arguing for.}\label{tab:trigger}\\
\toprule
\textbf{Proposal} & \textbf{Main instrument} & \textbf{Immediate benefit for GG-6} & \textbf{Why it is technically plausible} & \textbf{Broader benefit beyond GG-6}\\
\midrule
\endfirsthead
\toprule
\textbf{Proposal} & \textbf{Main instrument} & \textbf{Immediate benefit for GG-6} & \textbf{Why it is technically plausible} & \textbf{Broader benefit beyond GG-6}\\
\midrule
\endhead
Dedicated high-mass reduced stream & ATLAS TLA / CMS scouting & Retains far more high-mass hadronic events in the \(3\) to \(6\,\mathrm{TeV}\) window & Both collaborations already operate reduced-format high-rate streams \citep{ATLAS2025HighRate,CMS2025Scouting} & Benefits dijet, dark-sector, and generic broad-resonance programs.\\
Cluster-aware template storage & HLT / offline fit model & Preserves sensitivity to doublet or triplet structures around the first cluster & Requires software and analysis choices more than new hardware & Helps any model with nearby poles or non-trivial interference.\\
Boosted top + Higgs summary variables in scouting & Trigger objects + reduced event content & Improves \(\ttbar\) and \(hh\) sensitivity where \GG{} is strongest & Reuses existing top/Higgs tagging infrastructure with reduced payload & Benefits generic heavy resonances, vector-like quarks, and heavy Higgs sectors.\\
Timing-assisted substructure cleaning & HGTD / MTD + offline reconstruction & Stabilizes top and dijet mass/substructure variables under 200-pileup conditions & Built directly on official Phase-2 timing goals \citep{ATLAS2024HLUpgrade,CMS2019MTD} & Benefits essentially all boosted-object searches at HL-LHC.\\
Common public benchmark family & ATLAS/CMS reinterpretation note & Converts current proxy limits into a genuine \GG{} test & Requires collaboration-level benchmark agreement, not detector changes & Helps the community compare clustered or non-universal extra-dimensional models more systematically.\\
Joint channel combination with \(\ttbar\)+dijet+\(hh\)+\(\llbar\) & Analysis strategy & Matches the actual multi-channel logic of \GG{} rather than overvaluing any single clean channel & Statistically straightforward once channel-specific templates are defined & Improves discovery robustness for many non-universal resonance models.\\
\bottomrule
\end{longtable}\endgroup

Part~VII took stock of reality, in plain terms: nothing published excludes the octet at its derived rate --- the clean-channel limits that look threatening do not apply to a leptophobic, top-philic cluster, and the top-pair searches that do apply used templates mismatched to this broad, interfering shape; meanwhile the free-field horn of the old dichotomy is already dead, killed by exactly these data.  The kill criterion was priced (two inverse attobarns) and the timeline drawn.  But a frontier position and a timeline are still projections built on other people's analyses.  The next Part does something more direct: it simulates the discovery itself, at the fidelity of a real analysis, and asks when --- on the actual calendar --- the question gets answered.


\section[Part~VII in plain English]{Part~VII in plain English: what has already been excluded, and what has not been looked for}
This is the Part where the prediction meets the published record, and
the record is more interesting than a simple pass or fail.

Standard benchmarks in this mass range are strongly excluded, and the Part
says so first and without qualification. What has not been excluded is this
paper's target, because no published search has been designed for it. Every
existing limit was computed against a single resonance with a chosen width;
the prediction here is three states near one scale with fixed relative
positions, and an exclusion computed for one shape does not transfer to
another by rescaling.

That is not a loophole, and the Part is careful about the difference. A
loophole is what a theory with dials uses to survive a null result. This is
a mismatch between a template and a target, and the remedy is specific and
public: a dedicated reinterpretation of data that already exists.

The Part also states the kill criterion. There is a luminosity and a set of
channels beyond which a null result ends the reading, and the number is
printed here rather than left for the discussion. The trigger roadmap that
closes the Part is framed so that every change it asks for would help a
whole category of high-mass searches, because a trigger change justified by
one theory's target is a change no collaboration should make.

\GGPart[Simulating the discovery]{Simulating the Discovery}{What ATLAS and CMS will actually record --- the anchored HL-LHC simulation, the systematics wall, and the pre-registered search: 2026 / 2031 / 2033 / 2041}
\label{part:hllhcsim}

\section{What This Part Adds, and the Rules of the Game}
\label{sec:sim-intro}

Part~\ref{part:ttanatomy} derived the \emph{shapes} of the \(\ttbar\) signal --- line shape, interference sign, angular distributions, polarization --- and \S\ref{sec:near-term} placed the search on the calendar through the projected limit curves.  Between those two lies the question an experimentalist actually lives with, and this Part answers it: \emph{if the color octet exists at \(\Mlock=3.849\)~TeV with the derived rate, what do ATLAS and CMS record, bin by bin, at the end of 2026, 2031, 2033, and 2041 --- and when, exactly, can they claim discovery?}  The answer is quantitative, and parts of it are surprising: the excess is statistically overwhelming almost immediately, yet the discovery is gated not by luminosity but by \emph{background knowledge}; a reanalysis of data already on tape carries evidence-level sensitivity; and the fact that \GG{} pre-specifies mass, width, rate, sign, and angular shape changes the statistics of the search itself in ways the published pipeline does not exploit.

This Part is written deliberately at the level of a first-year graduate student joining an ATLAS or CMS analysis group.  Everything is derived or explained in single steps: how the anchor analysis actually works, cut by cut (\S\ref{sec:sim-anchor}); how our simulation is built on it (\S\ref{sec:sim-method}); the statistics of discovery from the Poisson likelihood up (\S\ref{sec:sim-stats}); how the toy is calibrated to the real analysis (\S\ref{sec:sim-calib}); why the eye wildly overestimates significance on a spectrum plot (\S\ref{sec:sim-ladder}); the four snapshots themselves (\S\ref{sec:sim-cases}); and the statistics of the pre-registered search --- look-elsewhere, matched templates, the angular dimension, and the two-experiment combination (\S\ref{sec:sim-prereg}).  A reader who has never run a resonance search should be able to follow every number; a reader who has run one should be able to check every number.

\paragraph{The rules of the game.}  Four framing rules keep this Part honest.  (i)~Everything experimental is anchored to the most relevant published analysis --- the ATLAS full-Run-2 \(\ttbar\)-resonance search \citep{ATLAS2025ttbar} --- whose backgrounds, efficiencies, resolution, binning, and systematics we reviewed item by item.  We do not invent detector performance; we borrow measured performance.  (ii)~Our simplified statistical machinery is \emph{calibrated} against that analysis's published expected limits before any projection is made; the calibration factor is reported, not hidden (\S\ref{sec:sim-calib}).  (iii)~Phase-2 detector performance is assumed equal to Run-2 --- a stated, conservative simplification, since the ATLAS ITk/HGTD and CMS MTD upgrades \citep{ATLAS2024HLUpgrade,CMS2019MTD} specifically target \emph{better} boosted-top performance under HL-LHC pile-up.  (iv)~Grades follow the cohort ladder: the signal model is [T/D\(^{*}\)] from Parts~\ref{part:ttanatomy} and earlier (nothing about the signal is adjusted here); the experiment model is [E], anchored; the projection scenarios are labeled where they enter.

\paragraph{How a resonance search works, in one paragraph.}  For the newcomer, the pipeline this Part simulates has five stages, and each gets its own subsection below.  \emph{Production}: protons collide; a parton pair (here \(q\bar q\)) either produces a \(\ttbar\) pair through the ordinary QCD continuum or through the resonance, coherently.  \emph{Trigger}: of the \(\sim\!40\)~MHz of bunch crossings, a hardware-plus-software cascade keeps \(\mathcal O(1)\)~kHz for permanent storage; an event that fails the trigger is lost forever, so the first design question of any search is which trigger it lives on.  \emph{Selection}: offline, a small set of cuts isolates the topology --- here, one lepton plus one large, fat jet containing the entire hadronic top.  \emph{Reconstruction}: from the selected objects the invariant mass \(m_{\ttbar}\) is rebuilt event by event, with a resolution that decides how sharply the resonance stands out.  \emph{Statistical fit}: the observed \(m_{\ttbar}\) histogram is compared to the background prediction by a binned profile-likelihood fit, and the excess (or its absence) is converted into a significance or a cross-section limit.  Every one of these stages costs signal, and the honest bookkeeping of those costs is the difference between a projection and a wish.

\section{The Anchor Analysis, Read Like an Analyst}
\label{sec:sim-anchor}

This section condenses our systematic review of the ATLAS full-Run-2 \(\ttbar\)-resonance search \citep{ATLAS2025ttbar} --- \(140\)~fb\(^{-1}\) at 13~TeV, one-lepton and two-lepton channels, resonance masses 0.4--5~TeV, eight signal regions fit simultaneously.  It is the knowledge base the simulation runs on, and it is worth reading closely: the projections of \S\ref{sec:sim-cases} inherit their realism from precisely these numbers.

\subsection{The dominant backgrounds, and how each is estimated}
\label{sec:sim-bkg}

At \(m_{\ttbar}>3\)~TeV the event rate is a few events per year per experiment, and the question ``what else can fake a 3.8~TeV \(\ttbar\) pair?'' dominates the analysis design.  The composition in the boosted (merged) one-lepton signal region --- the region that drives all high-mass sensitivity, and the one we anchor to --- is reproduced in Table~\ref{tab:sim-atlasbkg}.

\begin{table}[!t]
\centering\small
\caption[ATLAS merged-SR post-fit background at 140~fb$^{-1}$]{ATLAS merged-signal-region post-fit background at \(140\)~fb\(^{-1}\) (HEPData record of \citealp{ATLAS2025ttbar}), the anchor of this Part.  ``Post-fit'' means \emph{after} the profile-likelihood fit has adjusted all backgrounds within their uncertainties against the data; these are the yields and residual uncertainties the real analysis actually achieved.  ``Other'' collects \(t\bar tV\), \(t\bar tH\), diboson, and fake-lepton contributions.}
\label{tab:sim-atlasbkg}
\begin{tabular}{lcccccc}
\toprule
\(m_{\ttbar}\) bin [GeV] & SM \(\ttbar\) & single top & \(V\)+jets & Other & Total \(\pm\) syst & Observed\\
\midrule
3000--3500 & 60\% & 11\% & 24\% & 5\% & \(53.4\pm6.9\%\) & 47\\
3500--4200 & 47\% & 8\% & 34\% & 11\% & \(21.6\pm13.2\%\) & 18\\
4200--4900 (+ovf) & 40\% & 2\% & 40\% & 17\% & \(6.1\pm22.6\%\) & 6\\
\bottomrule
\end{tabular}
\end{table}

Three features of this table repay attention.  First, \emph{the composition drifts}: SM \(\ttbar\) falls from \(60\%\) to \(40\%\) across the tail while \(V\)+jets rises to \(\sim\!40\%\).  The physics is simple once seen: the \(\ttbar\) continuum at these masses comes from \(q\bar q\) annihilation on steeply falling antiquark distributions, while a \(W\)+jets event needs only one hard valence quark plus a gluon, so its spectrum falls less steeply.  A projection that scaled ``\(\ttbar\) only'' would therefore get the tail wrong by a factor approaching two; our anchoring to the measured \emph{total} avoids this trap automatically.  Second, \emph{the uncertainties explode with mass}: \(6.9\%\to13.2\%\to22.6\%\) across the three bins.  These are post-fit values --- the residual ignorance \emph{after} the fit has used every control region --- and they will turn out to be the single most important input to this entire Part (\S\ref{sec:sim-wall}).  Third, \emph{the estimation methods differ by component}, and knowing them tells us which uncertainties can shrink with more data: SM \(\ttbar\) is a Powheg+Pythia8 template, NNLO-reweighted in top \(p_T\) and \(m_{\ttbar}\), with its overall normalization left \emph{free} in the fit (the fit determines it from the data itself --- remember this when we come to \S\ref{sec:sim-contamination}); \(W\)+jets is a Sherpa template normalized in situ by the measured \(W\)-charge asymmetry (\(W^{+}/W^{-}\) production in \(pp\) is asymmetric because the proton holds two \(u\) valence quarks and one \(d\); the measured asymmetry pins the normalization to \(\pm3\%\), after which \(50\%\) normalization and heavy-flavor shape variations cover the tail); multijet fakes are estimated from data by the matrix method (loose-versus-tight lepton identification inverted into a fake probability); single top is taken from simulation with the \(Wt\)/\(\ttbar\) interference handled by diagram removal.

\subsection{The trigger, and why one lepton is enough}
\label{sec:sim-trigger}

The analysis records events with single-lepton triggers alone: thresholds \(p_T>26\)~GeV for muons and \(E_T>26\)~GeV for electrons through 2016--2018, with isolation requirements at the lowest thresholds; the offline requirement \(p_T>28\)~GeV then sits on the \emph{trigger plateau} --- the region where the trigger efficiency has flattened near its maximum, so that per-event efficiency no longer depends on the exact threshold and can be measured once and applied cleanly.

Why is a \(26\)~GeV lepton threshold comfortable for a \(3.85\)~TeV resonance?  Follow the kinematic chain, because it is a useful habit: each top from the decay carries \(E_t\simeq\Mlock/2\approx1.9\)~TeV.  The leptonic top splits its energy among \(b\), \(\ell\), and \(\nu\); in the top rest frame the lepton takes on average roughly a third of the \(W\) energy, and after the boost the typical laboratory lepton energy is \(\mathcal O(E_t/4)\approx500\)~GeV --- more than an order of magnitude above threshold.  Even the unluckiest decay orientations rarely push the lepton below \(50\)~GeV.  The single-lepton trigger is therefore effectively \(100\%\) efficient for this signal \emph{at any} \(m_{\ttbar}\) of interest, and no large-radius-jet or missing-energy trigger is needed.  This matters doubly for the HL-LHC projection: lepton thresholds are among the few trigger quantities expected to \emph{survive} the pile-up of 200 simultaneous collisions roughly unchanged (\(\sim\!25\)--\(30\)~GeV with the upgraded Level-0/HLT chains; \citealp{ATLAS2025HighRate}), so the trigger leg of our ``Run-2 performance'' assumption is if anything conservative.

\subsection{Event selection and reconstruction, cut by cut}
\label{sec:sim-selection}

The one-lepton selection, in the order an analyst would apply it: exactly one electron or muon with \(p_T>28\)~GeV (a second lepton vetoes the event into the dilepton channel); missing transverse momentum \(E_T^{\rm miss}>20\)~GeV and \(E_T^{\rm miss}+m_T^W>60\)~GeV, which together demand a genuine escaping neutrino and suppress multijet fakes (a fake lepton comes with no neutrino, so both quantities are small); then the \emph{merged-topology} requirements that define the high-mass regime.  At \(m_{\ttbar}\gtrsim2\)~TeV the hadronic top is so boosted that its three quark jets merge into a single cone: the analysis requires one anti-\(k_t\) \(R=1.0\) ``large-radius'' jet with \(p_T>350\)~GeV and \(|\eta|<2\), \emph{trimmed} (soft subjets removed, which strips pile-up while preserving the hard three-prong structure) and identified as a top by a deep-neural-network tagger at the \(80\%\)-efficiency working point --- the operating threshold chosen such that \(80\%\) of true boosted tops pass, at a light-jet rejection near \(10\) at these energies.  On the leptonic side, a small-radius jet within \(\Delta R<2.0\) of the lepton must be \(b\)-tagged (\(77\%\) working point), completing the \(t\to b\ell\nu\) hypothesis.

Reconstruction then rebuilds the resonance mass from four pieces: the tagged large-\(R\) jet (the hadronic top), the lepton, the \(b\)-jet, and the neutrino.  Only the neutrino's longitudinal momentum is unmeasured, and it is recovered by a constraint: demanding \((p_\ell+p_\nu)^2=m_W^2\) with \(\vec p_T^{\,\nu}=\vec E_T^{\rm miss}\) yields a quadratic equation in \(p_z^\nu\),
\begin{equation}
p_z^{\nu}
=\frac{\alpha\, p_z^{\ell}\pm E_\ell\sqrt{\alpha^{2}-(E_T^{\rm miss}p_T^{\ell})^{2}}}{(p_T^{\ell})^{2}},
\qquad
\alpha\equiv\frac{m_W^{2}}{2}+\vec p_T^{\,\ell}\!\cdot\!\vec E_T^{\rm miss},
\label{eq:sim-nupz}
\end{equation}
with the smaller-\(|p_z^\nu|\) solution taken (and the discriminant set to zero when resolution pushes it negative).  The result is a fully reconstructed \(m_{\ttbar}\) event by event.  The merged category is preferred over resolved-jet categories at high mass for exactly the reason it exists: one calorimetric object instead of three small jets means fewer combinatoric mistakes and a sharper mass peak.

\subsection{What \texorpdfstring{\(\varepsilon A=6.5\%\)}{eff x acc = 6.5\%} is made of}
\label{sec:sim-effacc}

The published signal efficiency-times-acceptance for a \(3.5\)--\(4\)~TeV resonance, counting \emph{all} \(\ttbar\) decay modes in the denominator, is \(6.0\)--\(6.8\%\); we use \(0.065\).  A newcomer should be able to reconstruct this number, so here is the accounting.  The one-lepton channel itself costs the branching ratio: \(\mathrm{BR}(\ttbar\to\ell+\mathrm{jets})=2\times\mathrm{BR}(W\to\ell\nu)\times\mathrm{BR}(W\to q\bar q')\approx2\times0.217\times0.674\approx0.29\) for \(\ell=e,\mu\) (the factor 2 because either top may decay leptonically).  Within the channel, the surviving fraction is roughly the product of the top-tag working point (\(0.80\)), the \(b\)-tag working point (\(0.77\)), and the kinematic-plus-fiducial acceptance of the remaining cuts (\(\sim\!0.36\); leptons or jets outside \(|\eta|\) coverage, the \(\Delta R\) matching, the \(E_T^{\rm miss}\) cuts).  The product \(0.29\times0.80\times0.77\times0.36\approx0.065\) closes the ledger.  This decomposition also identifies where HL-LHC improvements would land: tagger working points and \(b\)-tagging at 200 pile-up are exactly what the upgrades protect.

\subsection{Resolution, binning, and the published sensitivity}
\label{sec:sim-resolution}

The merged-channel mass resolution is \(\sigma(m_{\ttbar})/m_{\ttbar}\approx10\%\), dominated by the large-\(R\) jet energy scale and the neutrino reconstruction.  Ten percent of \(3.85\)~TeV is \(385\)~GeV: comparable to, but smaller than, the octet's intrinsic width \(\Gamma_{g^*}=561\)~GeV, so the observed line shape is intrinsic width convolved with a comparable experimental smearing --- a genuinely \emph{broad} object either way, roughly two analysis bins wide.  The analysis bins the spectrum coarsely in the tail (bin edges at 3.0, 3.5, 4.2, 4.9~TeV with overflow), matched to that resolution.  Its statistical machinery is a binned profile-likelihood fit over all signal regions simultaneously, with \(\mathcal O(100)\) nuisance parameters for the systematic uncertainties; the published expected 95\%~CL upper limits at our mass --- the numbers we calibrate against in \S\ref{sec:sim-calib} --- are \(8.93\)~fb (narrow template, \(3.5\)~TeV), \(6.41\)~fb (narrow, \(4.0\)~TeV), and \(\sim\!25\)~fb for the broad \(\Gamma/m=30\%\) KK-gluon template.

\section{The Simulation, Step by Step}
\label{sec:sim-method}

\subsection{Signal: from the derived line shape to events in bins}
\label{sec:sim-signal}

The truth-level signal is the coherent \(q\bar q\to\ttbar\) spectrum of Part~\ref{part:ttanatomy}, Eq.~\eqref{eq:tt-lineshape}: QCD continuum, Breit--Wigner, and interference with the theorem-fixed sign \(\xi_{\rm UV}g_V^{t}<0\) (constructive below the pole, destructive above), evaluated with CTEQ6 parton luminosities \citep{Pumplin2002CTEQ6} at both \(13\) and \(14\)~TeV, \(\Gamma_{g^*}/m=14.6\%\), and the channel-appropriate NLO K-factors of Part~\ref{part:ttanatomy} (\(K_{q\bar q}=1.3\) on the Breit--Wigner, \(\sqrt{K_{q\bar q}}\) on the interference, since interference is linear in the resonance amplitude while the squared term is quadratic --- a small but easily forgotten point).  Detector response enters in two steps that mimic the anchor analysis.  \emph{Smearing}: the truth spectrum is convolved with a Gaussian of width \(\sigma=0.10\,m\),
\begin{equation}
\frac{d\sigma^{\rm reco}}{dm}(m)=\int dm'\,
\frac{d\sigma^{\rm truth}}{dm'}(m')\,
\frac{1}{\sqrt{2\pi}\,(0.10\,m')}\exp\!\left[-\frac{(m-m')^{2}}{2(0.10\,m')^{2}}\right],
\label{eq:sim-smear}
\end{equation}
which redistributes events between bins exactly as the detector does --- note the visible consequence in Figure~\ref{fig:sim-spectra}: the smeared peak leaks into the 3.0--3.5~TeV bin and pulls the apparent maximum slightly below the pole.  \emph{Efficiency}: the smeared spectrum is multiplied by \(\varepsilon A=0.065\) and integrated over the ATLAS bin edges.  The event yield in bin \(i\) at integrated luminosity \(L\) is then \(s_i=L\,\varepsilon A\int_{\rm bin\,i}dm\,(d\sigma^{\rm reco}_{\rm BW}+d\sigma^{\rm reco}_{\rm int})/dm\), and we track the Breit--Wigner and interference pieces separately throughout because they respond differently to the fit (\S\ref{sec:sim-ladder}).

\subsection{Background: anchoring the total to the measured yields}
\label{sec:sim-bkgmodel}

We do not simulate \(W\)+jets, single top, and fakes from scratch --- that is a multi-year collaboration effort, and pretending otherwise would be theater.  Instead we exploit the fact that ATLAS has already \emph{measured} the total background in exactly the region we care about (Table~\ref{tab:sim-atlasbkg}), and we build a two-parameter model whose shape is physics-motivated and whose normalization is fit to those measurements:
\begin{equation}
B(m)=A\;\frac{d\sigma^{\rm reco}_{\rm SM,q\bar q}}{dm}\times\left(\frac{m}{3849\,\mathrm{GeV}}\right)^{\!n},
\label{eq:sim-bkgmodel}
\end{equation}
where the first factor is our own smeared LO \(q\bar q\to\ttbar\) continuum (which carries the correct convolution of parton luminosities and the correct resolution smearing) and the power-law tilt \((m/3849)^{n}\) absorbs everything that distinguishes the full cocktail from pure LO \(q\bar q\) \(\ttbar\): the \(V\)+jets admixture with its flatter slope, NLO shape corrections, and acceptance drifts.  Fitting \((A,n)\) to the three post-fit ATLAS yields reproduces all three within their uncertainties with \(n\approx0.76\) --- positive, exactly as the composition drift of \S\ref{sec:sim-bkg} predicts (the cocktail falls \emph{less} steeply than pure \(q\bar q\) \(\ttbar\)).  The same \((A,n)\) applied to the 14-TeV continuum shape gives the HL-LHC background.  The model is deliberately humble: it does not know more than ATLAS measured, and it is not allowed to.

\subsection{From 13 to 14 TeV, and ``14-TeV-equivalent'' luminosity}
\label{sec:sim-equiv}

Two collider energies enter.  The end-2026 snapshot is simulated \emph{natively at 13~TeV}: it is a reanalysis of the Run-2+3 dataset, \(0.466\)~ab\(^{-1}\) already on tape.  The HL-LHC snapshots run at 14~TeV, where the \(q\bar q\) parton luminosity at \(3.85\)~TeV is a factor \(\simeq1.55\) larger --- at these masses the colliding partons carry momentum fractions \(x_1x_2\gtrsim(3.85/13)^2\approx0.088\), deep on the steeply falling ends of the PDFs, so even a \(7\%\) energy increase buys a large rate gain.  For the HL-LHC snapshots the 13-TeV legacy is folded in as \(\approx\!303\)~fb\(^{-1}\) of ``14-TeV-equivalent'' data --- the 13-TeV yield divided by the signal-region rate ratio --- the luminosity of 14-TeV running that would yield the same signal statistics --- giving per-experiment totals of \(0.56\)~ab\(^{-1}\) (end-2031, about a year and a half of HL-LHC physics), \(1.09\)~ab\(^{-1}\) (end-2033), and \(3.29\)~ab\(^{-1}\) (end-2041), from the schedule model of \S\ref{sec:near-term} (LS3 to mid-2030, HL-LHC physics thereafter).  All results are single-experiment unless stated; the ATLAS+CMS combination is treated carefully in \S\ref{sec:sim-combo}.

\section{The Statistics of Discovery, from First Principles}
\label{sec:sim-stats}

Nothing in this section is original --- it is the standard machinery of \citet{CowanEtAl2011} --- but the projections cannot be understood, and certainly cannot be checked, without it, so we derive it in full.

\subsection{Counting, the likelihood ratio, and the meaning of \texorpdfstring{\(Z\)}{Z}}
\label{sec:sim-poisson}

A single bin records \(n\) events where \(b\) background events are expected.  ``Significance'' answers one question: \emph{if there were no signal, how improbable is a fluctuation at least this large?}  That probability \(p\) is conventionally quoted as the equivalent number of Gaussian standard deviations, \(Z=\Phi^{-1}(1-p)\), so \(Z=5\) means \(p=2.87\times10^{-7}\).  The optimal test statistic (by the Neyman--Pearson lemma) is the likelihood ratio.  For Poisson counting, \(\mathcal L(\mu)=e^{-(\mu s+b)}(\mu s+b)^{n}/n!\), and the discovery test compares \(\mu=0\) against the best-fit \(\hat\mu\):
\begin{equation}
q_0=-2\ln\frac{\mathcal L(0)}{\mathcal L(\hat\mu)}
=2\left[n\ln\frac{n}{b}-(n-b)\right]\quad(n>b),
\label{eq:sim-q0}
\end{equation}
where the second form follows by substituting \(\hat\mu s=n-b\) (the maximum-likelihood estimate simply saturates the observed count).  By Wilks' theorem \(q_0\) is distributed as \(\chi^2_1\) under the background-only hypothesis, so \(Z=\sqrt{q_0}\).  To get the \emph{median expected} significance one evaluates \(q_0\) not on a random dataset but on the \emph{Asimov dataset} --- the artificial dataset in which every observable equals its expectation, \(n=s+b\) (named by \citeauthor{CowanEtAl2011} after Asimov's story \emph{Franchise}, in which one representative voter replaces the electorate).  Substituting \(n=s+b\) into \eqref{eq:sim-q0}:
\begin{equation}
Z_{\rm stat}=\sqrt{2\left[(s+b)\ln\!\left(1+\frac{s}{b}\right)-s\right]}
\;\xrightarrow[\;s\ll b\;]{}\;\frac{s}{\sqrt b},
\label{eq:sim-zstat}
\end{equation}
where the limit follows from expanding the logarithm to second order --- and this is where the folklore formula \(s/\sqrt b\) comes from, along with its domain of validity (\(s\ll b\); at our signal-to-background ratios of order one, the full form matters and is what we use).

\subsection{A background systematic as a nuisance parameter}
\label{sec:sim-nuisance}

Now suppose \(b\) itself is uncertain by \(\pm\delta b\) --- in our case, the ATLAS post-fit fractional uncertainties of Table~\ref{tab:sim-atlasbkg}, \(\delta b=f_b\,b\).  The likelihood acquires a \emph{nuisance parameter}: the unknown true background \(b'\), constrained by a Gaussian of width \(\delta b\) (the auxiliary knowledge from control regions and calibrations).  The test statistic \emph{profiles} \(b'\) --- re-maximizes the likelihood over it, separately in numerator and denominator --- so that the fluctuation must beat not only Poisson noise but every background configuration the systematics allow.  The heuristic result can be guessed immediately: the two noise sources add in quadrature,
\begin{equation}
Z\approx\frac{s}{\sqrt{b+(f_b\,b)^{2}}},
\label{eq:sim-zheur}
\end{equation}
and the exact Asimov formula, obtained by carrying out the profiling analytically \citep{CowanEtAl2011}, is the one the simulation uses everywhere:
\begin{equation}
Z=\left\{2\left[(s+b)\ln\frac{(s+b)(b+\delta b^{2})}{b^{2}+(s+b)\delta b^{2}}
-\frac{b^{2}}{\delta b^{2}}\ln\!\left(1+\frac{\delta b^{2}\,s}{b\,(b+\delta b^{2})}\right)\right]\right\}^{1/2}.
\label{eq:sim-zcowan}
\end{equation}
Two limits confirm the formula against intuition: as \(\delta b\to0\) it collapses to \eqref{eq:sim-zstat}, and for \(\delta b\gg\sqrt b\) it saturates at \(Z\to s/\delta b\) --- the systematic wall.  Independent bins combine in quadrature, \(Z_{\rm tot}^2=\sum_i Z_i^2\), which is exact for the profile likelihood when the nuisances are bin-independent (our per-bin \(f_b\) treatment; correlations are absorbed into the calibration of \S\ref{sec:sim-calib}).

\subsection{The systematics wall, located precisely}
\label{sec:sim-wall}

Equation \eqref{eq:sim-zheur} contains the single most consequential piece of arithmetic in this Part.  Both \(s\) and \(b\) grow linearly with luminosity \(L\), so the Poisson term \(\sqrt b\) grows as \(\sqrt L\) while the systematic term \(f_b\,b\) grows as \(L\).  They cross when \(b=1/f_b^{2}\), i.e.\ at
\begin{equation}
L^{*}=\frac{1}{f_b^{2}\;b_{1{\rm fb}}}\,,
\label{eq:sim-lstar}
\end{equation}
with \(b_{1\rm fb}\) the background per fb\(^{-1}\).  Beyond \(L^{*}\) the per-bin significance saturates at the luminosity-\emph{independent} ceiling \(Z_{\max}=s/(f_b\,b)=({\rm signal\!-\!to\!-\!background})/f_b\): from then on, only better background knowledge --- smaller \(f_b\) --- moves the answer.  Evaluating \eqref{eq:sim-lstar} with the anchored backgrounds and the Run-2 post-fit \(f_b\): \(L^{*}=353\), \(554\), \(356\), \(485\), and \(1054\)~fb\(^{-1}\) for the 2.5--3.0, 3.0--3.5, 3.5--4.2, 4.2--4.9, and 4.9--5.6~TeV bins respectively.  Every one of these is of order the Run-3 dataset.  The conclusion deserves italics because nothing downstream makes sense without it: \emph{the high-mass \(\ttbar\) search leaves the statistics-limited regime during Run~3; the entire HL-LHC era is, for this search, a fight about systematics, not statistics.}  This single fact explains the shape of every sensitivity curve below.

\section{Calibration: Tying the Toy to the Real Analysis}
\label{sec:sim-calib}

Our likelihood is simplified: it ignores nuisance-parameter correlations, the floating \(\ttbar\) normalization's absorption of broad shapes, signal leakage across signal regions, the dilepton channel, and the multi-SR structure.  The cost of all this is measured, not guessed, by an \emph{injection test}: we set narrow test signals (Gaussians at the detector resolution, with the published narrow-template efficiencies) at \(140\)~fb\(^{-1}\) and compute our expected 95\% CL limits with the same machinery used for all projections.  We obtain \(1.46\)~fb at \(4.0\)~TeV and \(2.09\)~fb at \(3.5\)~TeV, against ATLAS's published \(6.41\) and \(8.93\)~fb: degradation factors \(4.40\) and \(4.27\) --- consistent at the two masses, which is what licenses promoting their mean to a single calibration constant,
\begin{equation}
F_{\rm cal}=4.33,\qquad Z_{\rm true}\simeq Z_{\rm naive}/F_{\rm cal},
\label{eq:sim-fcal}
\end{equation}
applied to every significance and limit quoted from here on.  This is a deliberately conservative choice: \(F_{\rm cal}\) is measured in exclusion-specific machinery and applied to discovery as well.  \emph{Validation on the shape that matters}: with \(F_{\rm cal}\) fixed and no further freedom, the framework predicts a \(140\)~fb\(^{-1}\) expected limit of \(8.4\)~fb for the \(\Gamma/m=15\%\) interference-free shape --- correctly ordered between ATLAS's published narrow (\(6.4\)~fb) and broad-\(30\%\) (\(\sim\!25\)~fb) template limits, exactly where a \(15\%\)-width shape must fall.  The framework also reproduces, without adjustment, ATLAS's own statement that the octet sits at the present frontier: injecting the predicted \(8.7\)~fb yields an expected excess just below the published exclusion.

One methodological remark for the statistically minded reader, because the temptation is real: one might think the toy should \emph{additionally} profile an explicit background normalization and shape tilt (as our diagnostic in \S\ref{sec:sim-ladder} does).  It should not, for a reason worth understanding: the ATLAS \(f_b\) of Table~\ref{tab:sim-atlasbkg} are \emph{post-fit} uncertainties --- they already encode the residual freedom of the background model \emph{after} the real fit has floated its normalizations and profiled its shapes.  Using post-fit \(f_b\) \emph{and} an explicit free normalization would double-count the background freedom and understate the sensitivity.  The post-fit-\(f_b\)-plus-calibration route is the validated one; the explicit-profile construction appears below strictly as a pedagogical instrument for locating where sensitivity is lost.

\subsection{What ``improved systematics'' means, quantitatively}
\label{sec:sim-impsyst}

Every projection below is quoted under two scenarios that differ in exactly one input: the per-bin fractional background uncertainty.
\begin{table}[htbp]
\centering
\small
\caption[What ``improved systematics'' means, stated as a number per bin rather than as an adjective.]{What ``improved systematics'' means, stated as a number per bin rather than as an adjective. The frozen row is the published post-fit background uncertainty of the ATLAS Run-2 search; the improved row halves it. Every projection in this Part is quoted under both, and the two rows differ in exactly one input, so a reader who thinks the improvement optimistic can read the frozen column throughout and still have a complete forecast.}
\label{tab:syst-bins}\small
\begin{tabular}{lccccc}
\toprule
\(m_{\ttbar}\) bin [TeV] & 2.5--3.0 & 3.0--3.5 & 3.5--4.2 & 4.2--4.9 & 4.9--5.6\\
\midrule
\textbf{frozen} (ATLAS post-fit, Run 2) & 5.0\% & 6.9\% & 13.2\% & 22.6\% & 30\%\\
\textbf{improved} (\(f_b/2\)) & 2.5\% & 3.45\% & 6.6\% & 11.3\% & 15\%\\
\bottomrule
\end{tabular}
\end{table}
The halving is the standard HL-LHC projection convention \citep{CERNYellowReport2019}, and it is decomposable rather than hopeful.  The ATLAS tail uncertainties contain three classes: (i)~template and control-region \emph{statistical} components (simulation sample sizes, the \(W\)-charge-asymmetry and \(Z\)-reweighting control regions), which scale as \(1/\sqrt L\) and alone shrink by \(\times2.8\) at \(1.1\)~ab\(^{-1}\) and \(\times4.9\) at \(3.3\)~ab\(^{-1}\); (ii)~object calibrations (jet energy scale, top- and \(b\)-tagging scale factors), data-driven and improving roughly as \(1/\sqrt L\) toward \(1\)--\(2\%\) floors; (iii)~theory-modeling terms (\(\ttbar\) generator and shower variations, the \(50\%\) \(W\)+heavy-flavor variations), which improve only with better simulation and auxiliary measurements.  Halving the total corresponds to classes (i)+(ii) improving as expected while (iii) improves modestly --- the plausible mid-course.  ``Frozen'' assumes \emph{no} improvement of any component over fifteen years: the pessimistic bound.  Reality should track between the two curves of Figure~\ref{fig:sim-zcurve}, closer to ``improved''.  Note one snapshot-specific subtlety used honestly below: at end-2026 roughly half of the ``improvement'' is \emph{automatic}, because the Run-2+3 dataset itself shrinks the class-(i) components by \(\sqrt{140/466}=0.55\) before anyone improves anything.

\section{Why the Eye Overestimates: The Significance Ladder}
\label{sec:sim-ladder}

Figure~\ref{fig:sim-spectra} \emph{looks} like an overwhelming discovery already at end-2026 --- and by pure counting it is: the excess over the drawn background is a \(23.6\sigma\) statistical effect over the full displayed range (\(18.6\sigma\) within the 3.0--4.9~TeV analysis window).  The honest calibrated significance is \(2.7\sigma\).  A first-year student should internalize where that factor of \(\sim\!9\) goes, because it is the difference between reading spectrum plots and understanding searches; Figure~\ref{fig:sim-absorb} decomposes it into three physically distinct steps.

\paragraph{Step 1 \(\to\) 2 (\(23.6\to12.8\sigma\)): the gray line has error bars too.}  The eye compares data points to the background curve as if the curve were exact; the correct per-bin comparison is \eqref{eq:sim-zheur}, and at \(0.466\)~ab\(^{-1}\) the systematic term \(f_b\,b\) has \emph{already overtaken} \(\sqrt b\) in every relevant bin (\S\ref{sec:sim-wall}).  Replacing \(s/\sqrt b\) with the full \eqref{eq:sim-zcowan} in each bin takes the counting ladder from \(23.6\sigma\) to \(12.8\sigma\).

\paragraph{Step 2 \(\to\) 3 (\(12.8\to4.0\sigma\)): the background shape is not God-given.}  The background curve of Figure~\ref{fig:sim-spectra} is drawn as if known in advance.  In the real analysis it is \emph{determined from the same data}: the SM \(\ttbar\) normalization floats freely and the shape nuisances are profiled.  A \(\Gamma/m=15\%\) resonance smeared by \(10\%\) resolution is a smooth, single-signed, two-bin-wide enhancement --- dangerously close to the null space of that fit.  The mechanism is worth seeing analytically once.  Let the fit have one free parameter, an overall normalization \(A\), determined by weighted least squares against data \(d_i=b_i+s_i\) with weights \(w_i=1/(b_i+(f_ib_i)^2)\); minimizing \(\sum_iw_i(d_i-Ab_i)^2\) gives
\begin{equation}
\hat A=1+\frac{\sum_i w_i\,b_i\,s_i}{\sum_i w_i\,b_i^{2}},
\label{eq:sim-absorb}
\end{equation}
i.e.\ the background estimate \emph{rises to meet the signal}, by an amount governed by how much the signal shape resembles the background shape in the metric \(w\).  A narrow spike (one bin) barely moves \(\hat A\); a smooth broad excess moves it a lot.  Our diagnostic fit with just \emph{two} parameters (normalization and a power-law tilt), profiled against the frozen uncertainties, absorbs \(75\%\) of the window signal --- the refit background climbs into the excess at negligible cost at low mass, where \(f_b=5\%\), because the tail bins where the signal lives allow \(13\)--\(30\%\) --- and takes \(12.8\sigma\) to \(4.0\sigma\) (Figure~\ref{fig:sim-absorb}, left).

\paragraph{Step 3 \(\to\) 4 (\(4.0\to2.7\sigma\)): the full machinery.}  The real fit has not two but \(\mathcal O(100)\) nuisance parameters --- per-component normalizations, jet energy scale eigenvectors, tagging scale factors, generator shape variations --- each with more absorbing power than a global tilt.  We do not model them individually; their aggregate cost is what the measured calibration \(F_{\rm cal}=4.33\) carries (\S\ref{sec:sim-calib}).

\paragraph{Two corollaries.}  (i)~\emph{The discovery variable is background knowledge, not luminosity}: this is the entire content of the frozen curve's plateau in Figure~\ref{fig:sim-zcurve}, and it is why the improved-systematics scenario, not more data, is what crosses \(5\sigma\).  (ii)~\emph{Absorption is blind to correlated, sign-changing structure}: a normalization-and-tilt fit can eat a single-signed bump, but it cannot generate the interference pattern --- excess below the pole, deficit above it, the summed spectrum dipping \emph{below} the SM beyond \(4.9\)~TeV.  A template fit carrying the interference degree of freedom therefore recovers much of what naive counting loses, which is one of the pre-registration advantages quantified in \S\ref{sec:sim-prereg}; the discovery numbers of this Part, which use counting only, are in that specific sense conservative.

\begin{figure}[!t]
\centering
\resizebox{\textwidth}{!}{\includegraphics{Arisaka-GG-P1-KK-hllhc-absorb.pdf}}
\caption[The significance ladder at end-2026]{Why the ``clean'' excess of Figure~\ref{fig:sim-spectra} is worth \(2.7\sigma\), not \(20\sigma\), at end-2026.  \textbf{Left:} the end-2026 spectrum with the frozen per-bin systematic band (gray) and the signal-free background refit with only two free parameters, normalization and tilt (blue dashed): the refit climbs into the excess and absorbs \(75\%\) of the window signal, at negligible cost at low mass where the band is tight.  The full ATLAS fit has \(\mathcal O(100)\) such degrees of freedom.  \textbf{Right:} the significance ladder of the identical dataset: perfect-background counting \(23.6\sigma\) \(\to\) per-bin systematics \(12.8\sigma\) \(\to\) two-parameter shape freedom \(4.0\sigma\) \(\to\) measured full-analysis calibration \(2.7\sigma\).}
\label{fig:sim-absorb}
\end{figure}

\section{The Four Snapshots: What the Detectors Record}
\label{sec:sim-cases}

Table~\ref{tab:sim-cases} gives the analysis-bin expectations and calibrated significances; Figure~\ref{fig:sim-spectra} draws the full simulated spectra over 1.5--7~TeV with statistical error bars (one Poisson pseudo-experiment per snapshot, fixed seed).  All yields are merged-SR, one-lepton, per experiment; end-2026 is 13~TeV, the rest 14~TeV.

\begin{table}[!t]
\centering\small
\setlength{\tabcolsep}{4.2pt}
\caption[The four snapshots: yields and significances]{The four snapshots.  Analysis bins 3.0--3.5 / 3.5--4.2 / 4.2--4.9~TeV (the ATLAS-anchored region).  \(S\): Breit--Wigner; \(I\): interference (signed --- note the deficit in the highest bin, the shape falsifier of Part~\ref{part:ttanatomy}); \(B\): background.  Significances are calibrated (\(F_{\rm cal}=4.33\)); scenarios as \S\ref{sec:sim-impsyst}.  Property rows use purity-corrected in-window counts.}
\label{tab:sim-cases}
\resizebox{\textwidth}{!}{%
\begin{tabular}{lcccc}
\toprule
 & \textbf{end-2026} & \textbf{end-2031} & \textbf{end-2033} & \textbf{end-2041}\\
 & \(0.466\) ab\(^{-1}\) (13 TeV) & \(0.56\) ab\(^{-1}\) & \(1.09\) ab\(^{-1}\) & \(3.29\) ab\(^{-1}\)\\
\midrule
\(S\) (3 bins) & \(81/113/32\) & \(144/211/63\) & \(279/409/121\) & \(843/1234/366\)\\
\(I\) (3 bins) & \(+64/+20/-5\) & \(+110/+35/-11\) & \(+212/+67/-21\) & \(+641/+203/-64\)\\
\(B\) (3 bins) & \(177/75/19\) & \(301/137/37\) & \(583/265/72\) & \(1758/798/216\)\\
\midrule
\(Z_{\rm disc}\) (frozen) & \(2.7\sigma\) & \(3.1\sigma\) & \(3.3\sigma\) & \(3.5\sigma\)\\
\(Z_{\rm disc}\) (improved) & \(\bm{3.7\sigma}\) & \(\bm{4.5\sigma}\) & \(\bm{5.4\sigma}\) & \(\bm{6.5\sigma}\)\\
\(Z_{\rm disc}\) (stat-only bound) & \(18.6\sigma\) & \(25.1\sigma\) & \(34.9\sigma\) & \(60.6\sigma\)\\
\midrule
signal events in window (3.0--4.9~TeV) & \(305\) & \(552\) & \(1068\) & \(3223\)\\
window purity & \(0.53\) & \(0.54\) & \(0.54\) & \(0.54\)\\
spin discrimination & \(5.2\sigma\) & \(7.1\sigma\) & \(9.9\sigma\) & \(17.3\sigma\)\\
\(\delta P_t\) (lepton slope) & \(0.27\) & \(0.19\) & \(0.14\) & \(0.08\)\\
interference sign (coarse bins) & \(\sim\!0.8\sigma\) & \(\sim\!0.9\sigma\) & \(\sim\!1.0\sigma\) & \(\sim\!1.0\sigma\)\\
\bottomrule
\end{tabular}}
\end{table}

\begin{figure}[!t]
\centering
\resizebox{\textwidth}{!}{\includegraphics{Arisaka-GG-P1-KK-hllhc-spectra4.pdf}}
\caption[Simulated spectra at the four snapshots]{Simulated merged-SR \(m_{\ttbar}\) spectra, 1.5--7~TeV, at the four snapshots (one experiment; end-2026 at 13~TeV, others at 14~TeV).  Points: Poisson pseudo-data with \(\pm\sqrt N\) statistical error bars.  Gray: ATLAS-anchored background (anchored in 3.0--4.9~TeV, extrapolated outside) with the improved-scenario systematic band.  Red solid: background \(+\) octet with the derived interference; red dotted: Breit--Wigner only.  Three coherent-structure signatures are visible at this mass range: the \(10\)--\(15\%\) interference enhancement of the continuum extending far below the pole (solid above dotted below \(\sim\!3.8\)~TeV), the constructive shoulder at 3.0--3.5~TeV, and the destructive high side, with the solid curve dipping below the Breit--Wigner-only expectation above the pole and below the SM itself beyond \(\simeq\!4.9\)~TeV.  The visual size of the excess greatly overstates its significance --- see \S\ref{sec:sim-ladder} and Figure~\ref{fig:sim-absorb} for where the sensitivity actually goes.}
\label{fig:sim-spectra}
\end{figure}

\begin{figure}[!t]
\centering
\resizebox{0.85\textwidth}{!}{\includegraphics{Arisaka-GG-P1-KK-hllhc-zcurve.pdf}}
\caption[Calibrated significance versus luminosity]{Calibrated discovery significance versus integrated luminosity (14-TeV-equivalent, per experiment) for the three systematics scenarios; stars mark the end-2026 13-TeV reanalysis case.  The saturation of the frozen curve is the central lesson of \S\ref{sec:sim-wall}: beyond \(\sim\!0.5\)~ab\(^{-1}\) the search is entirely systematics-limited.  In the improved scenario \(5\sigma\) arrives at \(0.80\)~ab\(^{-1}\) --- data on tape during 2032; frozen Run-2 systematics cap a single experiment at \(3.6\sigma\) forever.}
\label{fig:sim-zcurve}
\end{figure}

\paragraph{End-2026: the reanalysis case.}  This snapshot needs no new collisions: it is the full Run-2+3 dataset, on tape since June 2026.  At the derived rate the sample contains \(\sim\!300\) octet events (Breit--Wigner plus net interference) in the 3.0--4.9~TeV window against \(\sim\!270\) background --- a signal-to-background near \emph{one}, and statistically a \(19\sigma\) effect --- worth \(2.7\sigma\) (frozen) to \(3.7\sigma\) (improved) per experiment once background systematics are profiled.  In plain terms: \emph{evidence-level sensitivity to the \GG{} octet may already exist in recorded data}, accessible through exactly the dedicated \(\Gamma/m\simeq15\%\) interference-aware reinterpretation that this paper requests (\S\ref{sec:operational}).  The same reanalysis, if null at the predicted rate, would put the composite scenario under severe pressure --- the exclusion mode of \S\ref{subsec:killwindow}.

\paragraph{End-2031 (\(\sim\)1.5 years of HL-LHC).}  The 14-TeV data roughly doubles the effective sample.  Improved-scenario significance reaches \(4.5\sigma\) per experiment; spin discrimination reaches \(7\sigma\); \(\delta P_t=0.19\) separates the predicted \(P_t=+0.91\) from zero at \(\sim\!5\sigma\).  The earliest defensible single-experiment \(5\sigma\) (improved scenario) arrives at \(0.80\)~ab\(^{-1}\): data on tape during \textbf{2032}.

\paragraph{End-2033 (two years of HL-LHC).}  \(5.4\sigma\) per experiment (improved); the frozen scenario is already deep into saturation (\(3.3\sigma\)), demonstrating that everything now hinges on background-model improvements, not luminosity.  Properties: spin at \(9.9\sigma\); \(\delta P_t=0.14\), establishing the right-handedness of the top coupling at \(\sim\!6\sigma\) if the state is there.

\paragraph{End-2041 (full program).}  All discrete properties saturate: spin at \(17\sigma\); \(\delta P_t=0.08\), resolving the interior of the cascade-depth band \(P_t\in[+0.88,+0.92]\).  Frozen-scenario ceiling: \(3.5\sigma\) per experiment; improved: \(6.5\sigma\).

\paragraph{The interference sign in coarse bins.}  At every snapshot the coarse-binned sign visibility stalls near \(1\sigma\): the constructive low-side excess is smooth and single-signed --- precisely the shape a floating \(\ttbar\) normalization absorbs (Eq.~\ref{eq:sim-absorb}).  The sign lives in the correlated low-surplus/high-deficit pattern, including the spectrum falling \emph{below} the SM beyond \(4.9\)~TeV, and a template fit with the interference degree of freedom converts the \(600\)--\(3000\)-event lever arm into a decisive measurement.  That is the experimental case for the closing request of Part~\ref{part:ttanatomy}, now with numbers attached.

\section{The Pre-Registered Search: What the Exact Prediction Buys}
\label{sec:sim-prereg}

Everything above used the projection language of a generic search.  But the \GG{} octet is not a generic hypothesis: the mass (\(3.849\)~TeV, from \(\tfrac12M_{\rm Pl}e^{-35}\), zero free parameters), the width (\(\Gamma/m=14.6\%\), T-KK-2), the rate (\(8.7\)~fb, Corollary T-KK-2c), the interference sign (a theorem), and the angular distribution (\(\tfrac38(1+\cos^2\theta^*)\), Part~\ref{part:ttanatomy}) are all specified \emph{before looking}.  Registered where?  In print: the public posting of this paper fixes the mass, the width, the rate, the sign, and the angular shape before the analysis exists, and any later adjustment would be visible to everyone as a revision.  That is the entire mechanism, and it is sufficient.  A search for \emph{this} hypothesis differs statistically from the published bump hunt in three separable ways.  We quantify each --- and, equally important, state which are already banked in Table~\ref{tab:sim-cases} and which are genuinely additional.

\subsection{Advantage 1: no look-elsewhere penalty (banked)}
\label{sec:sim-lee}

When a search scans many mass hypotheses, a fluctuation anywhere counts, so the probability of a fake excess is inflated by the number of independent places one looked --- the \emph{trials factor}, or look-elsewhere effect.  The estimate is elementary and worth doing in one's head: two mass hypotheses are statistically independent when separated by roughly the resolution, so the number of independent cells in a scan from \(m_{\rm lo}\) to \(m_{\rm hi}\) with resolution \(\sigma(m)=0.10\,m\) is
\begin{equation}
N_{\rm cells}\simeq\int_{m_{\rm lo}}^{m_{\rm hi}}\frac{dm}{0.10\,m}
=10\,\ln\frac{m_{\rm hi}}{m_{\rm lo}}
=10\,\ln\frac{5000}{400}\approx25,
\label{eq:sim-trials}
\end{equation}
and with \(\sim\!3\) width hypotheses the published scan carries \(N_{\rm trials}\approx75\) (the rigorous up-crossings treatment of \citealp{GrossVitells2010} gives the same order).  For small \(p\), the global probability is \(p_{\rm glob}\simeq N_{\rm trials}\,p_{\rm loc}\), and the discount is steepest exactly where it hurts: a local \(5\sigma\) (\(p=2.9\times10^{-7}\)) deflates to a global \(4.1\sigma\); a global \(5\sigma\) claim requires local \(5.8\sigma\).  A pre-registered single point pays nothing: local \emph{is} global.  And near the systematics plateau, where significance grows by only \(\sim\!1\sigma\) per \emph{tripling} of luminosity (Figure~\ref{fig:sim-zcurve}), the \(0.8\sigma\) that pre-registration saves is worth roughly a \emph{doubling of the HL-LHC dataset} --- of order five years of running.  Two honesty notes.  (i)~The significances quoted throughout this Part are already local ones (the calibration is to per-mass-point expected limits), so this advantage is \emph{already banked} in Table~\ref{tab:sim-cases}: it is a gain relative to the published pipeline's discovery quote, not an addition to our numbers.  (ii)~Look-elsewhere never applied to \emph{exclusion}, so the falsification timeline of \S\ref{subsec:killwindow} is untouched.

\subsection{Advantage 2: the matched template (banked); what the published pipeline actually tests (not this)}
\label{sec:sim-matched}

The published analysis tests generic narrow-\(Z'\) and \(\Gamma/m\simeq30\%\) KK-gluon templates.  \emph{No published template has the \GG{} shape}: \(\Gamma/m=15\%\) with coherent QCD interference at the derived sign, the low-side enhancement, and the high-side deficit.  The cost of the mismatch is measured inside our own validated framework: the expected limit on the matched \(15\%\)-width coherent shape is \(8.4\)~fb at \(140\)~fb\(^{-1}\), while the closest published broad template delivers \(\sim\!25\)~fb --- \emph{a factor \(\sim\!3\) dilution in cross-section sensitivity} for the mismatched pipeline.  Our simulation always injected and counted the exact \(S+I\) template, so this advantage too is already banked in Table~\ref{tab:sim-cases}.  The operational content is the converse: \emph{the published pipeline, run as-is on Run-3 data, would under-report this signal by a factor \(\sim\!3\) in rate and discount whatever excess it sees by the trials factor} --- both removed, at no experimental cost, by the dedicated analysis this paper requests.

\subsection{Advantage 3: the angular dimension and the exact rate (not banked)}
\label{sec:sim-angular}

Two genuinely additional handles remain outside Table~\ref{tab:sim-cases}.  (i)~\emph{Angular information}: the discovery numbers bin only in \(m_{\ttbar}\).  The signal is central --- integrating the spin-1 form of Part~\ref{part:ttanatomy}, \(\int_{-1/2}^{1/2}(1+c^2)\,dc\big/\int_{-1}^{1}(1+c^2)\,dc=\tfrac{13/12}{8/3}=0.41\), so \(41\%\) of the signal sits at \(|\cos\theta^*|<0.5\) --- while the high-mass background cocktail (40\% \(V\)+jets, with forward-peaked \(t\)-channel topologies) is not.  In the systematics-limited regime the per-bin ceiling is \(s/(f_b\,b)\), so any selection or two-dimensional template that improves \(s/b\) converts \emph{directly} into significance, with no luminosity price.  Lacking anchored angular shapes for the full background cocktail, we bound rather than bank: a further \(10\)--\(30\%\) in \(Z\) is plausible, and a detector-level two-dimensional study should replace this bound.  (ii)~\emph{The rate is predicted}: the hypothesis test can be sharpened from ``some \(\mu>0\)'' to the simple-versus-simple test of \(\mu=1\), and an unbinned matched filter can replace the coarse ATLAS binning; each is worth \(\mathcal O(10\%)\).  We bank neither.

\subsection{The two-experiment combination, and a calibrated verdict}
\label{sec:sim-combo}

The remaining multiplier is decisive: \emph{two} experiments.  In the statistics-limited regime an ATLAS+CMS combination gains exactly \(\sqrt2\); in the systematics-limited regime the gain depends on whether the dominant uncertainties are correlated between the experiments (theory modeling: yes; jet calibrations, tagging, control-region statistics: no), so Table~\ref{tab:sim-combo} quotes both extremes, between which reality lies.

\begin{table}[!t]
\centering\small
\caption[Dedicated-search significance, with ATLAS+CMS combination]{Calibrated discovery significance of the dedicated, pre-registered, matched-template search (no angular information banked).  ``corr''/``uncorr'' = ATLAS+CMS combination with fully correlated / fully uncorrelated background systematics.  Improved scenario as \S\ref{sec:sim-impsyst}; at end-2026 roughly half of the improvement is automatic (control-region statistics shrink with the Run-3 dataset itself).}
\label{tab:sim-combo}
\begin{tabular}{lcccccc}
\toprule
 & \multicolumn{3}{c}{\textbf{frozen syst}} & \multicolumn{3}{c}{\textbf{improved syst}}\\
 & 1 expt & comb.\ corr & comb.\ uncorr & 1 expt & comb.\ corr & comb.\ uncorr\\
\midrule
end-2026 & \(2.7\sigma\) & \(3.1\sigma\) & \(3.9\sigma\) & \(3.7\sigma\) & \(4.6\sigma\) & \(5.2\sigma\)\\
end-2031 & \(3.1\sigma\) & \(3.3\sigma\) & \(4.3\sigma\) & \(4.5\sigma\) & \(5.4\sigma\) & \(6.3\sigma\)\\
end-2033 & \(3.3\sigma\) & \(3.5\sigma\) & \(4.7\sigma\) & \(5.4\sigma\) & \(6.1\sigma\) & \(7.6\sigma\)\\
end-2041 & \(3.5\sigma\) & \(3.6\sigma\) & \(5.0\sigma\) & \(6.5\sigma\) & \(6.9\sigma\) & \(9.1\sigma\)\\
\bottomrule
\end{tabular}
\end{table}

The verdict on the natural question --- \emph{could Run-3 data alone already deliver \(5\sigma\)?} --- stated with the calibration behind it: per experiment, no (\(2.7\)--\(3.7\sigma\): evidence, not discovery).  For a combined ATLAS+CMS dedicated analysis with improved systematics: \(4.6\)--\(5.2\sigma\) before any angular information --- \(5\sigma\) from data already on tape is a \emph{live possibility}, not a guarantee; the bounded angular gain would move the bracket to \(5.1\)--\(6.7\sigma\), but that piece awaits a real detector-level study.  The central expectation is therefore: a combined dedicated reanalysis of existing data yields strong evidence (\(\gtrsim\!4\sigma\)), with discovery either from that same data if the systematics and angular handles break favorably, or with the first \(\sim\!1\)~ab\(^{-1}\) of HL-LHC otherwise.

\subsection{A warning the dedicated fit must respect: the sideband is contaminated}
\label{sec:sim-contamination}

One subtlety cuts \emph{against} naive optimism, and the dedicated analysis must be designed around it.  Take the ratio of the interference term to the QCD continuum in Eq.~\eqref{eq:tt-lineshape}: both share the same parton luminosities and color factors, so far from the pole the ratio reduces to the propagator structure,
\begin{equation}
\frac{d\sigma_{\rm int}}{d\sigma_{\rm SM}}
\;\simeq\;2\,\xi_{\rm UV}\,\frac{g_V^t}{g_s}\;\frac{\hat s}{\hat s-\Mlock^{2}}
\qquad(|\hat s-\Mlock^2|\gg\Mlock\Gamma),
\label{eq:sim-contam}
\end{equation}
which is \emph{positive} below the pole (both factors negative) and falls off only as \(\hat s/(\hat s-\Mlock^2)\) --- slowly.  Numerically the coherent enhancement of the total background is \(\sim\!16\%\) at \(1.75\)~TeV and \(\sim\!10\%\) at \(2.25\)~TeV: precisely the region a fitter would instinctively use to pin the floating \(\ttbar\) normalization.  A smooth single-signed lift is partially degenerate with a normalization shift (Eq.~\ref{eq:sim-absorb}), and the degeneracy actively hurts: in the explicit two-parameter profile of \S\ref{sec:sim-ladder}, \emph{removing} the low-side interference from the injected signal \emph{raises} the profiled significance (from \(4.1\) to \(4.7\), frozen, uncalibrated), because an uncontaminated sideband pins the background better than a contaminated one.  The prescription follows from \eqref{eq:sim-contam}: the contamination falls to \(\sim\!4\%\) at \(1\)~TeV and \(\sim\!1\%\) below \(700\)~GeV, so \emph{the \(\ttbar\) normalization must be anchored at \(m_{\ttbar}\lesssim1\)~TeV (or in the dilepton channel), and the fit must carry the interference degree of freedom explicitly, sign free}.  The low-side enhancement is also the template component most sensitive to our leading-order treatment: an NLO computation of the interference K-factor is the single theory input the dedicated analysis should commission first.

\section{What This Means for F-KK-2B, and the Caveat Ledger}
\label{sec:sim-meaning}

The snapshot sequence rewrites the operational content of F-KK-2B (\S\ref{subsec:killwindow}) as a calendar.  \emph{By end-2026}: a dedicated reanalysis of data already on tape carries evidence-level sensitivity per experiment and possibly discovery-level sensitivity in combination --- in either direction: at the predicted rate it shows a strong excess, and a null at that rate puts the composite scenario under decisive pressure.  \emph{By 2032}: with improved systematics, single-experiment \(5\sigma\) luminosity is on tape.  \emph{By end-2033}: discovery or exclusion is decided in practice for the improved scenario, and the property program (spin at \(9.9\sigma\), chirality at \(\sim\!6\sigma\)) is already underway.  \emph{By 2041}: every discrete property of the octet hypothesis is measured to saturation, including the interior of the cascade-depth band via \(\delta P_t=0.08\); and if nothing is there, the frozen-scenario combined exclusion has long since closed the window.  In both directions the message to the experiments is the same one this paper has carried from the start, now with the full simulation behind it: \emph{the dedicated, interference-aware, pre-registered \(\ttbar\) analysis at \(3.85\)~TeV is disproportionately valuable relative to its cost, and the data for its first decisive pass already exist.}

\paragraph{The caveat ledger.}  Every simplification with sign and rough size, for the reader who will improve on this Part.  (1)~LO\(\times\)K rates with \(\sqrt K\) on the interference; an NLO interference calculation could move the low-side template by tens of percent (direction unknown; \S\ref{sec:sim-contamination}).  (2)~Run-2 detector performance frozen through 2041; the Phase-2 upgrades target better boosted-top performance under pile-up, so this is conservative.  (In particular, the boosted-top tagging efficiency above \(\sim\!3.5\)~TeV is an extrapolation of the published working points; the Phase-2 upgrades target exactly this regime.)  (3)~The background model is a two-parameter anchored extrapolation; below \(2.5\)~TeV and above \(4.9\)~TeV it is an extrapolation, clearly flagged in Figure~\ref{fig:sim-spectra}.  (4)~The calibration factor \(F_{\rm cal}\) is measured on exclusion machinery at two masses and applied to discovery; its consistency (\(4.40\) vs \(4.27\)) and the broad-shape validation are the evidence for its transferability.  (5)~Per-bin systematics are treated as uncorrelated; correlations are absorbed into \(F_{\rm cal}\) rather than modeled.  (6)~The property projections (spin, \(P_t\)) use purity-corrected event counts with the acceptance-degraded separations of Part~\ref{part:ttanatomy}, not a detector-level angular analysis.  (7)~The ATLAS+CMS combination brackets correlation extremes rather than modeling the shared nuisances.  (8)~All pseudo-data use a single fixed random seed; the Asimov significances are seed-independent.  (9)~Electroweak corrections are not applied: at \(m_{\ttbar}\simeq4\)~TeV the electroweak Sudakov logarithms are a negative \(\mathcal O(10\text{--}20\%)\) effect on continuum and signal alike.  Because the background normalization is anchored to data in situ, the bulk of this common suppression cancels in the excess-over-anchor observable; the residual shape distortion belongs, with the NLO interference calculation, to the theory tasks of Appendix~O.  The full numerical workbook behind this Part (parton-luminosity grids, the anchored background fit, the calibration injections, and every table entry) is archived with the cohort's build materials at \url{https://arisaka-gg.org/} and is reproducible from the quoted inputs.


Part~VIII turned the prediction into dates, and its plain-English summary is startling enough to bear repeating: the statistical power to see this particle may already exist, unread, in the data recorded since 2015 --- evidence-level for either experiment alone, possibly discovery-level for the two combined --- and the obstacle is not luminosity but background knowledge plus a template no one has yet fitted.  The systematics wall, the calibrated ladder from naive counting to honest significance, and the pre-registration dividends were each located and priced.  The story to 2041 is thereby fixed.  What of the decades after?  The next Part maps the machines beyond the HL-LHC --- what each can measure, what each would decide, and which questions must wait for which ring.


\section[Part~VIII in plain English]{Part~VIII in plain English: what the detectors would actually record, worked out at analysis fidelity}
The claim that gives this Part its point is that the collisions which
decide the question may already be on tape.

To make that credible the Part does not estimate; it simulates, anchored to
a published ATLAS top-pair search so that every step can be checked against
something real. Backgrounds, selection, binning, and the per-bin
background uncertainty all come from the anchor. The arithmetic is then
shown bin by bin for one dataset so a reader can extend the same table to
any other.

Two scenarios run throughout, differing in exactly one input: whether the
background systematic stays at its published value or is halved. Both are
carried to the end. A reader who thinks the improvement optimistic can read
the frozen column and still have a complete forecast, which is the reason
for quoting two rather than one.

The Part also spends a section on why the eye overestimates a bump, and
that section is there for the paper's own protection. A spectrum that looks
convincing at a glance may carry a modest significance once the systematic
is folded in, and a paper forecasting a discovery has an obligation to say
so before its own plots are read.

The output is a set of dates with numbers attached, pre-registered: what
the existing dataset can reach, and what the high-luminosity run adds. If
the cluster is there, it is findable in this decade. If it is not, the same
schedule says when that becomes certain.

\GGPart[Beyond the HL-LHC]{Beyond the HL-LHC}{The 2035--2070 program: facility statuses, the post-discovery checklist, and machine-by-machine falsification scenarios}
\label{part:beyond}

\section{The Question This Part Answers}
\label{sec:beyond}

Part~\ref{part:hllhcsim} ends with the HL-LHC deciding F-KK-2B: on the simulated calendar, discovery-grade sensitivity is reached between roughly 2032 and 2035 (combined, improved systematics), and the frozen-systematics exclusion closes the window not much later if nothing is there.  Either way, by \(\sim\!2035\) the yes/no question of this paper is answered --- and an entirely different program begins.  If the cluster is there, the remaining task is \emph{spectroscopy}: measuring everything the discovery machine cannot, from the direct spin-2 anchor to the second radial family to the exact zero of the leptophobic branching ratio.  If it is not, the same future machines become falsification engines of increasing severity for whatever residue of the framework survives.  This Part maps that post-2035 program facility by facility and decade by decade, on the \emph{actual} status of the world's collider projects as of July 2026 --- not the aspirational landscape of a decade ago --- and it works out, with the same machinery used throughout this paper, what each machine can measure, including the subdominant channels that only become relevant once the discovery is in hand.

Two framing rules.  (i)~Facility statuses and dates below are taken from the primary strategy documents of 2025--2026 --- the FCC Feasibility Study Report \citep{FCC2025FSR}, the 2026 Update of the European Strategy for Particle Physics \citep{ESPP2026Adoption}, the CEPC Technical Design Reports \citep{CEPC2025TDR}, and the International Muon Collider Collaboration's design reports \citep{IMCC2024Interim} --- and every schedule is quoted as those documents state it, not as optimism would prefer it.  (ii)~Rates at future machines are computed with this paper's own parton-luminosity machinery where the couplings are derived (the octet and the spin-2 anchor), and clearly labeled as estimates where they are not (the \(n=2\) radial family, whose overlap integrals are not derived here).

\section{The Facility Landscape as It Actually Stands}
\label{sec:beyond-landscape}

The single most important fact about the post-LHC landscape is easily stated: \emph{as of July 2026, exactly one collider beyond LHC Run 3 is approved, funded, and under construction --- the HL-LHC.}  Everything else is a recommendation, a design report, or an R\&D program.  This is why Parts~VII--VIII of this paper center the HL-LHC so insistently: it is not merely the sharpest test available, it is the only one guaranteed to happen.  Table~\ref{tab:facilitystatus} summarizes the landscape; the prose below records the decisive facts per facility.

\begin{table}[!t]
\centering\small
\setlength{\tabcolsep}{3.5pt}
\caption[Status of future collider facilities, July 2026]{The future-collider landscape as of July 2026, from the primary strategy documents \citep{ESPP2026Adoption,FCC2025FSR,CEPC2025TDR,IMCC2024Interim}.  ``Earliest physics'' quotes the reference schedules of those documents.  Only the HL-LHC is approved and under construction.}
\label{tab:facilitystatus}
\begin{tabular}{>{\raggedright\arraybackslash}p{2.13cm} >{\raggedright\arraybackslash}p{2.23cm} >{\raggedright\arraybackslash}p{4.67cm} >{\raggedright\arraybackslash}p{2.03cm} >{\raggedright\arraybackslash}p{3.35cm}}
\toprule
\textbf{Facility} & \textbf{Type / energy} & \textbf{Status (July 2026)} & \textbf{Earliest physics} & \textbf{Next decision gate}\\
\midrule
HL-LHC & \(pp\), 14 TeV & \textbf{Approved, funded, under construction}; Long Shutdown 3 under way & \(\sim\)2030 & none --- proceeding\\
FCC-ee & \(e^+e^-\), 88--365 GeV, 91-km ring & \textbf{Recommended}: ESPP 2026 Update, adopted unanimously by the CERN Council (Budapest, 22 May 2026), names FCC-ee the preferred next flagship; \(\sim\!15.3\)~BCHF; four interaction points; not yet approved or funded & mid-2040s & CERN Council decision by 2028, informed by funding plan and host-state consultations\\
FCC-hh & \(pp\), \textbf{84--85 TeV}, same tunnel & Concept within the FCC integrated program; FSR baseline 14-T Nb\(_3\)Sn dipoles; HTS options studied & early-to-mid 2070s (FSR reference); earlier LTS variants appear in FSR option studies & follows the FCC-ee decision and the high-field magnet program\\
CEPC & \(e^+e^-\) Higgs/Z factory, 100-km ring & Technical Design Reports complete (October 2025); \textbf{not selected} for China's 15th Five-Year Plan (March 2026); resubmission intended \(\sim\)2030, or merger of effort into FCC if approved & mid-2040s at the earliest, contingent on 2030 approval & Chinese national planning, \(\sim\)2030\\
Muon collider & \(\mu^+\mu^-\), staged 3 \(\to\) 10 TeV & International design study (IMCC); ESPP 2026 names muon-collider R\&D a priority; ionization-cooling demonstrator planned for the 2030s & stage-1 (\(\sim\!3\) TeV) \(\sim\)2050; 10 TeV \(\sim\)2070 & demonstrator results, mid-2030s\\
ILC & \(e^+e^-\), 250 GeV linear & Design mature (TDR 2013); Japanese government non-committal (February 2026); international technology network (ITN) R\&D running 2026--28 & undetermined & Japanese ministerial roadmap decisions\\
CLIC & \(e^+e^-\), 0.38 \(\to\) 3 TeV linear & Mature design; \textbf{not selected} as the preferred ESPP option; maintained as an alternative & undetermined & contingent on FCC-ee outcome\\
C\(^3\) / wakefield (HALHF etc.) & compact \(e^+e^-\) & R\&D concepts; encouraged by the U.S. P5 report and ESPP inputs & 2040s demonstrations at the earliest & technology milestones\\
LHeC & \(ep\), \(\sim\!1.2\) TeV & Concept reusing the HL-LHC beam; positioned as a bridge project & 2040s & CERN program decisions\\
\bottomrule
\end{tabular}
\end{table}

Five facts from this table shape everything below.  \emph{First}, Europe has chosen its path: the ESPP 2026 Update, adopted unanimously in May 2026, names FCC-ee the preferred flagship, with a Council decision by 2028 \citep{ESPP2026Adoption}.  The realistic default future is therefore FCC-ee physics in the mid-2040s followed by FCC-hh in the 2070s.  \emph{Second}, the FCC-hh baseline energy in the Feasibility Study Report is \(84\)--\(85\)~TeV, not the 100~TeV of the older literature \citep{FCC2025FSR}; every 100-TeV number in the field --- including the ones quoted earlier in this paper --- must be rescaled by roughly the parton-luminosity factor \(\sim\!0.7\) at our masses, which changes no conclusion but should be stated plainly.  \emph{Third}, the Chinese alternative has receded: the CEPC's technical design is complete and excellent, but the project was not selected for the 15th Five-Year Plan and will not be reconsidered before \(\sim\!2030\) \citep{CEPC2025TDR}.  \emph{Fourth}, the muon collider is real R\&D with an honest schedule: a 10-TeV machine is a \(\sim\!2070\) prospect, with a possible \(\sim\!3\)-TeV first stage around 2050 \citep{IMCC2024Interim}.  \emph{Fifth}, the linear-collider options (ILC, CLIC, C\(^3\), wakefield hybrids) survive as designs and R\&D but none currently has a committed host or funding line.  The planning horizon of this Part is therefore built on three anchors: HL-LHC (certain, 2030--2042), FCC-ee (probable, mid-2040s onward), FCC-hh (the reference energy-frontier machine, 2070s) --- with the muon collider as the wild card that could compress the third anchor's timeline.

\section{The Post-Discovery Checklist: What Is Left After 2035}
\label{sec:beyond-checklist}

Assume now the discovery future: the octet is found at \(3.85\)~TeV, per Part~\ref{part:hllhcsim}, between 2032 and 2035.  By end-2041 the HL-LHC will have delivered the full property program of Table~\ref{tab:sim-cases}: the mass to better than a percent, the width at the \(10\%\)-resolution ceiling, spin-1 versus spin-2 discrimination at \(17\sigma\), the top-coupling chirality \(P_t\) to \(\pm0.08\) (resolving the interior of the derived band \([+0.88,+0.92]\)), and the interference sign through the dedicated template fit.  That is a remarkable harvest --- and it leaves exactly six items open, each of which is precisely one future machine's job.

\begin{enumerate}
\item \textbf{The direct spin-2 anchor.}  \(\sigma\times\mathrm{BR}(G_1\to\ttbar)=0.045\)~fb at 13~TeV at the derived \(\kappa_{gg}\) (Corollary T-KK-2c): even \(3\)~ab\(^{-1}\) of HL-LHC yields a handful of reconstructed events under a \(10^3\)-fold background.  The state that \emph{names the geometry} is invisible at the machine that finds the cluster.
\item \textbf{Resolving the degenerate spin-1 pair.}  \(g^*\) and \(Z'_B\) sit at the same mass by theorem (same boundary-value problem); they are separated not by peak position but by color and channel decomposition --- the octet-to-singlet ratio in dijets versus \(\ttbar\), color-flow observables, and the small \(Z'_B\) fraction (0.2--6.8\% of the cluster \(\ttbar\) sum across the \(g_B\) menu).  HL-LHC statistics and jet systematics will not cleanly unfold this.
\item \textbf{The exact zero of leptophobia.}  \(\mathrm{BR}(Z'_B\to\ell^+\ell^-)=0\) \emph{exactly} is a theorem-grade prediction; the HL-LHC can bound the cluster's dilepton fraction only at the \(10^{-2}\)--\(10^{-3}\) level relative to \(\ttbar\).  An exact zero deserves a much harder test.
\item \textbf{The second radial family at \(7.05\)~TeV.}  The tower ratio \(x_2/x_1=1.831\) puts the next spin-2 mode at \(7.05\)~TeV; at 14~TeV the \(gg\) luminosity there is a factor \(\sim\!800\) below the first mode's ---  \(\sigma(G_2)\sim10^{-4}\)~fb: hopeless.  Yet this single measurement --- \emph{is the second family exactly where the Bessel spectrum of the RG throat demands?} --- is the cleanest conceivable test of the non-spatial geometry, worth more than any single property of the first cluster.
\item \textbf{Sub-resolution line-shape precision.}  The interference zero-crossing at \(\simeq4.9\)~TeV, the \(-\mathcal O(100)\)~GeV centroid pull, and the width itself are all resolution-limited at hadron machines with \(10\%\) mass resolution; percent-level line-shape work needs either enormous statistics (FCC-hh) or a lepton-machine scan.
\item \textbf{The scalar-sector null.}  \GG{} predicts \emph{no radion} --- the radial length is the fixed integer \(K\), not a modulus \citep{ArisakaA4Euclid} --- so the Higgs sector must be exactly Standard-Model-like in its couplings apart from the derived \(G_1\to hh\) line.  The decisive test of a null is precision, and the precision lives at Higgs factories.
\end{enumerate}

\section{Machine by Machine: What Each Facility Decides}
\label{sec:beyond-machines}

\subsection{FCC-hh at 84--85 TeV: the spectroscopy machine}
\label{sec:beyond-fcchh}

At the FSR baseline energy the parton luminosities at our masses grow spectacularly: relative to 13~TeV, the \(q\bar q\) luminosity at \(3.85\)~TeV rises by a factor \(\simeq330\) and the \(gg\) luminosity by \(\simeq3.2\times10^{3}\) (computed with this paper's CTEQ6 machinery; low-\(x\) PDF uncertainties of a few tens of percent apply and are immaterial to any conclusion).  The consequences, state by state:

\paragraph{The octet becomes a precision laboratory.}  \(\sigma\times\mathrm{BR}(g^*\to\ttbar)\approx2.8\)~pb \([2.3\text{--}4.0]\) at 85~TeV.  At the FSR reference luminosity (\(\sim\!20\)~ab\(^{-1}\) per experiment over the program) that is \(\sim\!5\times10^{7}\) signal events before acceptance: the line shape, the interference pattern including its zero-crossing near \(4.9\)~TeV, the width, and the angular distributions all become systematics- and theory-limited measurements rather than statistics-limited ones.  The NLO-interference calculation requested in \S\ref{sec:sim-contamination} graduates from desirable to mandatory.

\paragraph{The spin-2 anchor falls in the first weeks.}  Scaling the derived 13-TeV rate by the \(gg\)-luminosity ratio: \(\sigma(pp\to G_1)\approx0.13\)~pb at 85~TeV (the historical 100-TeV reference value \(\approx205\)~fb used earlier in this paper rescales to \(\approx0.7\) of itself at the FSR baseline; the early-running-discovery conclusion is untouched, and we quote both to avoid any appearance of a moving target).  With \(\Gamma/m=6.3\%\) --- comfortably narrower than the octet --- and the derived branching pattern \(t\bar t:W_LW_L:Z_LZ_L:hh\simeq0.69{:}0.15{:}0.08{:}0.08\), the anchor delivers four correlated resonance lines at one mass.  The longitudinal-diboson dominance among the bosonic channels (\(W_LW_L\) at \(15\%\)) is itself diagnostic: it is the signature of a graviton coupling to the infrared-localized electroweak sector, and no spin-1 impostor reproduces it together with the \(\ttbar\) angular form.
\paragraph{The second radial family is decided.}  At 85~TeV the \(gg\) luminosity at \(7.05\)~TeV is \(7.8\%\) of its value at \(3.85\)~TeV, so a same-coupling estimate gives \(\sigma(G_2)\sim10\)~fb --- of order \(10^{5}\) events: an unmistakable second line.  (We label this an estimate, not a derivation: the \(n=2\) overlap integrals are not computed in this paper.  The \emph{position}, however, is a theorem-grade ratio: \(x_2/x_1=7.0156/3.8317=1.831\), i.e.\ \(7.05\)~TeV, testable at FCC-hh to per-mille precision.)  Vector recurrences accompany it at \(\mathcal O(10^{2})\)~fb in \(\ttbar\) if the composite spectrum spaces them near the radial families.  Measuring \emph{two} (ideally three: \(n=3\) at \(10.2\)~TeV, still within a machine whose \(s\)-channel reach extends to \(\sim\!40\)~TeV) radial masses turns the single number \(3.849\)~TeV into a measured \emph{spectrum} --- the point at which one is no longer testing a prediction but reading the geometry.
\paragraph{The pair is unfolded and leptophobia is pushed to \(10^{-6}\).}  With \(10^{7}\)-event statistics, the octet/singlet decomposition (item 2 of the checklist) proceeds through dijet-to-\(\ttbar\) ratios and color-flow observables, and the dilepton null at the cluster mass is probed to branching fractions of order \(10^{-6}\) relative to the hadronic channels --- a serious test of an exact zero.

\subsection{FCC-ee: the parameter-ledger machine, and the no-radion test}
\label{sec:beyond-fccee}

FCC-ee never touches the cluster directly: its top energy (\(365\)~GeV) is an order of magnitude below \(\Mlock\), and because the cluster is leptophobic and lepton-poor, even its indirect electroweak footprint is loop-suppressed rather than tree-level.  It would be easy to conclude that the machine most likely to be built next is irrelevant to \GG{} --- and wrong.  FCC-ee's role in this program is different in kind: it is the precision arm of the \emph{cohort ledger} that this paper's mass sits inside.  Three examples define the class.  (i)~The \GG{} structural value of the strong coupling is \(\alpha_s(M_Z)=1/(\pi e)=0.11710\) \citep{ArisakaA9SM}; the current world average, \(0.1180\pm0.0009\) \citep{PDG2024}, sits \(\sim\!1\sigma\) above it.  FCC-ee's projected determination at the few-\(10^{-4}\) level converts this from a tension-free coexistence into a decisive test --- arguably the sharpest single number the machine will produce for this framework.  (ii)~The top-quark mass from the threshold scan (\(\pm\mathcal O(20)\)~MeV) and the \(Z\)-pole program sharpen every top-overlap quantity in this paper --- the band \(r_{t_R}\in[4,5]\) that sets the octet's width, branching ratios, and polarization.  (iii)~The Higgs-coupling program at the per-mille-to-percent level executes the \emph{no-radion} test (checklist item 6): \GG{} requires the observed Higgs to be unmixed with any radial scalar, because no such scalar exists; a coupling pattern showing radion-like mixing would falsify the scalar sector outright, discovery or no discovery.  A machine that cannot produce a single cluster particle can therefore still falsify the framework --- which is exactly the property one wants from a complementary program.  (The CEPC, if revived in 2030, carries the identical program; we do not treat it separately.)

\subsection{A muon collider: the leptophobia machine}
\label{sec:beyond-muc}

The muon collider inverts the logic of every other entry: its decisive \GG{} measurement is a \emph{null}.  The cluster couples to quarks and gluons and shuns leptons --- \(Z'_B\) exactly (theorem), \(G_1\) strongly (UV-localized lepton overlaps).  A generic \(3.85\)-TeV \(Z'\) or KK gluon with electroweak-strength lepton couplings would announce itself in \(\mu^+\mu^-\to f\bar f\) contact-interaction tails even at a \(3\)-TeV stage-1 machine, and would dominate an \(s\)-channel scan at a 10-TeV machine able to sit on \(\sqrt s=3.849\)~TeV.  \GG{} predicts the opposite, and sharply: \emph{the cluster that roars in \(\ttbar\) at hadron colliders is nearly silent in the \(s\)-channel of a lepton collider} --- no \(Z'_B\) line at all, a \(G_1\) line suppressed by two powers of the UV lepton overlap, and Standard-Model-like contact-interaction observables at stage-1.  If the HL-LHC has found the cluster, this null is as decisive a fingerprint as any positive signal: observing a leptonic \(s\)-channel line at \(3.85\)~TeV would falsify the \(\mathrm{U}(1)_B\) assignment and the localization geometry in one measurement.  What the machine does measure directly: \(G_1\) production through \(W\)-fusion and \(\gamma\gamma\) fusion (fed by the \(15\%+8\%\) longitudinal-diboson couplings), at femtobarn-class rates at \(10\)~TeV --- a clean, low-background environment for the bosonic side of the branching pattern, complementary to FCC-hh's hadronic side.  On the IMCC schedule these tests sit at \(\sim\!2050\) (stage-1 nulls) and \(\sim\!2070\) (10-TeV program) \citep{IMCC2024Interim}.

\subsection{Linear colliders, the LHeC, and the supporting cast}
\label{sec:beyond-linear}

ILC- or CLIC-class machines carry the same ledger-and-null program as FCC-ee (Higgs precision, top precision, contact-interaction nulls), with CLIC's 3-TeV stage adding modest direct sensitivity to the cluster's virtual exchange in \(e^+e^-\to t\bar t\) --- again predicted \emph{absent} at leading order by leptophobia.  Their current status (Table~\ref{tab:facilitystatus}) makes them contingency options rather than planning anchors.  The LHeC deserves one specific remark: its deep-inelastic program would pin the high-\(x\) quark and gluon densities that dominate the uncertainty bands of every rate in this paper --- the octet's \([7.0\text{--}12]\)~fb band at 13~TeV is PDF-limited --- so a single \(ep\) machine would quietly convert the \GG{} rate test from factor-level to percent-level without ever producing a cluster particle.

\section{Concrete Scenarios, 2035--2070}
\label{sec:beyond-scenarios}

\paragraph{Scenario D (discovery, the Part-\ref{part:hllhcsim} calendar).}
\emph{2032--2035}: combined ATLAS+CMS dedicated analysis crosses \(5\sigma\); the discovery paper's mass is \(3.85\pm0.04\)~TeV and the width is broad; the world notices.  \emph{2035--2041}: the HL-LHC property program saturates (spin \(17\sigma\), \(P_t\pm0.08\), interference template); the theory side delivers the NLO interference calculation; the \(hh\)/diboson subdominant searches at \(3.85\)~TeV sharpen from curiosities into targeted corroborators.  \emph{2041--2045}: LHC legacy combinations; FCC-ee construction proceeds on the post-2028 approval; the discovery reshapes the FCC-hh case --- a confirmed 3.85-TeV cluster with a predicted second family at 7.05~TeV is precisely the ``guaranteed-physics'' argument that historically accelerates energy-frontier machines, and the FSR's earlier-LTS-magnet options give the strategy bodies a concrete lever.  \emph{Mid-2040s--2050s}: FCC-ee runs; \(\alpha_s(M_Z)\) versus \(1/(\pi e)\) is decided; the no-radion Higgs-precision test is executed; the muon-collider demonstrator reports and, in the favorable branch, a stage-1 machine near 2050 delivers the leptophobia null.  \emph{2070s}: FCC-hh (or a 10-TeV muon collider, whichever materializes) observes the spin-2 anchor in its first weeks, resolves the four-line branching pattern, unfolds the degenerate pair, pushes leptophobia to \(10^{-6}\), and --- the endpoint of this entire program --- measures the second and third radial families at \(7.05\) and \(10.2\)~TeV, reading the Bessel spectrum of the non-spatial throat directly.  One should be honest about the gap this timeline contains: on current schedules the \emph{direct} spin-2 confirmation arrives roughly four decades after the octet discovery.  That gap is not a physics necessity but a programmatic choice, and a 2030s discovery would be the strongest imaginable argument for compressing it.

\paragraph{Scenario X (exclusion).}
If instead the combined HL-LHC program excludes the octet band --- decided, per Part~\ref{part:hllhcsim}, at \(\lesssim\!2\)~ab\(^{-1}\), i.e.\ by the mid-2030s --- the collider implementation of \GG{} is retired, and this paper's interest in the post-LHC machines reduces to the discipline it has kept throughout: there is no parameter to slide, so there is no zombie phase in which the same prediction reappears at \(6\)~TeV.  The framework's remaining exposure then lives entirely in the cohort's non-collider arms (the fifth-force window and the \(8.1\,\mu\)eV dark-matter line of the companion papers), and the future colliders inherit only the generic duty of exploration.  We state this plainly because the asymmetry is the point: the machines of Table~\ref{tab:facilitystatus} are worth building in either future, but only in Scenario~D do they inherit a fully specified, decade-by-decade program with derived rates waiting for them.

\section{Summary Matrix}
\label{sec:beyond-matrix}

Table~\ref{tab:beyondmatrix} condenses this Part.  The taxonomy is worth keeping in mind: the HL-LHC is the \emph{verdict machine} --- the yes/no lives there, and nowhere else, before 2045.  FCC-hh is the \emph{spectroscopy machine}: the anchor, the pair, the radial tower.  The muon collider is the \emph{leptophobia machine}: its decisive contribution is a predicted silence.  FCC-ee and the linear colliders are the \emph{ledger machines}: they test the same integer bookkeeping from the precision side, and carry the no-radion null.  And the LHeC is the \emph{sharpening machine}: it buys precision on everyone else's predictions.  The sharpest yes/no test lives at the LHC first; everything else in this Part is what victory would purchase.

\begin{table}[!t]
\centering\small
\setlength{\tabcolsep}{3pt}
\caption[GG falsification program at future facilities]{The \GG{} program beyond the HL-LHC, assuming discovery by \(\sim\!2035\) (Scenario D).  Rates with derived couplings are computed with this paper's machinery; the \(G_2\) entry is a same-coupling estimate (position, however, is theorem-grade). The table is written under an assumption stated in its own first line, which is that the discovery happens. Its purpose is not to argue that it will but to show that the program has somewhere to go if it does, and that the next targets are already positioned rather than to be chosen later. The one same-coupling estimate is marked as such.}
\label{tab:beyondmatrix}
\begin{tabular}{>{\raggedright\arraybackslash}p{1.94cm} >{\raggedright\arraybackslash}p{2.45cm} >{\raggedright\arraybackslash}p{3.68cm} >{\raggedright\arraybackslash}p{3.27cm} >{\raggedright\arraybackslash}p{3.07cm}}
\toprule
\textbf{Facility} & \textbf{Earliest \GG{} physics} & \textbf{Direct reach into \GG{} targets} & \textbf{Decisive deliverable} & \textbf{Key subdominant channels}\\
\midrule
HL-LHC & now--2041 & octet 8.7\,\(\to\)\,13.6 fb (13\,\(\to\)\,14 TeV); anchor and \(n{=}2\) family out of reach & the verdict (F-KK-2B); spin, chirality, interference sign & \(hh\), diboson corroborators at \(3.85\) TeV; dijet angularity\\
FCC-ee & mid-2040s & none (365 GeV \(\ll\Mlock\); leptophobic cluster) & \(\alpha_s(M_Z)\) vs \(1/(\pi e)\) at \(4\sigma\)-class; no-radion Higgs precision & \(Z\)-pole ledger; \(m_t\) to \(\pm20\) MeV\\
Muon collider & \(\sim\)2050 (3 TeV); \(\sim\)2070 (10 TeV) & \(s\)-channel scan of 3.85 TeV possible at 10 TeV & the leptophobia null: no \(Z'_B\) line, suppressed \(G_1\), SM-like contact terms & \(W\)-fusion/\(\gamma\gamma\)-fusion \(G_1\) (fb-class); bosonic BR pattern\\
FCC-hh (84--85 TeV) & 2070s & octet \(\approx2.8\) pb; \(G_1\approx0.13\) pb; \(G_2\sim10\) fb (est.); reach to \(\sim\!40\) TeV & spin-2 anchor + four-line BR pattern; radial tower at 7.05/10.2 TeV; pair unfolding; leptophobia to \(10^{-6}\) & \(W_LW_L\), \(Z_LZ_L\), \(hh\) lines; vector recurrences \(\mathcal O(10^2)\) fb\\
LHeC / linear options & 2040s+ (contingent) & none direct & PDF precision (percent-grade rate tests); ledger + contact-interaction nulls & high-\(x\) \(g,q\) densities; \(e^+e^-\to t\bar t\) tails\\
\bottomrule
\end{tabular}
\end{table}

Part~IX mapped the far future onto the facilities that actually exist on paper, in their actual states of approval: the verdict belongs to the HL-LHC alone; the spectroscopy --- the anchor, the overtones at their Bessel ratios, leptophobia at the part-per-million level --- belongs to FCC-hh in the 2070s; the precision ledger belongs to FCC-ee; and the muon collider contributes the strangest measurement of all, a predicted silence.  Every derivation, simulation, and schedule the paper owes is now delivered.  What remains is judgment: what all of this \emph{means}, how it compares to the roads not taken, and what would count as being wrong.  The next Part steps back and argues.


\section[Part~IX in plain English]{Part~IX in plain English: what happens after 2035, on either answer}
This Part is written for the world in which the previous one succeeds,
and it starts from an unglamorous fact: as of now exactly one collider
project beyond the high-luminosity upgrade is actually approved.

That fact governs everything else. The Part surveys the real status of each
proposed machine from the primary strategy documents rather than from
ambition, and it is explicit about which entries are decisions already
taken and which are proposals awaiting one. A roadmap built on the
aspirational landscape of a decade ago would be a different and much more
comfortable document.

Given the real machines, the Part works out what each would decide. A
discovery in the top channel would fix a mass and little else; the
follow-up questions --- the spin of the anchor, the couplings of the
partners, the position of the next recurrence --- each need a specific
capability, and the Part matches question to machine.

There is a version of this Part that would have read as advocacy for a
particular facility. This one tries not to be: the checklist is written so
that a reader can see which questions would go unanswered under each
funding outcome, including the outcome where nothing after the
high-luminosity run is built at all.

\GGPart[Discussions]{Discussions}{Significance and uniqueness of the result; the classic collider puzzles addressed; comparison with other theories and models; the structural innovations; predictions and falsification; open questions; and the completed cohort}
\label{part:discussions}

The derivations are finished; this Part steps back and argues.  Five discussions follow --- the significance of the result; its uniqueness, nobility, and innovations; the comparison with the other roads beyond the Standard Model; the predictions for falsification; and the open questions --- closed by the paper's place in the completed cohort and a summary.  The reference tables that condense the whole collider sector into three pages (the master ledger, the falsification hierarchy, and the discovery calendar) live with the other lookup material in Appendix~D, where a reader can consult them without the argument, and the argument can proceed without the furniture.


\section{Significance of the result}
\label{sec:disc-significance}

What this paper delivers, reduced to its irreducible content, is a single sentence: \emph{a falsifiable, zero-parameter collider target --- one mass, one width, one rate, one interference sign, one angular shape, one branching fingerprint --- with a dated calendar for its decision.}  Each clause of that sentence is individually rare in the beyond-Standard-Model literature; their conjunction is, to our knowledge, unprecedented.  This section unpacks why, deliberately slowly, because the significance of a result of this shape is easy to misjudge in both directions --- easy to inflate, if one forgets how much of it is conditional on upstream theorems, and easy to deflate, if one has been trained by fifty years of adjustable predictions to assume that every collider target is secretly a scan.

\paragraph{The result in the fifty-year arc.}  Place the claim against the decades traversed in \S1.  In the 1980s, the community searched for proton decay at the rates grand unification suggested; the prediction had order-of-magnitude freedom, and when Kamiokande and IMB saw nothing, the models moved.  In the 1990s and 2000s, supersymmetry's ``natural'' window slid from LEP to the Tevatron to the LHC, a moving target by construction, because more than one hundred continuous parameters can absorb any null.  In the 2010s, after the Higgs, the extra-dimensional and composite programs offered benchmarks --- honest, useful benchmarks --- whose masses and couplings were nevertheless inputs, so that exclusion plots grew steadily without any exclusion being fatal.  Against that history, the present claim is structurally different in kind, not degree: the mass \(3.849\)~TeV cannot move (an integer has no neighborhood); the rate \(8.7\)~fb cannot be lowered (the coupling \(1/\sqrt K\) is the same object that fixes the mass); the width, sign, polarization, and branching pattern are outputs of the same three theorems; and the paper has pre-committed, in writing, to the luminosity at which a null retires the construction.  Whether or not nature cooperates, this is what the community spent five decades asking theory to produce.

\paragraph{What ``zero parameters'' buys, epistemically.}  The absence of continuous freedom is not an aesthetic garnish; it changes what an experimental outcome \emph{means}.  With a dial, a positive result confirms little (the dial was set after the fact) and a negative result kills nothing (the dial turns).  Without one, both outcomes become maximally informative: a discovery at \(3.849\pm0.10\)~TeV with the correlated pattern would confirm a chain that reaches from collider phenomenology back through the electroweak ledger to the axioms, and a clean exclusion falsifies the same chain outright --- there is no fallback version of this paper at 6~TeV.  The statistics inherit the same virtue in a precisely quantifiable way (\S\ref{sec:sim-prereg}): no look-elsewhere discount (worth \(0.8\sigma\), equivalently a doubling of dataset near the systematics plateau) and no template-mismatch dilution (a factor \(\sim\!3\) in rate).  Pre-registration is usually discussed as a virtue of experimental design; here it is a consequence of the theory's structure, and it converts directly into sensitivity.

\paragraph{The over-constraint, told in full.}  A skeptic granting everything above could still ask: why believe the chain \emph{before} the collider verdict?  The paper's strongest answer predates any collider data: the same integer that fixes \(\Mlock\) is spent a second time on a number measured to four digits decades ago.  The chain runs \(K=35\Rightarrow\Mlock=\tfrac12M_{\rm Pl}e^{-35}=3.849\)~TeV \(\Rightarrow v=\Mlock(2/5)^3=246.3\)~GeV, and the second arrow is not a separate assumption but the Wilson-line readout of \citet{ArisakaA9SM}.  One integer, two scales, one of them already confirmed: the construction walks into the collider test having already survived a four-digit audit.  This is what distinguishes the present wager from a lottery ticket --- the ticket has already matched half the draw --- and it is also what raises the stakes, because a collider null does not merely eliminate a candidate resonance; it breaks a chain that presently accounts for the electroweak scale.

\paragraph{The calendar as part of the result.}  Part~\ref{part:hllhcsim} is not an appendix to the claim; it is half of it.  A falsifiable statement without a decision procedure is a promissory note, and the simulation converts the note to a schedule: expected evidence-level sensitivity (\(2.7\)--\(3.7\sigma\) per experiment) already resident in the recorded Run-2+3 data; combined ATLAS+CMS sensitivity of \(4.6\)--\(5.2\sigma\) under realistically improved systematics --- five-sigma from tape a live possibility; single-experiment five-sigma luminosity on tape during 2032; the frozen-systematics exclusion closing the window by the mid-2030s if nothing is there.  The dates matter beyond logistics: they place the verdict within the professional lifetime of the paper's addressed reader (the Preface's graduate student) and within the strategic planning horizon of the laboratories now deciding the post-LHC program (Part~\ref{part:beyond}).  A prediction whose test is generations away is philosophy with equations; this one is a thesis proposal.

\paragraph{Significance for each community, separately.}  For the \emph{experimental collaborations}, the significance is operational: a fully specified template, a priced reanalysis of data already on tape, and a deliverable that is valuable under every outcome --- the world's strongest limit on any broad top-philic resonance if the answer is no.  For \emph{collider phenomenology}, the significance is methodological: an existence proof that a derived rate, a derived sign, and a derived angular shape can be produced from a geometric construction with no scan, and a worked demonstration (Parts~\ref{part:ttanatomy}--\ref{part:hllhcsim}) of what the corresponding analysis chain looks like end to end.  For \emph{the theory community}, the significance is the inversion described in \S\ref{subsec:notKKRS}: the first framework to field both of Weyl's geometrizations --- scale and phase --- as forced, non-spatial dimensions, with the moduli, landscape, and supersymmetry obligations dissolved as consequences rather than assumptions.  And for \emph{strategic planning}, the significance is conditional but sharp: a 2030s discovery at \(3.849\)~TeV, with a predicted second family at \(7.047\)~TeV, is precisely the ``guaranteed-physics'' input that historically accelerates energy-frontier machines, and Part~\ref{part:beyond} maps the resulting program to 2070.

\paragraph{The two-scenario resolution as significance of a second kind.}  One episode inside the paper deserves separate billing, because it demonstrates the framework's epistemic behavior under fire.  When the free-field placement of the spin-1 pair at \(2.461\)~TeV was confronted with existing data, the derived rate (224~fb) sat a factor 3.6--11 \emph{above} the published exclusion: Scenario~A died (\S\ref{sec:operational}).  The framework did not slide; it \emph{learned} --- the throat vectors are composite, not free-field --- and the surviving claim became sharper, not vaguer.  A theory whose exclusions teach it structural facts, rather than merely relocating its expectations, is operating in a different epistemic regime from the parameter scans of \S1, and the episode is now part of the paper's evidence about itself: the falsification machinery has been observed to function, once, in the kill direction.

\paragraph{The discovery, read on the anatomy: one resonance, two fibers.}
A last register of significance has been made exact by the cohort's
recent campaign (\textsf{GG-W1-Symmetry}; \citealp{ArisakaA2GG6};
\citealp{ArisakaP5CPT}). The two non-spatial directions are fibers
of opposite topological kind --- \emph{holonomy requires closed
loops, and an interval has none}: the compact circle $S^1_\theta$
carries all of the bulk's holonomy --- quantized, protected, a
group --- while the scale interval $I_r$ carries none --- one-way
transport, a semigroup. The lock cluster is these two fibers in a
single measurement. Everything \emph{quantized} about the target is
the circle's signature: the tower's existence, the winding origin
of the charges, the spin content, the branching fingerprint fixed
by ledger integers --- holonomy-class data on the fiber that
remembers, which is why none of it is adjustable. And everything
about the target's \emph{location} is the interval's signature: the
depth $\Mlock = \tfrac12 M_{\mathrm{Pl}}\, e^{-K}$ is thirty-five
e-folds of one-way descent, the address the fiber that forgets
assigns to the first rung. A discovery at $3.849$~TeV would
therefore be a simultaneous reading of both fibers --- the
resonance's \emph{identity} written on the circle, its
\emph{address} written by the interval --- and, in the campaign's
terms, the first laboratory instrument ever to read a non-spatial
fiber directly: the circle made audible, at the energy the interval
sets.

\section{Uniqueness, nobility, and innovation}
\label{sec:disc-unique}

Three words that a physics paper should use carefully, so we define them operationally before earning them.  \emph{Uniqueness}: what this paper does that no neighboring framework does, stated so that a referee can check the negative.  \emph{Nobility}: the epistemic virtues --- pre-commitment, honest grading, the refusal to retreat --- that make the result trustworthy independent of whether it is true.  \emph{Innovation}: the specific technical moves, registered as I-KK-1 through I-KK-10, that did not exist before this paper.  The three are related but not the same, and conflating them is how strong claims come to sound like salesmanship.

\subsection{The classic collider puzzles, resolved or dissolved}
\label{sec:disc-puzzles}

Each entry below states a puzzle the field has carried for decades, in the form a practitioner would recognize, and then what this paper does to it.  ``Resolved'' means an answer is derived; ``dissolved'' means the question is shown to have been wrongly posed.

\paragraph{PS-KK-1: Why has no new particle appeared in fifty years of searching?}  The puzzle, stated honestly, has haunted every strategy report since LEP: if the TeV scale hides new physics, why have the cleanest channels --- dileptons, diphotons, the golden signatures that found the \(Z\) and the Higgs --- stayed empty through three orders of magnitude in luminosity?  The conventional answers are either despairing (there is nothing) or evasive (it is heavier).  This paper's answer is neither: the derived coupling texture --- UV-localized light fermions at \(\xi_{\rm UV}=-0.194\), a flat photon, an IR-localized top at \(r_{t_R}\approx4.5\) --- makes the cluster \emph{loud precisely where searching is hardest} (broad, interfering, \(\ttbar\)) and \emph{silent precisely where searching is easiest}.  On this reading the fifty-year silence is not evidence of an empty TeV scale; it is a statement about which channels were listened to, and PS-KK-1 dissolves into an instruction: listen in boosted top pairs, with the template of Part~\ref{part:ttanatomy}.

\paragraph{PS-KK-2: The hierarchy problem, as an output.}  For four decades ``why is \(v/M_{\rm Pl}\sim10^{-16}\)?'' has been the field's engine, driving supersymmetry, technicolor, warping, and anthropics in turn --- every proposed answer a new mechanism added to protect a ratio.  Here the ratio is not protected; it is \emph{computed}: \(e^{-35}\) times derived rational factors, where 35 is a count.  The hierarchy is the length of the scale throat, and asking why it is large is asking why \(LJ=M^2+N^2+L^2=35\) --- a question with a Diophantine answer \citep{ArisakaA3BI6}, not a dynamical one.  Resolved, conditional on the cascade; and the resolution is testable, because the same count sits at \(3.849\)~TeV.

\paragraph{PS-KK-3: The radion and moduli problem.}  Every spatial extra dimension ever proposed drags the same embarrassment behind it: the size of the new dimension is a field, the field must be stabilized, and the stabilizer (Goldberger--Wise and its descendants; \citealp{GoldbergerWise1999}) is an addition whose only job is to hold the stage still.  String compactifications multiply the embarrassment by hundreds of moduli.  \GG{} dissolves the problem at the root: a dimension whose length is a counting number has no fluctuation to stabilize --- there is no radion field at all --- and the dissolution is falsifiable, because a hidden modulus always leaks into Higgs couplings (P-KK-9, the Higgs-precision null of Part~\ref{part:beyond}).

\paragraph{PS-KK-4: The look-elsewhere pathology.}  Resonance hunting has a statistical tax and a sociological one.  The statistical tax is the trials factor: scan enough mass points and fluctuations are guaranteed, so every local excess is discounted (\(N_{\rm trials}\approx76\) for the published \(\ttbar\) scan, \S\ref{sec:sim-lee}).  The sociological tax is worse: because scanned theories relocate after every null, the community has learned to treat all BSM excitement as noise --- the 750~GeV diphoton episode being the canonical cautionary tale.  A pre-registered single point pays neither tax.  The statistical discount vanishes by arithmetic; the sociological one vanishes because the prediction was in print, with its rate and sign, before the data were unblinded.  Dissolved --- for this search alone, which is rather the point.

\paragraph{PS-KK-5: The ``KK graviton'' normalization ambiguity.}  A subtle, insider's puzzle: the warped literature's first-pole mass \(m_1\simeq x_1 ke^{-k\pi r_c}\) depends on conventions (which brane, which boundary condition, which normalization of \(k\)) that a data analysis cannot check, so ``the KK graviton at \(m_1\)'' has never been a sharp target.  This paper's statement is constructed to be convention-proof: the \emph{centroid} of a multi-field cluster is the theorem-level object (T-KK-1), the fine structure is derived separately (T-KK-2), and the search instruction changes accordingly --- hunt the cluster, not a lone pole.  Resolved by reformulation.

\paragraph{PS-KK-6: Why should new physics sit near the weak scale at all?}  Naturalness said: because something must cancel the Higgs divergence, and it must be nearby.  The LHC has spent fifteen years eroding that argument, and its erosion is routinely counted as evidence against \emph{any} TeV-scale expectation.  This paper's chain is immune to the erosion because it never invoked naturalness: \(v\) and \(\Mlock\) are outputs of one integer chain, so their proximity is an identity, not a hope.  The failure of naturalness-based expectations is evidence against naturalness --- and says nothing about a prediction that never used it.  Dissolved.

\subsection{The structural innovations}
\label{sec:disc-innovations}

Ten registered moves that did not exist before this paper, each with its before-and-after stated so the claim of novelty can be audited.

\begin{description}
\item[I-KK-1 (the zero-parameter collider mass).]  \emph{Before}: every extra-dimensional benchmark mass was an input --- RS's \(kr_c\) tuned to the hierarchy, ADD's \(M_D\) scanned per analysis.  \emph{After}: \(\Mlock=\tfrac12M_{\rm Pl}e^{-K}\) with \(K\) a Diophantine integer, doubly spent through \(v=\Mlock(2/5)^3\).  To our knowledge the first collider mass in the BSM era derived with no continuous freedom anywhere in its chain.
\item[I-KK-2 (the two-scenario fine structure).]  \emph{Before}: fine structure of KK spectra quoted with \(\mathcal O(1)\) placeholders.  \emph{After}: exact free-field eigenvalues \emph{and} an honest discrete dichotomy (free-field versus composite-hardened), both horns carrying complete mass-plus-rate predictions --- one now excluded by data, the survivor sharpened by the exclusion.  The innovation is the \emph{form}: uncertainty packaged as a finite decision problem rather than a band.
\item[I-KK-3 (the derived production coupling).]  \emph{Before}: KK-graviton collider studies parametrized \(k/\bar M_{\rm Pl}\) and scanned it.  \emph{After}: \(\kappa_{gg}=2(1+J_2(x_1))/(Kx_1^2J_2(x_1))=0.47/K\), a closed Bessel form with no inputs to tune (Corollary T-KK-2c; derived in full in Appendix~I) --- and with it, the first parameter-free retro-diction of a published null (the diphoton limit, by a factor of several thousand).
\item[I-KK-4 (the interference-sign theorem).]  \emph{Before}: resonance--continuum interference treated as a nuisance with a sign to fit.  \emph{After}: \(\xi_{\rm UV}g_V^t<0\) forced, making the line shape --- enhancement below, deficit above, the summed spectrum crossing \emph{below} the Standard Model near \(4.9\)~TeV --- a parameter-free shape falsifier, drawn quantitatively in Figure~\ref{fig:lhc-signature}.
\item[I-KK-5 (leptophobia as an exact zero).]  \emph{Before}: leptophobic \(Z'\)s built by charge assignment chosen to evade dilepton limits.  \emph{After}: \(\mathrm{BR}(Z'_B\to\ell\ell)=0\) because leptons carry no baryon number --- the evasion is a theorem, not a choice, and it is testable to \(10^{-6}\) (\S\ref{sec:beyond-fcchh}).
\item[I-KK-6 (the multi-field cluster signature).]  \emph{Before}: searches target single poles; even multi-state models scan the splittings.  \emph{After}: a forced near-degeneracy of three species at one derived mass --- spin-2 anchor plus two vectors --- with the search instruction changed to match.  The cluster, not any member, is the 6D fingerprint.
\item[I-KK-7 (pre-registered-search statistics).]  \emph{Before}: look-elsewhere and template dilution treated as unavoidable costs of BSM searching.  \emph{After}: both quantified and \emph{eliminated} by derivation --- no trials factor, no mismatch dilution --- converting existing data into \(4.6\)--\(5.2\sigma\) of combined dedicated sensitivity (\S\ref{sec:sim-prereg}).  The innovation is the demonstration that theoretical rigidity is worth luminosity.
\item[I-KK-8 (the anchored-simulation method).]  \emph{Before}: BSM sensitivity projections routinely extrapolate with \(\sqrt L\) and idealized detectors.  \emph{After}: every experimental input anchored to a published analysis, the statistical machinery calibrated to its published limits (\(F_{\rm cal}=4.33\), validated on an independent shape), and the systematics wall located analytically (\(L^*=1/f_b^2b\)) --- a projection whose error budget is itself audited (Part~\ref{part:hllhcsim}).
\item[I-KK-9 (the falsification calendar).]  \emph{Before}: falsifiability claimed in principle, rarely dated.  \emph{After}: the kill criterion carries a luminosity (\(\lesssim2\)~ab\(^{-1}\)) and therefore a decade (the 2030s), with the property-measurement ladder scheduled behind it snapshot by snapshot.
\item[I-KK-10 (the facility-role taxonomy).]  \emph{Before}: future-collider physics cases argued machine by machine, each claiming everything.  \emph{After}: a division of labor derived from the couplings themselves --- verdict machine (HL-LHC), spectroscopy machine (FCC-hh), leptophobia machine (the muon collider, whose decisive \GG{} measurement is a predicted \emph{silence}), ledger machines (FCC-ee and the linear options) --- each with named, dated deliverables (Part~\ref{part:beyond}).
\end{description}

\subsection{The nobility clause}
\label{sec:disc-nobility}

``Nobility'' is not a word for the theory; it is a word for the \emph{conduct} the paper binds itself to, and it can be stated as four commitments, each checkable.  \emph{Pre-commitment}: every falsifier is stated with its decision rule and its price before the data arrive --- F-KK-2B's kill window was in print before the reanalysis it invites.  \emph{Honest grading}: every quantitative claim carries [T]/[D\(^*\)]/[M]/[E], and the least flattering grades are displayed most prominently (the \(g_B\) menu is on the first summary table, not in a footnote).  \emph{No retreat}: the response to Scenario~A's exclusion was retirement, not relocation --- and the paper's structure makes future relocation impossible in public, because the integers are load-bearing across twenty-two documents.  \emph{Symmetric stakes}: the paper commits to what failure means (the electroweak derivation falls with the collider claim) as explicitly as to what success means.  These commitments cost rhetorical safety and buy something the fifty-year history shows to be scarcer than cleverness: a claim whose author has arranged, in advance, to be unable to escape.

\section{Comparison with the other roads beyond the Standard Model}
\label{sec:disc-comparison}

Table~\ref{tab:kkrsgg} (\S\ref{subsec:notKKRS}) carried the structural comparison with KK and RS; this section widens the lens to every major road the field has walked since the 1970s and sharpens the question to the operational one: \emph{if that framework, rather than \GG{}, owned the TeV scale, what would the four snapshots of Part~\ref{part:hllhcsim} record --- and which of this paper's registered predictions would it fail?}  Table~\ref{tab:discrimination} collects the verdicts across fifteen observables; the prose takes the frameworks in turn, each with the respect its history has earned and the precision its comparison requires.

\paragraph{Versus 5D Randall--Sundrum and the warped benchmarks.}  RS is this paper's closest relative and its most instructive foil --- the mathematics of Appendix~J is shared, which is exactly why the differences are clean.  A canonical RS graviton at \(3.85\)~TeV with \(k/\bar M_{\rm Pl}\sim0.1\)--\(1\) would have announced itself \emph{first} in dileptons and diphotons, the channels where \GG{} is structurally silent; its single narrow pole carries no mass-degenerate vector pair, no exact leptophobia, and a rate dialed by \(k/\bar M_{\rm Pl}\); its stabilized radion mixes into the Higgs sector at levels a Higgs factory sees; and its second KK level sits at \(x_2/x_1\) \emph{only if} the 5D boundary conditions are pure --- a convention, not a theorem (PS-KK-5).  At the snapshots: RS would show clean-channel lines by 2026 or nothing at all, since its couplings large enough for \(\ttbar\) visibility are excluded in \(\ell\ell\).  Fails P-KK-6 (leptophobia), P-KK-9 (no radion), and the cluster degeneracy; shares (in weakened form) the overtone ratio.

\paragraph{Versus composite Higgs and KK-gluon models.}  These frameworks share this paper's top-philic octet --- indeed the boundary-composite reading of Scenario~B \emph{is} a compositeness statement, and the overlap is acknowledged debt, not coincidence.  The differences are the dials: composite octet masses are scanned (the exclusion plots run from 1 to 6~TeV precisely because nothing fixes the mass), widths are adjustable through the compositeness angle, there is generically no spin-2 anchor mass-locked to the octet, and no second family at a fixed ratio.  At the snapshots: a composite octet \emph{could} imitate end-2026 through end-2041 in \(\ttbar\) alone --- this is the one framework that can gate-crash the first bump --- but it fails the post-discovery program: no anchor at the same mass at FCC-hh (F-KK-1 Tier 2), no \(1.831\times\) overtone, free \(P_t\) rather than the band \([+0.88,+0.92]\).  The discrimination is therefore scheduled rather than immediate, and Part~\ref{part:beyond} is its calendar.

\paragraph{Versus generic \(Z'\) models.}  The workhorse of BSM searching, and the cleanest contrast.  A \(Z'\) at \(3.85\)~TeV with electroweak-strength lepton couplings is already excluded; a leptophobic \(Z'\) survives but only by construction, and it comes alone --- no color octet locked to its mass, no graviton, no interference with QCD (a color singlet cannot interfere with the gluon channel), and no reason for its mass.  At the snapshots: a leptophobic \(Z'\) tuned to \(8.7\)~fb would track the octet's counting significance but fail the \emph{shape} program immediately --- no low-side enhancement, no below-SM crossing (P-KK-4), the wrong dijet-to-\(\ttbar\) decomposition --- and the interference dichotomy of O-KK-28 separates the two at the template-fit level with the same data.

\paragraph{Versus supersymmetry.}  The comparison is methodological rather than observational, because the MSSM makes no sharp statement about this channel at all --- which is itself the point \S1 labored.  A century of collective effort produced, in the MSSM, a framework whose \(>\!100\) parameters can accommodate any \(\ttbar\) spectrum the LHC records; it is not falsifiable by this measurement, and symmetrically, it cannot claim it.  The structural contrast is sharper: \GG{}'s bulk admits \emph{no} super-Poincar\'e extension (no translation generators along \(r,\theta\); \S\ref{subsec:notKKRS}), so fifty years of superpartner nulls --- unexplained embarrassments under naturalness --- are, here, confirmations of a theorem.  One framework predicts the silence the other must excuse.

\paragraph{Versus large extra dimensions (ADD).}  The other great extra-dimensional road fields \emph{spatial} dimensions at sub-millimeter-to-TeV scales, with a quasi-continuum KK tower and a fundamental scale \(M_D\) scanned per search.  Its collider signatures --- monojets from graviton escape, non-resonant dijet/dilepton excesses rising with energy --- are qualitatively different from a resonant cluster, and its towers are picket-fence continua (Appendix~J, Step 1), not an organ pipe.  At the snapshots: ADD would show smoothly rising tails, no peak, no interference structure; it fails every resonant clause of the P-KK list and is separated from \GG{} by the first spectrum plot.  The deeper contrast is the one of \S\ref{subsec:notKKRS}: ADD's dimensions carry moduli and require stabilization; \GG{}'s carry neither.

\paragraph{Versus the string landscape.}  String theory is not a competitor at the level of a 3.85-TeV template --- it makes no such statement --- and the honest comparison is the one A2 draws: two consistent strategies with their irreducible assumptions at opposite ends of the energy axis.  What belongs \emph{here} is the falsifiability asymmetry.  A landscape with \(10^{500}\) vacua can absorb any collider outcome, positive or negative; the present framework is killed by one.  If the cluster is found with its correlated pattern, string model-building will retro-fit it within the decade --- such is the flexibility --- but the \emph{prediction} will have belonged to the framework that made it when it could still have been wrong.  Priority of risk, not priority of publication, is the scientifically meaningful ordering, and it is the one quantity a landscape cannot manufacture after the fact.

\begin{table}[!t]
\centering\small
\setlength{\tabcolsep}{3pt}
\renewcommand{\arraystretch}{1.28}
\caption[Framework discrimination matrix]{Framework-by-observable discrimination matrix, transposed for completeness: observables as rows, frameworks as columns.  Each cell: does the framework generically reproduce the observable?  (\(\checkmark\) = yes / forced; \(\sim\) = achievable by parameter choice; \(\times\) = fails or forbidden; --- = no statement.)  The MSSM column is mostly ``---'' because the framework makes no sharp claim in this channel --- which is the methodological point of \S1; the string-landscape column is ``\(\sim\)'' throughout because any cell is fillable after the fact (priority of risk, see text).  The final row names the earliest measurement separating each framework from \GG{}.}
\label{tab:discrimination}
\small
\setlength{\tabcolsep}{5pt}
\begin{tabular}{>{\raggedright\arraybackslash}p{5.6cm} c c c c c c c}
\toprule
\textbf{Observable} & \textbf{\GG{}} & \textbf{RS} & \textbf{CHM} & \textbf{\(Z'\)} & \textbf{MSSM} & \textbf{ADD} & \textbf{Str.}\\
\midrule
Mass derived (no scan) & \(\checkmark\) & \(\times\) & \(\times\) & \(\times\) & --- & \(\times\) & \(\sim\)\\
Rate derived (no coupling dial) & \(\checkmark\) & \(\times\) & \(\times\) & \(\times\) & --- & \(\times\) & \(\sim\)\\
EW scale tied to new mass & \(\checkmark\) & \(\sim\) & \(\sim\) & \(\times\) & \(\sim\) & \(\times\) & \(\sim\)\\
Cluster degeneracy \(m_{g^*}{=}m_{Z'_B}\) & \(\checkmark\) & \(\times\) & \(\times\) & \(\times\) & --- & \(\times\) & \(\sim\)\\
Exact leptophobia (P-KK-6) & \(\checkmark\) & \(\times\) & \(\sim\) & \(\sim\) & --- & \(\times\) & \(\sim\)\\
Width \(\Gamma/m\) predicted & \(\checkmark\) & \(\times\) & \(\times\) & \(\times\) & --- & \(\times\) & \(\sim\)\\
Interference sign (P-KK-4) & \(\checkmark\) & \(\sim\) & \(\checkmark\) & \(\times\) & --- & \(\times\) & \(\sim\)\\
\(P_t\in[+0.88,+0.92]\) (P-KK-5) & \(\checkmark\) & \(\sim\) & \(\sim\) & \(\sim\) & --- & \(\times\) & \(\sim\)\\
BR fingerprint \(70{:}3.5{:}1\) (P-KK-2) & \(\checkmark\) & \(\times\) & \(\sim\) & \(\sim\) & --- & \(\times\) & \(\sim\)\\
No radion / moduli (P-KK-9) & \(\checkmark\) & \(\times\) & \(\checkmark\) & \(\checkmark\) & \(\times\) & \(\times\) & \(\times\)\\
No angular satellites (P-KK-7) & \(\checkmark\) & \(\checkmark\) & \(\checkmark\) & \(\checkmark\) & --- & \(\times\) & \(\sim\)\\
Overtone at \(1.831\times\) (P-KK-8) & \(\checkmark\) & \(\sim\) & \(\times\) & \(\times\) & --- & \(\times\) & \(\sim\)\\
Hierarchy as output & \(\checkmark\) & \(\sim\) & \(\sim\) & \(\times\) & \(\sim\) & \(\sim\) & \(\sim\)\\
SUSY absence explained & \(\checkmark\) & --- & --- & --- & \(\times\) & --- & \(\times\)\\
Lepton-collider silence (P-KK-10) & \(\checkmark\) & \(\times\) & \(\sim\) & \(\sim\) & --- & \(\times\) & \(\sim\)\\
\bottomrule
\end{tabular}
\par\smallskip
{\small\raggedright \emph{Column heads}: RS = 5D Randall--Sundrum benchmark; CHM = composite-Higgs / KK-gluon models; \(Z'\) = generic leptophobic \(Z'\); Str.\ = string landscape.  \emph{Earliest measurement separating each framework from \GG{}}: RS --- the \(\ell\ell/\gamma\gamma\) lines it requires and \GG{} forbids (already decided by existing data); CHM --- the spin-2 anchor and the \(1.831\times\) overtone (FCC-hh, Part~\ref{part:beyond}); \(Z'\) --- the interference shape, inside the discovery fit itself; MSSM --- none (no claim in this channel); ADD --- resonant versus non-resonant spectrum shape (the first plot); string landscape --- priority of risk (see text).\par}
\end{table}

\section{Predictions for falsification}
\label{sec:disc-predictions}

A falsifier is a prediction plus a decision rule plus a price; a prediction alone is only a hope with units.  This section restates the paper's three falsifiers as narratives --- where each was declared, what has happened to it since, and what ends it --- and then walks the ten registered predictions individually, because a table row (the summary remains as Table~\ref{tab:predictions}) cannot carry what a referee needs: for each claim, \emph{what} is predicted, \emph{where} it was derived, \emph{how} it will be measured, and \emph{what outcome kills it}.

\paragraph{F-KK-1, the two-tier cluster falsifier: a history.}  Declared as the paper's original headline --- the spin-2 anchor at the lock scale --- F-KK-1 was requalified, in public and by this paper's own Corollary T-KK-2c, when the graviton's gluon coupling was computed and came out at \(0.47/K\): the anchor's LHC rate fell to invisibility, and the falsifier restructured into two tiers.  \emph{Tier 1 (HL-LHC, decisive)}: the lock-scale implementation dies on a null in the cluster-aware \(\ttbar\)/dijet combination within \(3.85\pm0.10\)~TeV at \(\lesssim2\)~ab\(^{-1}\), carried entirely by the un-suppressible octet.  \emph{Tier 2 (FCC-hh, confirmatory)}: the direct spin-2 anchor at the same mass, \(\Gamma/m=6.3\%\), branching \(0.69{:}0.15{:}0.08{:}0.08\), rate \(\approx0.13\)~pb at 85~TeV --- discovery in the first weeks or falsification of the cluster reading.  The division-of-labor clause is itself falsifiable: finding either tier without the other kills the interpretation.  The requalification episode belongs in this narrative because it demonstrates the register the paper operates in: when a derivation moved a falsifier, the falsifier was restated \emph{harder}, not retired.

\paragraph{F-KK-2A: fired.}  The free-field scenario --- spin-1 pair at \(0.639\,\Mlock=2.461\)~TeV, \(\sigma\times\mathrm{BR}=224\,[180\text{--}325]\)~fb, an unmissable rate --- was confronted with the published Run-2 record in \S\ref{sec:operational} and lost: the prediction sits a factor 3.6--11 above the observed exclusion, outside its band by any reading.  The closure is graded provisional on template shape alone (no plausible line-shape correction recovers a factor of three), and its structural yield is recorded in \S\ref{subsec:tkk2}: the throat vectors are composite.  One of this paper's falsifiers has therefore already executed, in the kill direction, on schedule --- the strongest possible evidence that the others are load-bearing.

\paragraph{F-KK-2B: live, and the center of everything.}  The composite-scenario octet at \(\Mlock\): \(8.7\,[7.0\text{--}12]\)~fb at 13~TeV, \(13.6\)~fb at 14, width \(14.6\%\), interference sign fixed.  Its decision procedure is the entire content of Part~\ref{part:hllhcsim}; its price is stated (\(\lesssim2\)~ab\(^{-1}\)); its earliest possible execution is a reanalysis of tape that already exists.  Every other claim in the paper is downstream of this one's verdict.

\begin{table}[H]
\centering
\caption[The numbered predictions of this paper, with grades.]{The numbered predictions of this paper, with grades ([T] theorem; [D\(^*\)] derived-conditional; [M] menu; [E] external-input). Four grades and no unlabeled rows. A theorem entry falls with its theorem, a derived-conditional entry falls with the condition printed beside it, a menu entry is one of a finite listed set, and an external-input entry is somebody else's number this paper consumes. Mixing the four is how a prediction list comes to look stronger than it is, and the grade column exists to stop that.}
\label{tab:predictions}
\small
\usebox{\GGscPredP}
\end{table}

\paragraph{The ten predictions, walked individually.}
\emph{P-KK-1 (the rates).}  \(\sigma\times\mathrm{BR}(g^*\to\ttbar)=8.7\,[7.0\text{--}12]\)~fb at 13~TeV, \(13.6\,[11\text{--}19]\) at 14, \(\approx2.8\)~pb at 85 --- derived in \S\ref{sec:operational} from \(\xi_{\rm UV}\) and the parton chain of Appendix~K; measured by the dedicated template fit; killed by a fitted rate outside the band in either direction (an \emph{excess} beyond \(\sim\!12\)~fb falsifies as surely as a null).
\emph{P-KK-2 (the octet fingerprint).}  \(\ttbar:b\bar b:jj\approx70:3.5:1\), \(\mathrm{BR}(\ttbar)=0.94\,[0.93\text{--}0.95]\) --- from the width table of Appendix~I; measured by the dijet and \(b\bar b\) companions once the \(\ttbar\) line exists; killed by any ratio off by more than the band, most sharply by a dijet line \emph{stronger} than \(\ttbar\)/20.
\emph{P-KK-3 (the anchor pattern).}  \(G_1\) at the same mass with \(\ttbar:W_LW_L:Z_LZ_L:hh=0.69:0.15:0.08:0.08\) and \(\Gamma/m=6.3\%\) --- from T-KK-2c and the stress-tensor couplings; measured at FCC-hh across four channels; killed by a missing anchor, a wrong width, or transverse-dominated diboson decays (the longitudinal dominance is the Higgs-sector signature).
\emph{P-KK-4 (the sign).}  Constructive below the pole, destructive above, SM crossing at \(\simeq4.9\)~TeV, centroid pulled \(-\mathcal O(100)\)~GeV --- the theorem of Part~\ref{part:ttanatomy}; measured by the sign-floated template fit of the discovery analysis itself; killed by a wrong-sign or no-interference preference at HL-LHC statistics.
\emph{P-KK-5 (polarization and asymmetry).}  \(P_t=+0.91\), band \([+0.88,+0.92]\); \(A_{FB}=0\) --- from the chirality structure \(r_{t_R}\gg r_{t_L}\) and the vector-like light-quark coupling respectively; measured through the lepton-analyzer distributions to \(\delta P_t=0.08\) by 2041; killed by low polarization (a vector-like top coupling) or any significant \(A_{FB}\) (an axial light-quark coupling), either of which breaks T-KK-2 directly.
\emph{P-KK-6 (leptophobia and degeneracy).}  \(\mathrm{BR}(Z'_B\to\ell\ell)=0\); \(m_{Z'_B}=m_{g^*}\) exactly --- assignments and shared boundary-value problem; measured by the dilepton null (to \(10^{-6}\) at FCC-hh) and the two-component line-shape decomposition; killed by any dilepton line at the cluster mass or any resolved splitting.
\emph{P-KK-7 (no angular satellites).}  The \(\theta\) tower at \(\sim\EGUT\), hence \emph{nothing} decorating the cluster at percent-level mass offsets --- the non-spatial reading's sharpest negative (\S\ref{subsec:notKKRS}); measured for free in any high-resolution scan of the peak region; killed by satellite structure, which would simultaneously revive the spatial-6D reading this paper rejects.
\emph{P-KK-8 (the overtone).}  \(G_2\) at \(1.831\times\Mlock=7.047\)~TeV, position theorem-grade, rate \(\sim\!10\)~fb at 85~TeV (estimate) --- the Bessel ratio of \S\ref{sec:t-kk-1}; measured at FCC-hh spectroscopy; killed by a position-null at FCC-hh statistics (O-KK-18 records the no-excuses clause) or by an overtone at the \emph{wrong} ratio, which would falsify the cavity itself.
\emph{P-KK-9 (no radion).}  Higgs couplings Standard-Model-like except the derived \(G_1\to hh\) line --- from the integer length (PS-KK-3); measured at the per-mille level by FCC-ee; killed by a scalar-mixing pattern in Higgs precision, discovery or no discovery at the LHC.
\emph{P-KK-10 (the lepton-collider silence).}  No \(s\)-channel cluster line in \(\mu^+\mu^-\); SM-like contact terms at a 3-TeV stage --- from the same localization geometry as PS-KK-1; measured whenever a muon collider runs; killed by any leptonic line at \(3.849\)~TeV, which would falsify the \(\UB\) assignment and the localization in one stroke.

\paragraph{The falsification lattice.}  The ten predictions are not independent shots; they form a lattice with the three theorems at its nodes, and the lattice is the real object under test.  A wrong sign (P-KK-4) breaks T-KK-2's mode structure and takes P-KK-5 with it; satellites (P-KK-7) break the non-spatial reading and take P-KK-8's ratio with them; a radion (P-KK-9) breaks the integer length and takes the mass itself.  Conversely a discovery that confirms the lattice confirms it \emph{jointly} --- which is why \S\ref{sec:disc-comparison} could promise that no neighboring framework survives the full list: frameworks can imitate entries, but the correlations are the geometry.

\section{Open questions}
\label{sec:disc-open}

Honesty requires the complement, at the same depth as the claims.  Six items remain open; for each we state why it is open (not merely that it is), what has been tried, the closing condition, and who is best placed to close it.  Appendix~O carries the same list as a project roadmap.

\paragraph{1. L-KK-1: the \(g_B\) readout menu.}  \emph{Why open}: the \(\UB\) coupling is fixed by \GDOS{} counting in principle, but the counting has four defensible readout conventions, and this paper declines to pick one without the derivation --- hence a menu \(\{0.14, 0.23, 0.36, 0.57\}\)-class spread and \(Z'_B\) rates quoted per point.  \emph{Tried}: the direct transcription of the \(\alpha_s=1/(\pi e)\) counting; it fixes the structure but not the convention.  \emph{Closing condition}: the same GDOS derivation carried once more, carefully, for the Abelian line --- a bounded theory exercise.  \emph{Impact}: single-valued \(Z'_B\) phenomenology and a sharpened fifth-force window \citep{ArisakaA9SM}.
\paragraph{2. The NLO interference K-factor.}  \emph{Why open}: the interference term is linear in the resonance amplitude, so its K-factor is \emph{not} the Breit--Wigner's \(K_{q\bar q}=1.3\), and no published computation covers a broad color-octet interfering with the \(\ttbar\) continuum at this mass.  \emph{Tried}: the \(\sqrt K\) prescription used throughout, standard but uncontrolled at the tens-of-percent level.  \emph{Closing condition}: a conventional NLO calculation --- the one entry on this list a standard QCD group could adopt tomorrow, with immediate experimental consumers (\S\ref{sec:sim-contamination}).  \emph{Impact}: the low-side template firmed; the largest single theory systematic of the discovery fit retired.
\paragraph{3. The \(n\ge2\) overlap couplings.}  \emph{Why open}: Corollary T-KK-2c was evaluated at \(x_1\); the same machinery at \(x_2, x_3\) is straightforward for diagonal couplings but the inter-mode mixing needs care.  \emph{Closing condition}: Eq.~\eqref{eq:appJ-kappa} at higher zeros --- an afternoon for the diagonal terms, a short paper done properly.  \emph{Impact}: P-KK-8's rate promoted from estimate to derivation before FCC-hh needs it.
\paragraph{4. The composite-scenario recurrence spacing.}  \emph{Why open}: Scenario~B fixes the lowest states by matching, but the spacing of the vector recurrences is a property of the strongly coupled boundary theory, and no weak-coupling trick reaches it.  \emph{Closing condition}: a dual (holographic or lattice-inspired) computation, or the data themselves at FCC-hh.  \emph{Impact}: search targets for the \(\mathcal O(10^2)\)~fb recurrences of Part~\ref{part:beyond}.
\paragraph{5. The detector-level angular analysis.}  \emph{Why open}: the \(10\)--\(30\%\) discovery gain from \(\cos\theta^*\) requires the real background cocktail's angular shapes, which live inside the collaborations.  \emph{Closing condition}: a full-simulation 2D study --- thesis-sized, and the natural companion to the reanalysis itself.  \emph{Impact}: the bounded gain of \S\ref{sec:sim-angular} banked, potentially advancing the five-sigma crossing by a year.
\paragraph{6. The composite width band.}  \emph{Why open}: \(\Gamma/m=14.6\%\) carries the free-field overlaps; the composite reading could shift it within a band no current calculation pins.  \emph{Closing condition}: self-closing --- the HL-LHC line-shape fit measures \(\Gamma/m\) to a few percent, converting the band into a cross-check of the overlap structure.  \emph{Impact}: a measured width becomes a second-generation test of T-KK-2's interior.

None of the six shields the verdict: the falsifier stands on the theorem-grade spine --- mass, degeneracy, sign, leptophobia --- alone.  That is the design: the open questions sharpen the program; they do not prop it up.


\section{The completed cohort and future prospects}
\label{sec:disc-cohort}

This paper is one of twenty-two, and its place in the cohort is load-bearing in both directions.  \emph{Imports:} the bulk and its canonical metric (\citealp{ArisakaA2GG6}, Theorems 0 and 2), the \OGNfive{} integer cascade (\citealp{ArisakaA3BI6}), the lock-scale chain and the electroweak over-constraint (\citealp{ArisakaA9SM}), the open-system reading of detection (\citealp{ArisakaA5GDOS}; \citealp{ArisakaA6Gi4}), and the no-radion theorem (\citealp{ArisakaA4Euclid}).  \emph{Exports:} the requalified falsifier F-SM-1 of \citet{ArisakaA9SM} now points at this paper's computed rates; the \(g_B\) menu feeds the fifth-force window; and the collider verdict, when it arrives, propagates to the constitutional core --- in both directions.  The near-term prospect is stated once more because it is the paper's operational request: \emph{a dedicated, interference-aware, pre-registered reinterpretation of the \(\ttbar\) data already on tape}, jointly by ATLAS and CMS, which Part~\ref{part:hllhcsim} shows may already contain the discovery.  The far-term prospects are Part~\ref{part:beyond}'s: the anchor, the overtones, and the leptophobic silence, distributed across the machines of 2035--2070.

\section{Summary of the Discussion}
\label{sec:disc-summary}

The strengths, stated plainly: a derived, immovable target whose integer has already survived a four-digit audit at the electroweak scale; a fully specified signal (mass, width, rate, sign, polarization, fingerprint); a calibrated calendar with evidence-level sensitivity already on tape; statistics that reward the derivation itself (no look-elsewhere, no template dilution); an exclusion already converted into structural knowledge (Scenario~A, the observed functioning of the kill machinery); a discrimination lattice that no neighboring framework survives intact (Table~\ref{tab:discrimination}); and a falsifier whose failure mode is clean retirement.  The weaknesses, stated with equal plainness: the rate band is PDF- and K-factor-limited until LHeC-class input arrives; the interference template is leading-order, with the NLO task named but not done; one coupling remains a four-point menu; the \(n\ge2\) couplings are estimates; and no public collaboration reinterpretation of the exact benchmark yet exists --- the paper's central request.  The posture these imply is the one the paper has kept throughout: aggressively experimental, theoretically disciplined --- and, with Part~\ref{part:hllhcsim} in hand, \emph{urgent}, because the evidence may already be sitting on tape in Geneva, waiting for a template that was only written this year.

Part~X made the arguments.  In one breath each: the result is significant because both of its possible endings are informative, which fifty years of adjustable predictions could never say; it is unique in kind, not degree, and its innovations are auditable one by one; no neighboring framework survives the full fifteen-observable fingerprint; the falsifiers are priced, dated, and --- in one case already --- executed; and the open questions sharpen the program without propping it up.  The argument of the paper is now complete.  What follows is its ending in the older sense of the word: conclusions, a personal accounting, and the questions a result of this shape forces one to ask about number, nature, and the act of knowing.


\section[Part~X in plain English]{Part~X in plain English: what the result is worth, set against the alternatives}
This is the Part where the paper argues with itself and with everyone
else.

The claim to significance is narrow and should be stated narrowly: not that
the framework is right, but that it has produced a collider number with no
adjustable parameter behind it, which the alternatives have not. That is a
claim about method. It is worth something only if the number is wrong in a
way that would be visible, and the Part is careful to say what visible
means here.

The comparison with the other roads beyond the Standard Model is done
property by property rather than by verdict, because the roads were built
for different purposes. Supersymmetry addressed a naturalness problem this
framework does not have. String theory operates at an energy no collider
reaches. Judging either against a 3.85 TeV prediction would be answering a
question they were not asking, and the table says which cells are empty for
that reason rather than by failure.

What the Part does claim as new is structural: the absence of dials, the
fact that the same integer that fixes this mass also fixes the electroweak
scale, and the pre-registration of a number before the measurement exists.
The open questions are listed as work orders with closing conditions
attached, and one of them --- an interference calculation --- is the input
the proposed reanalysis needs first.

\GGPart[Conclusions]{Conclusions}{What the paper has established; the objective limitations; a personal perspective from fifty years on the experimental front line; the vision of a geometric century; and what either verdict would mean --- for physics, and for the way human beings come to know the world}
\label{part:conclusions}

\section{What the paper has established}
\label{sec:concl-established}

The results of this paper, in the order they were earned.  \emph{Theorem T-KK-1}: the first observable Kaluza--Klein resonance of the six-dimensional bulk is the radial lock cluster at \(\Mlock=\tfrac12 M_{\rm Pl}e^{-35}=3.849\)~TeV, with no continuous freedom.  \emph{Theorem T-KK-2}: the cluster's fine structure --- the exactly degenerate spin-1 pair, the exact free-field eigenvalues, the coupling \(\xi_{\rm UV}=-0.194\) with its sign, the two-scenario dichotomy --- derived, with Scenario~A subsequently excluded by existing data and the dichotomy resolved to the degenerate cluster.  \emph{Corollary T-KK-2c}: the graviton--gluon coupling computed in closed form, \(\kappa_{gg}=0.47/K\), moving the direct spin-2 test to FCC-hh and concentrating the entire LHC falsifier on the color octet.  \emph{Part~\ref{part:ttanatomy}}: the complete derived anatomy of the \(\ttbar\) signal --- angular forms, polarization, the interference-sign theorem and its line shape.  \emph{Part~\ref{part:hllhcsim}}: the anchored, calibrated simulation converting the prediction into a calendar --- evidence-level sensitivity already on tape, combined discovery-grade sensitivity a live possibility, single-experiment five-sigma luminosity during 2032, and the pre-registration advantages quantified.  \emph{Part~\ref{part:beyond}}: every remaining measurement assigned to a named future machine, through 2070.

Now the same results once more, in plain English, because a conclusion should be readable by the colleague down the hall who is not a collider physicist.  A theory with no adjustable numbers --- its entire freedom is five counting numbers --- says there is a new particle, and says everything about it: where it is (3.85~TeV, about four thousand times heavier than a hydrogen atom's proton), how wide it is, how often the LHC should have made it already (roughly ten thousand of them are sitting, unrecognized, in the data recorded since 2015), what it decays into (top quarks, overwhelmingly), what it refuses to decay into (electrons and muons --- exactly the particles our cleanest searches watch), and even the subtle way it interferes with ordinary background, down to the sign.  The same five counting numbers already produce the known strength of the weak force scale --- that part of the ledger was checked decades ago, by people with no stake in this theory.  What this paper adds is the practical half: proof, at the fidelity of a real analysis, that the existing data may already contain the evidence, and a step-by-step recipe by which the experiments can decide the question --- either way --- within this decade.

The practical collider statement in one sentence: the first experimentally relevant new-physics structure in \GG{} should be a \emph{spin-2-anchored, octet-led resonance cluster centered near \(3.849\,\mathrm{TeV}\), with top-rich and dijet-friendly branching, a mild leptophobic admixture, and a second \emph{radial} family near \(7\,\mathrm{TeV}\)} --- under real pressure from existing clean-channel limits (pressure the derived coupling texture explains), and decided by exactly the analysis this paper specifies.

\section{The objective limitations and the next technical tasks}
\label{sec:concl-limitations}

The paper's honest boundary, stated as a task list rather than a confession --- and prefaced by the reason the list exists at all.  A framework that graded its own claims generously would have nothing to put here; this one carries a menu, several estimates, and a leading-order template, all labeled at their point of use, and the discipline of \emph{saying so} is what separates a falsifiable program from an advertising campaign.  \emph{Theory side:} the rates are LO\(\times\)K with PDF-limited bands (task: NLO signal and, above all, the NLO interference K-factor of \S\ref{sec:sim-contamination} --- the single largest theory systematic of the discovery fit); one coupling remains a discrete menu (task: close L-KK-1 by the GDOS counting that fixed \(\alpha_s\)); the \(n\ge2\) couplings are same-coupling estimates (task: run the T-KK-2c machinery at \(x_2,x_3\) --- an afternoon for the diagonal terms); the composite-scenario recurrence spacing is undetermined (task: the strong-coupling dual calculation, the genuinely hard one).  \emph{Experiment side:} no public collaboration reinterpretation of the exact benchmark exists (task: the pre-registered \(\Gamma/m=15\%\) interference-aware template analysis, jointly ATLAS+CMS, on data already recorded --- the paper's central request); the angular dimension is bounded but unbanked (task: a detector-level 2D study with the anchored background cocktail); the sideband-contamination prescription must be built into the fit design from the start.  None of these tasks blocks the verdict; all of them sharpen it.

\section{A personal perspective: fifty years on the experimental front line}
\label{sec:concl-personal}

A word is owed about why \emph{this} paper, of the twenty-two, is the one closest to its author's life.  I have spent fifty years as an experimental particle physicist --- at Kamiokande, waiting for the proton decay that grand unification promised and that never came; on $C\!P$ violation in the kaon and $B$-meson systems; on CMS at the Large Hadron Collider, where I watched the Higgs \emph{complete} the Standard Model rather than break it; on the Pierre Auger Observatory at the ultra-high-energy frontier; and on XENON, in the direct search for dark matter.  Five front lines, fifty years, and on every one of them the same experience: theory offered us targets that could always retreat, and nature offered us confirmations of a model written before I began.

Let me say what that felt like from the inside, because the sociology matters as much as the physics.  An experimentalist's working life is spent converting other people's hopes into shift schedules.  You calibrate for two years, run for five, unblind with your heart pounding --- and when the plot is empty, the theorists whose prediction you tested explain, sometimes within the week, why the interesting region was always just above your reach.  After the third or fourth such cycle you stop being angry and start being careful: you learn to ask, before committing a decade, \emph{what exactly will you concede if we see nothing?}  Fifty years of asking that question, and almost never receiving an answer, is the biographical origin of every discipline in this paper.  I have been on the receiving end of theoretical predictions often enough to know precisely what an experimentalist needs and almost never gets: one mass, not a scan range; a rate with a derivation, not a coupling dial; a width, a sign, an angular shape; and a statement of what would kill the theory, in advance, in writing.  The reinterpretation card of Appendix~M, the pre-registered template of Part~\ref{part:hllhcsim}, the dated kill window --- these are the artifacts I wished for across five decades of searches, written now from the other side of the table.

When the Standard Model's own list was exhausted in 2012, I left the field for the better part of a decade and worked on the nervous system of \emph{C.~elegans} and the architecture of human vision.  \GGTheory{} grew, unexpectedly, out of that detour --- the story is told in \citet{ArisakaA9SM} and \citet{ArisakaA7Qi4} --- and when it returned me to particle physics, it returned me with a number: \(3.849\)~TeV.  The graduate student addressed in this paper's Preface is, of course, myself in 1976, offered the choice I wish I had been given: a prediction that cannot move, testable within one PhD, on data partly already recorded.  The calendar of Part~\ref{part:hllhcsim} runs through the mid-2030s.  I intend to watch the verdict arrive, and I have written this paper so that either verdict --- the cluster, or its clean exclusion --- will be worth the fifty years.

\section{The vision: what a geometric century could look like}
\label{sec:concl-vision}

Permit me one section of unguarded ambition, in plain English, before the philosophy and the close.  Suppose the octet is there.  Not as advocacy --- Part~\ref{part:hllhcsim} has stated the odds as honestly as I know how --- but as a planning exercise, because visions are how fields decide what to build.

The twentieth century learned that matter is fields and that fields are geometry's tenants: they live \emph{on} spacetime.  The claim under test in this paper is one notch deeper --- that the tenants are the landlord's own rooms; that the energy scale at which you probe the world, and the gauge phase that makes charged-matter interference possible, are not bookkeeping devices but \emph{directions}, as real as north; and that once you count those directions correctly, the numbers we have spent a century measuring --- the strength of the strong force, the mass of the weak scale, and now, testably, a resonance at 3.849~TeV --- stop being inputs and become readings.  If that claim survives its collider test, the twenty-first century's physics acquires a different daily texture.  The question at the frontier stops being ``what new particle might exist?''\ --- a question that has, for fifty years, admitted too many answers --- and becomes ``what does the geometry say, and to how many digits can we check it?''\ --- a question with the texture of nineteenth-century spectroscopy, which is meant as the highest compliment: spectroscopy is the mode in which physics has historically been most honest, most cumulative, and most kind to its young people, because a student with a good idea and a data set can be \emph{right} in a way no committee can argue with.

In that century, the machines of Part~\ref{part:beyond} are not lottery tickets but instruments in the older sense: FCC-hh does for the scale throat what the Balmer series did for the atom --- reads its overtones; the muon collider's silence at 3.849~TeV is as informative as any peak; the Higgs factories audit the ledger digit by digit.  And the two cultures whose fifty-year estrangement \S1 recounted --- the theorists who could not commit and the experimentalists who could not forgive --- have, for the first time in living memory, a shared object neither can do without: a number that theory cannot move and experiment cannot ignore.  My deepest professional hope for this paper is not that it is right; it is that it makes that kind of physics imaginable again to a twenty-five-year-old.

\section{The trailhead, not the summit: a final reflection}
\label{sec:concl-reflection}

If a \(3.85\,\mathrm{TeV}\) spin-2-anchored cluster is discovered with the correlated pattern described here, the reward will be extraordinary: not merely another BSM particle, but direct evidence that the deep ultraviolet of nature is better described by a six-dimensional geometry--gauge-locking stage than by a purely four-dimensional field ledger.  And yet discovery would be a beginning, not an end: the anchor, the overtones at the Bessel ratios, the exact zero of leptophobia, the silence of the lepton machines --- the entire spectroscopy of a non-spatial geometry, mapped across the machines of Part~\ref{part:beyond} --- would lie ahead, the way the hydrogen spectrum lay ahead of Balmer's formula.  The LHC is the trailhead of that program, not its summit.

If instead no such structure appears and the full combined-channel \GG{} corridor is excluded, the present implementation of the theory will have failed one of the clearest collider tests imaginable --- cleanly, permanently, and without a zombie phase, because there is no parameter with which to retreat.  Either way, the experiment is worth doing.  That is the mark of a good falsifier; it was the missing property of the fifty years this paper's introduction recounted; and it is the property this paper was written to restore.

\section{What it would mean: matter, number, and the legibility of nature}
\label{sec:concl-philosophy}

A paper that asks two collaborations for person-years of work owes them more than a template; it owes them an account of what the answer would \emph{mean}.  This closing section attempts that account at the depth the question deserves, in four movements.

\paragraph{On falsifiability as an ethic, not a slogan.}  Popper made falsifiability a criterion of demarcation; half a century of practice has shown that as a mere criterion it is too easy to satisfy in the letter and evade in the spirit --- every scanned model is ``falsifiable'' at each parameter point and immortal as a family.  What this paper has tried to practice is falsifiability as an \emph{ethic}: the deliberate construction of a position from which one's future self cannot escape.  The integers are load-bearing across twenty-two documents precisely so that no quiet revision is possible; the kill window is priced in inverse attobarns so that no schedule slip can blur it; the grades are printed so that no later reader can be told the menu was always a theorem.  There is a moral structure here older than science: it is the structure of a \emph{promise}.  A theory that cannot break its word to the data is behaving, in the only sense available to a mathematical object, honorably --- and fifty years of BSM history are best summarized as the slow discovery of how hard, sociologically, that honor is to maintain when careers depend on survival.  If the cluster is found, the finding will vindicate a geometry.  Whatever is found, I hope the \emph{form} of the wager outlives the wager itself.

\paragraph{On why counting numbers, and the old Pythagorean question.}  Wigner famously called mathematics ``unreasonably effective'' in physics --- a puzzle, because there is no evident reason why nature should honor our notation.  The framework under test dissolves the puzzle in the most direct way imaginable: if the constants of nature are \emph{counts} --- dimensions counted, thresholds counted, windings counted --- then the effectiveness of number is not unreasonable but tautological, because the world's deep structure would \emph{be} arithmetic enacted in geometry.  This is the oldest speculation in Western natural philosophy: the Pythagoreans' ``all is number,'' which the continuum revolution of the seventeenth century seemed to retire permanently in favor of the real line and the differential equation.  It is worth savoring the historical irony that the retirement may have been premature at exactly the deepest level: the continuum machinery survives in \GG{} --- fields, actions, renormalization --- but every place where the continuum seemed to demand an arbitrary \emph{choice} (a coupling, a mass ratio, a mixing angle) is, on this account, occupied by an integer instead.  A discovery at 3.849~TeV would not prove Pythagoras right; nothing proves a metaphysics.  But it would establish, as an experimental fact, that at least one number formerly believed continuous and arbitrary --- the mass scale of new physics --- was a count all along.  Physics has changed its mind about the continuum before, exactly once, in 1900, and we named the resulting century after the change: \emph{quantum}.  The wager of this cohort is that the twentieth century stopped one dimension short.

\paragraph{On a world that measures itself.}  The strangest feature of the framework, judged by classical intuitions, is not the extra dimensions; it is the claim (\S\ref{sec:central-claim}; \citealp{ArisakaA5GDOS}; \citealp{ArisakaA8QM}) that \emph{observation is not something we add to nature but something nature does} --- that the projection from bulk to boundary \emph{is} measurement, and that quantum mechanics is its grammar rather than a separate rulebook.  Philosophy has circled this possibility for a century under many names (the measurement problem, the participatory universe, the view from nowhere), always blocked by the same objection: such claims seemed permanently untestable, differences of interpretation rather than of physics.  What the lock cluster does --- and this may be its deepest significance, beyond any particular geometry --- is drag one consequence of that picture into a detector.  If registered observables are boundary readouts of one projection, then their values are collectively fixed, and the cluster at 3.849~TeV is simply the next unread line of a ledger whose previous lines (the weak scale, the strong coupling) have already been read and found consistent.  A positive verdict would therefore be evidence, indirect but real, for the most philosophically radical clause in the entire cohort: that the ancient partition between the world and the knowing of the world was drawn in the wrong place --- that physics, pursued far enough, does not merely describe nature but participates in nature's own self-description.  I am aware how large that sentence is.  It is printed here, rather than in a philosophy journal, because for the first time in my lifetime a sentence of that size has a luminosity attached to it.

\paragraph{On either verdict, and the legibility of the world.}  Let me end with the question beneath all the others: is nature legible --- not merely describable in retrospect, but \emph{readable in advance}, by finite minds, down to its constants?  Fifty years of frontier physics have quietly taught most of us to answer no: to expect that the deepest layer is either an accident (the landscape), or forever beyond our energy reach, or intelligible only statistically.  Every result in this paper is a bet against that resignation.  If the cluster is there, the answer to the legibility question becomes \emph{yes, at least one layer deeper than we had any right to expect} --- and every young physicist inherits a world in which the constants are a text and the instruments are how one turns its pages.  If the cluster is not there, the answer is subtler but not empty: we will have learned that \emph{this} reading fails, learned it cleanly, at a stated cost, with the manner of the failure itself a contribution --- the first zero-parameter exclusion in the history of the energy frontier, a permanent lesson in what such theories must do better.  Legibility, after all, has two requirements: a world with structure, and readers with the discipline to be corrected by it.  Only the first is nature's responsibility.  The second is ours, and it is the one thing this paper could guarantee.  The mass is the bet; the LHC is the table; the cards are already partly dealt, sitting on tape in Geneva.  Let the data speak --- and let us have the grace to listen, whichever way they answer.

What remains after conclusions is service: the appendices that teach the material from scratch, the objections answered one by one, and the references with their reasons --- the parts of a paper that work for the reader long after the argument has ended.


\section[Part~XI in plain English]{Part~XI in plain English: what is being staked, and what it would mean to lose}
A prediction that cannot lose is worth nothing, so it is worth being
plain about what losing looks like here.

If the dedicated search is run on the data that already exists and finds
nothing, and the high-luminosity run finds nothing after it, then the
reading in this paper is finished. There is no version of it with the mass
somewhere else. The integer is not adjustable, the cluster spacing is not
adjustable, and the paper has printed the luminosity at which silence
becomes fatal. That is the whole content of the wager.

If instead the cluster is there, what has been found is not one particle
but evidence that a mass can be derived from counting rather than measured
and accommodated. The specific number would matter much less than that.

The limitations are restated one last time because a conclusion that
quietly drops them is not a conclusion. The rate depends on four named
assumptions. One interference calculation is owed and has not been done.
The composite scenario's recurrence spacing is open. Each is written into
the grade of the prediction it affects.

What is unusual about the position is only its timing. The question does
not need a new machine or a new generation of physicists; it needs a
dedicated analysis of collisions already recorded, and a decade that has
already begun. The cards are dealt and they are sitting on tape in
Geneva.

\GGAcknowledgments

\GGPart[Technical Appendices]{Technical Appendices}{A graduate-level course in the collider physics of the lock cluster --- from-scratch derivations of the KK spectrum, the parton-luminosity cross-section chain, and the statistics of discovery, with worked numerical examples throughout --- preceded by the reference dictionaries and the conceptual primers, and followed by the reinterpretation walkthrough and the promotion roadmap}
\label{part:appendices}

The body of this paper has emphasized the theorem chain and its experimental confrontation.  The present appendices supply the detailed technical content that supports each step, and taken together they are intended to be read as \emph{a graduate-level course in the collider physics of the lock cluster}, addressed to an incoming ATLAS or CMS graduate student who will carry the search forward --- the reader of the Preface, met again at working depth.  We organize them into four groups.  The \emph{reference} appendices (A--D) provide the tabulated abbreviations, key symbols, the cohort naming convention, and the comprehensive summary tables, and serve as the reader's quick lookup.  The five \emph{conceptual primers} (E--I) come next, and a newcomer should read them first: born of questions actually asked, they explain how the gluon and \(g^*\) are one field (Appendix~E), where the baryonic \(Z'_B\) comes from (Appendix~F), what the ``graviton'' of \GG{} is (Appendix~G), why \(g^*\) dominates the cluster's rates and is so broad (Appendix~H), and why one integer controls both the rate hierarchy and the \(\ttbar\) dominance (Appendix~I).  The \emph{constructive} appendices (J--L) are the course proper: a self-contained sequence covering the three competencies this search actually requires --- where a KK spectrum comes from (Appendix~J), how a hadron-collider cross section is computed from parton distributions (Appendix~K), and how a discovery significance is built from the Poisson likelihood up (Appendix~L) --- each treated in \emph{three steps}.  First comes a full from-scratch derivation of the standard result, in the form a graduate student would meet it in a textbook, carried through with no skipped manipulations and closed by a worked numerical example.  Then comes a frank statement of what that standard treatment \emph{assumes but cannot fix} --- the inputs it takes from outside and the questions it leaves open.  Finally comes the \GG{} step, in which the same quantity descends as a derived output of the geometry, and the worked example becomes a number from this paper's tables.  The \emph{operational} appendices give the reinterpretation card with its implementation walkthrough (Appendix~M) and the collider dictionary (Appendix~N); the promotion roadmap for the open items closes the Part (Appendix~O).

Throughout these appendices we adhere to the two stylistic commitments of the cohort's appendix courses \citep{ArisakaA9SM}.  First, every derivation is presented step by step with no skipped intermediate manipulations: a graduate student should be able to reproduce each calculation with paper and pencil (and, for Appendices J--K, with a dozen lines of code).  Second, formal steps are accompanied by parenthetical \emph{physics intuition} notes that translate the algebra into a mental picture.  The combination is deliberate: the appendices have no page budget, because this is where the next generation is meant to learn the material on its own.

\paragraph{Prerequisites and recommended textbooks.}
The course assumes the standard first-year graduate sequence and is meant to be read \emph{alongside} the canonical texts, not in place of them.  Recommended companions, each cited again where it is best matched: \citet{HalzenMartin1984} (\emph{Quarks and Leptons}) for the cross-section formalism of Appendix~K at its most accessible; \citet{EllisStirlingWebber1996} (\emph{QCD and Collider Physics}) as the definitive source for factorization, parton luminosities, and resonance production; \citet{PeskinSchroeder1995} for the field-theoretic underpinnings, including the KK mode decompositions of Appendix~J; \citet{Thomson2013} (\emph{Modern Particle Physics}) for the experiment-aware first pass at everything; and \citet{Cowan1998} (\emph{Statistical Data Analysis}) together with \citet{CowanEtAl2011} for the statistics of Appendix~L.  The recommended use of each constructive appendix: read the matching textbook chapter first, return here for the compact from-scratch recap of Step~1, and then proceed to Steps~2 and~3, where the standard result acquires a derived payload.

\appendix

\section{Summary of Abbreviations}
\label{app:abbrev}

This appendix collects every abbreviation used in the present paper.  All program-defined names are rendered in \emph{sans-serif type} (\GGTheory{}, \GG{}, \GDOS{}, \BIsix{}, \OGNfive{}, \GRIP{}, etc.) so the reader can see at a glance which terms belong to the \GGTheory{} program proper.  Standard physics and collider abbreviations imported from the wider literature (KK, RS, PDF, JES, etc.) are rendered in plain uppercase Roman, following the convention of the major journals; collider terms are expanded further in Appendix~N.  The third column gives the \emph{primary link}: the cohort paper, or the section of the present paper, where each term is introduced in detail.

{\small
\begin{longtable}{@{}>{\raggedright\arraybackslash}p{0.16\linewidth} >{\raggedright\arraybackslash}p{0.56\linewidth} >{\raggedright\arraybackslash}p{0.24\linewidth}@{}}
\caption[Summary of abbreviations used in the present paper.]{Summary of abbreviations used in the present paper. The abbreviations are collected here rather than glossed at first use because the paper is read in more than one order, and a reader entering at the simulation Part should not have to find the introduction to learn what a reduced stream is. Where an abbreviation belongs to the cohort rather than to collider physics, the third column points at the paper that defines it.}
\label{tab:appA-abbrev}\\
\toprule
\textbf{Symbol} & \textbf{Meaning} & \textbf{Primary link}\\
\midrule
\endfirsthead
\multicolumn{3}{c}{\itshape Table~\ref{tab:appA-abbrev} continued}\\
\toprule
\textbf{Symbol} & \textbf{Meaning} & \textbf{Primary link}\\
\midrule
\endhead
\midrule \multicolumn{3}{r}{\itshape continued on next page}\\
\bottomrule
\endfoot
\bottomrule
\endlastfoot
\BIsix{} & The tensor-mediated anomaly-locking Bianchi identity & \AThree{}; \S\ref{subsec:notKKRS}\\
BR & Branching ratio & standard\\
BSM & Beyond the Standard Model & \S1\\
BVP & Boundary-value problem & \S\ref{subsec:tkk2}\\
BW & Breit--Wigner resonance line shape & Appendix~K\\
CEPC & Circular Electron--Positron Collider (China) & Part~\ref{part:beyond}\\
CL & Confidence level & Appendix~L\\
CLIC & Compact Linear Collider & Part~\ref{part:beyond}\\
CTEQ & The CTEQ parton-distribution family (CTEQ6 used here) & \citet{Pumplin2002CTEQ6}\\
DNN & Deep neural network (the boosted-top tagger) & \S\ref{sec:sim-selection}\\
ESPP & European Strategy for Particle Physics & \citet{ESPP2026Adoption}\\
FCC-ee/hh & Future Circular Collider, lepton / hadron stages & Part~\ref{part:beyond}\\
FSR & FCC Feasibility Study Report (2025) & \citet{FCC2025FSR}\\
\textsf{GA} & Geometric Attractor & \ASixGRIP{}\\
\GDOS{} & The open-system boundary law & \AFive{}\\
\GG{} / \textsf{GG-6} & Geometry--Gauge (locking); the six-dimensional bulk & \ATwo{}\\
\textsf{GGEP} & Geometry--Gauge Equivalence Principle (AX-2) & \AOne{}\\
\GRIP{} & The runtime cascade; regimes \textsf{GRIP-In}/\textsf{GRIP-Out} & \ASixGRIP{}\\
\textsf{GU-5} & The five Geometric Universal constants & \AOne{}; \S\ref{sec:t-kk-1}\\
HGTD & High-Granularity Timing Detector (ATLAS Phase-2) & \citet{ATLAS2024HLUpgrade}\\
HL-LHC & High-Luminosity LHC & \citet{CERN2023HLLHC}\\
HLT & High-Level Trigger & \S\ref{sec:sim-trigger}\\
ILC & International Linear Collider & Part~\ref{part:beyond}\\
IMCC & International Muon Collider Collaboration & \citet{IMCC2024Interim}\\
IR / UV & Infrared / ultraviolet ends of the throat (\(r=K\) / \(r=0\)) & \S\ref{subsec:notKKRS}\\
ITk & ATLAS Inner Tracker (Phase-2) & \citet{ATLAS2024HLUpgrade}\\
JES & Jet energy scale & \S\ref{sec:sim-impsyst}\\
K-factor & QCD higher-order normalization (\(K_{q\bar q}=1.3\), \(K_{gg}=1.6\)); unrelated to the \OGNfive{} integer \(K=35\) & \S\ref{sec:operational}\\
KK & Kaluza--Klein & \S\ref{subsec:notKKRS}; Appendix~J\\
LEE & Look-elsewhere effect & \S\ref{sec:sim-lee}\\
LHeC & Large Hadron--electron Collider (concept) & Part~\ref{part:beyond}\\
LO / NLO & Leading / next-to-leading order & Appendix~K\\
LS3 & Long Shutdown 3 (2026--2030) & \S\ref{sec:sim-equiv}\\
MTD & MIP Timing Detector (CMS Phase-2) & \citet{CMS2019MTD}\\
NWA & Narrow-width approximation & Appendix~K\\
\textsf{OGGEP} & Open-system \textsf{GGEP}: the bulk-to-boundary projection (AX-3) & \AOne{}; Appendix~G\\
\OGNfive{} & The five-integer ledger \((M,N,L;J,K)=(1,3,5;7,35)\) & \AThree{}\\
PDF & Parton distribution function & Appendix~K\\
PDG & Particle Data Group & \citet{PDG2024}\\
RG & Renormalization group & \S\ref{subsec:notKKRS}\\
RS & Randall--Sundrum & \citet{RandallSundrum1999}; \S\ref{subsec:notKKRS}\\
\textsf{SIP} & System instance packet (a full measurement chain in \GDOS{} language) & \AFive{}; \S\ref{sec:gdos-section}\\
SM & Standard Model & standard\\
SR & Signal region & \S\ref{sec:sim-anchor}\\
TDR & Technical Design Report & Part~\ref{part:beyond}\\
TLA & Trigger-level analysis (ATLAS) & Table~\ref{tab:trigger}\\
WP & Working point (of a tagger) & \S\ref{sec:sim-selection}\\
\end{longtable}
}

\section{Glossary of key symbols}
\label{app:symbols}

\small
\setlength{\tabcolsep}{4pt}
\begin{longtable}{>{\raggedright\arraybackslash}p{2.5cm} >{\raggedright\arraybackslash}p{8.2cm} >{\raggedright\arraybackslash}p{2.8cm}}
\caption[The symbols of this paper at a glance, with the section or companion that defines each.]{The symbols of this paper at a glance, with the section or companion that defines each. It is meant to be read beside the paper rather than before it, and the third column is the part that earns its place: a symbol whose home is a companion paper is one this paper consumes rather than derives, and the distinction matters for every grade in the provenance table. Symbols introduced and discharged inside a single derivation are left out.}\label{tab:appA-symbols}\\
\toprule
\textbf{Symbol} & \textbf{Meaning} & \textbf{Home}\\
\midrule
\(\Mlock\) & The lock-cluster mass, \(\tfrac12 M_{\rm Pl}e^{-K}=3.849\)~TeV & T-KK-1, \S\ref{sec:t-kk-1}\\
\(K,J\) & \OGNfive{} integers: throat length \(K=35\); \(J=7\) & \S\ref{sec:t-kk-1}\\
\(r,\ \theta\) & The non-spatial coordinates: RG-scale \(r=\ln(\EGUT/\mu)\); phase angle & \S\ref{subsec:notKKRS}\\
\(x_n\) & Bessel zeros of \(J_1\): standing-wave conditions; \(x_1=3.8317\) & Eq.~\eqref{eq:tkk1-tower}\\
\(m_n\) & Radial tower masses \((x_n/x_1)\Mlock\): 3.849, 7.047, 10.219~TeV & \S\ref{sec:t-kk-1}\\
\(\xi_{\rm UV}\) & Light-quark (UV-endpoint) coupling amplitude, \(-0.194\) & T-KK-2, \S\ref{subsec:tkk2}\\
\(r_{t_L}, r_{t_R}\) & Top localization band: \(1.0\), \([4,5]\) (central 4.5) & \S\ref{subsec:tkk2}\\
\(g_V^t, g_A^t\) & Top vector/axial couplings, \(2.75\,g_s\), \(1.75\,g_s\) & Eq.~\eqref{eq:tt-gVgA}\\
\(\kappa_{gg}\) & Graviton--gluon coupling, \(0.47/K=0.0136\) (derived) & T-KK-2c, \S\ref{subsec:kappagg}\\
\(g_B\) & \(\UB\) coupling; discrete menu (open lemma L-KK-1) & \S\ref{subsec:gbmenu}\\
\(\Gamma_{g^*},\ \Gamma_{G_1}\) & Widths: 561~GeV (\(\Gamma/m=14.6\%\)); 243~GeV (\(6.3\%\)) & Part~\ref{part:ttanatomy}\\
\(\beta_t\) & Top velocity at the pole, \(0.99597\) & Part~\ref{part:ttanatomy}\\
\(\theta^*\) & Production angle in the \(\ttbar\) rest frame & Part~\ref{part:ttanatomy}\\
\(P_t,\ A_{FB}\) & Top polarization \(+0.91\,[0.88,0.92]\); forward--backward asymmetry \(=0\) & Part~\ref{part:ttanatomy}\\
\(S, I, B\) & Breit--Wigner, interference, background yields per bin & Part~\ref{part:hllhcsim}\\
\(f_b\) & Fractional per-bin background systematic (post-fit) & \S\ref{sec:sim-nuisance}\\
\(Z,\ q_0\) & Discovery significance; profile-likelihood test statistic & \S\ref{sec:sim-stats}\\
\(F_{\rm cal}\) & Calibration factor to the real analysis, \(4.33\) & \S\ref{sec:sim-calib}\\
\(L^*\) & Statistics-to-systematics crossover luminosity, \(1/(f_b^2 b_{1\rm fb})\) & \S\ref{sec:sim-wall}\\
\(\varepsilon A\) & Signal efficiency \(\times\) acceptance, \(0.065\) & \S\ref{sec:sim-effacc}\\
\(N_{\rm trials}\) & Look-elsewhere trials factor, \(\approx75\) for the generic scan & \S\ref{sec:sim-lee}\\
\(v\) & Electroweak scale, \(\Mlock(2/5)^3\approx246\)~GeV & \S\ref{sec:t-kk-1}\\
\(\EGUT,\ M_{\rm Pl}\) & UV endpoint \(\approx2.28\times10^{16}\)~GeV; Planck energy & \S\ref{sec:constitutional-route}\\
\bottomrule
\end{longtable}


\section{Global naming convention for cross-cohort citation}
\label{app:naming}

Following \citet{ArisakaA9SM} (its Appendix~C) and beginning with \citet{ArisakaA2GG6}, each load-bearing item of the \GGTheory{} program carries a globally unique identifier of the form \emph{Prefix-Suffix-N}, where the Prefix names the item type and the Suffix names the paper:
\begin{itemize}[leftmargin=2em, itemsep=2pt]
\item \textbf{T-Suffix-N} for the \(N\)th Theorem (e.g., T-KK-1 is the lock-cluster mass theorem of this paper).
\item \textbf{L-Suffix-N} for the \(N\)th Lemma (e.g., L-KK-1 is the open \(g_B\)-menu selection lemma).
\item \textbf{C-Suffix-N} for the \(N\)th Corollary (this paper's T-KK-2c retains its historical theorem-attached name).
\item \textbf{D-Suffix-N} for the \(N\)th Definition.
\item \textbf{R-Suffix-N} for the \(N\)th Remark.
\item \textbf{P-Suffix-N} for the \(N\)th Prediction (P-KK-1\,\ldots\,P-KK-10, \S\ref{sec:disc-predictions}).
\item \textbf{O-Suffix-N} for the \(N\)th Objection (O-KK-1\,\ldots\,O-KK-58, Part~\ref{part:objections}).
\item \textbf{F-Suffix-N} for the \(N\)th Falsifier (F-KK-1 and the two-scenario pair F-KK-2A/2B).
\item \textbf{I-Suffix-N} for the \(N\)th Innovation (I-KK-1\,\ldots\,I-KK-10, \S\ref{sec:disc-innovations}).
\item \textbf{PS-Suffix-N} for the \(N\)th Problem-Solved entry (PS-KK-1\,\ldots\,PS-KK-6, \S\ref{sec:disc-puzzles}).
\end{itemize}
The canonical Suffix list for the twenty-two cohort papers is: \emph{Constitution} (A1), \emph{GG6} (A2), \emph{BI6} (A3), \emph{Euclid} (A4), \emph{GDOS} (A5), \emph{GRIP} (A6), \emph{Qi4} (A7), \emph{QM} (A8), \emph{SM} (A9), \emph{Cosmos} (A10), \emph{Origin} (A11), together with the phenomenology preprints \emph{KK} (P1, the present paper), \emph{N3} (P2), \emph{Electron} (P3), \emph{Proton} (P6), the foundations preprints \emph{Real} (P4) and \emph{CPT} (P5), and the consequence preprints \emph{Kaxion} (C1), \emph{Hubble} (C2), \emph{BH} (C3), and the white papers \emph{Symmetry} (W1) and \emph{Holonomy} (W2).  The four axioms (AX-1\,\ldots\,AX-4) and seven postulates (POS-1\,\ldots\,POS-7) of \citet{ArisakaA1Constitution} keep their existing cohort-wide names; the global identifier is the same item viewed from outside its home paper.

\section{Comprehensive summary tables}
\label{sec:summarytables}

This appendix gathers the paper's load-bearing conclusions into three tables, so that the paper can serve as a working reference document: the phenomenology ledger of the low-lying collider sector (Table~\ref{tab:master1}), the falsification hierarchy (Table~\ref{tab:falsification}), and the discovery calendar established by the simulation of Part~\ref{part:hllhcsim} (Table~\ref{tab:calendar-recap}).  A fourth summary object --- the framework discrimination matrix --- lives with the comparison discussion (Table~\ref{tab:discrimination}, \S\ref{sec:disc-comparison}).  Because these tables will be consulted by readers who have not read the Parts that generated them, each is introduced with a short guide to what it contains, where its entries were derived, and how it should --- and should not --- be read.  For the reader with one minute, Table~\ref{tab:glance} is the paper.

\begin{table}[H]
\centering\small
\caption[The ten numbers at a glance]{The ten numbers at a glance: the paper's load-bearing quantities, each with its provenance.  Every entry is derived or bracketed as indicated; none is fitted. Ten numbers is the number a reader should be able to carry away, and the provenance column is why the list is short. Every entry can be traced to the theorem or the assumption that produces it, and an entry that could not be traced would not be in the table. None is fitted, which is a statement about the method rather than a boast about the result.}
\label{tab:glance}
\begin{tabular}{@{}rll@{}}
\toprule
 & \textbf{Quantity} & \textbf{Provenance}\\
\midrule
1 & \(\Mlock=\tfrac12 M_{\rm Pl}e^{-35}=3.849\)~TeV & T-KK-1 (\S\ref{sec:t-kk-1})\\
2 & \(v=\Mlock(2/5)^3=246.3\)~GeV & over-constraint (\S\ref{sec:constitutional-route})\\
3 & \(\xi_{\rm UV}=-0.194\) (light quarks) & T-KK-2 (\S\ref{subsec:tkk2})\\
4 & \(+2.75\,g_s\) (top) & Read-out II bracket (\S\ref{sec:scenarioB})\\
5 & \(\Gamma/m=14.6\%\); \(\mathrm{BR}(\ttbar)=94\%\) & width table (Appendix~I)\\
6 & \(\sigma\times\mathrm{BR}(\ttbar)=8.7\)~fb (13~TeV); \(13.6\)~fb (14~TeV) & T-KK-2c chain (\S\ref{sec:operational})\\
7 & \(Z'_B\): \(0.030\times(g_B/0.2)^2\)~fb (13~TeV) & L-KK-1 menu (\S\ref{subsec:gbmenu})\\
8 & \(G_1\): \(0.045\)~fb (13~TeV); \(\kappa_{gg}=0.47/K\) & T-KK-2c (\S\ref{subsec:kappagg})\\
9 & radial tower \(3.849/7.047/10.219\)~TeV & \(x_n/x_1\) (\S\ref{sec:t-kk-1})\\
10 & kill window \(\lesssim2\)~ab\(^{-1}\); decision 2026--2035 & F-KK-2B (\S\ref{subsec:killwindow})\\
\bottomrule
\end{tabular}
\end{table}


\paragraph{How to read the master ledger (Table~\ref{tab:master1}).}  The ledger answers, for each member of the low-lying spectrum, the four questions an analyst asks first: what is it, where is it, how does it decay, and why does it matter to the search.  Three reading rules.  \emph{First}, every mass in the table is Scenario~B --- the composite-hardened reading that survived the data --- with Scenario~A's excluded 2.46-TeV placement retained only in the historical record (\S\ref{subsec:tkk2}, \S\ref{sec:operational}).  \emph{Second}, the ``Best channels'' column encodes the coupling texture, not a preference: the octet decays to \(\ttbar\) at 94\% because the top is the only fermion at the infrared tip (Appendix~I), and the same geometry empties the dilepton and diphoton columns that a reader trained on RS benchmarks will look for.  \emph{Third}, the two entries an experimentalist should internalize are the first and the last: the octet row is the entirety of the near-term falsifier (F-KK-2B), and the angular-tower row is a prediction of \emph{absence} --- the row is in the table precisely because finding anything there would falsify the non-spatial reading (P-KK-7).

\paragraph{How to read the falsification hierarchy (Table~\ref{tab:falsification}).}  The hierarchy orders null results by what they would actually kill, and its purpose is to prevent the two symmetric misreadings of this program: premature celebration (``a dilepton null already tests \GG{}'' --- it does not; the framework predicts that null) and premature burial (``the octet band is crossed, so the theory survives on technicalities'' --- it does not; level four is written to be terminal).  The levels ascend in decisiveness because they ascend in \emph{coupling-texture coverage}: a single clean channel constrains one overlap; the top-plus-dijet-plus-\(hh\) combination constrains the pattern; the full cluster-aware combination across the geometry-allowed corridor is the kill (F-KK-2B's operational form).  The fifth row is the mirror case that a reader may miss: a \emph{discovery} of the wrong morphology --- a single clean RS-like pole with none of the cluster correlates --- would challenge this paper as sharply as any null, because the fingerprint, not the bump, is the claim.

\paragraph{How to read the calendar (Table~\ref{tab:calendar-recap}).}  The calendar compresses Part~\ref{part:hllhcsim} to four rows, and its columns carry the whole logic of that Part: the per-experiment pair (frozen/improved) brackets the background-systematics future, the combined column is the realistic ATLAS+CMS decision variable with its correlation bracket, and the milestone column converts luminosity to consequence.  Two warnings for the casual reader.  The differences \emph{within} rows dwarf the differences \emph{between} rows --- moving from frozen to improved systematics is worth more than a decade of additional running (the wall of \S\ref{sec:sim-wall}) --- so the table is an argument about analysis investment, not about patience.  And the first row is not a projection: end-2026 is data already on tape, which is why the paper's operational request (the dedicated reanalysis) carries the urgency it does.

\begin{table}[H]
\centering
\caption[Master phenomenology ledger for the low-lying \GG{} collider sector.]{Master phenomenology ledger for the low-lying \GG{} collider sector (all entries at their derived values; Scenario~B throughout, Scenario~A being excluded). Scenario~B throughout, because Scenario~A is excluded by the public limits of Part~VII, and the table says so in its own first line rather than leaving the reader to work out which branch is live. Every entry is at its derived value, so the ledger can be checked row by row against the sections that derive it.}
\label{tab:master1}
\small
\begin{tabularx}{\textwidth}{>{\raggedright\arraybackslash}p{3.0cm} >{\centering\arraybackslash}p{1.1cm} >{\centering\arraybackslash}p{2.1cm} >{\raggedright\arraybackslash}p{3.4cm} X}
\toprule
\textbf{Object} & \textbf{Spin} & \textbf{Mass} & \textbf{Best channels} & \textbf{Role}\\
\midrule
Color-octet KK gluon \(g^*\) & 1 & \(\Mlock=3.849\)~TeV (T-KK-2) & \(\ttbar\) (BR 0.94), dijets & The un-suppressible member: \(\sigma\times\mathrm{BR}(\ttbar)=8.7\)~fb (13~TeV); carries the LHC falsifier F-KK-2B.\\
Baryonic vector \(Z'_B\) & 1 & \(=m_{g^*}\) exactly (same BVP) & \(\ttbar\), dijets; \emph{no leptons} & Exactly leptophobic (theorem); 0.2--6.8\% of the cluster \(\ttbar\) sum across the L-KK-1 menu.\\
Graviton anchor \(G_1\) & 2 & \(\Mlock\) exactly (T-KK-1) & \(\ttbar:W_LW_L:Z_LZ_L:hh\) = 0.69:0.15:0.08:0.08 & Names the geometry; \(\kappa_{gg}=0.47/K\) (T-KK-2c) makes it FCC-hh physics (0.13~pb at 85~TeV).\\
Second radial family \(G_2\) & 2 & \(1.831\,\Mlock=7.047\)~TeV (Bessel ratio) & \(\ttbar\), dijets & The overtone that confirms the cavity; FCC-hh spectroscopy (Part~\ref{part:beyond}).\\
Angular \(\theta\) tower & --- & \(\sim\EGUT\) & none & Predicted \emph{absent} at the TeV scale: no angular satellites (a falsifiable absence).\\
\bottomrule
\end{tabularx}
\end{table}

\begin{table}[H]
\centering
\caption[Falsification hierarchy for the collider program.]{Falsification hierarchy for the collider program.  The hierarchy distinguishes between a single-channel null result, which is usually not enough, and a combined geometry-aware null result, which is decisive. The distinction the hierarchy draws is the one collider nulls are most often read wrongly against. A single channel returning nothing constrains one production mode at one set of couplings; the cluster reading survives that and should. What it does not survive is a geometry-aware combination across the members, and the table states the threshold at which that becomes decisive.}
\label{tab:falsification}
\small
\begin{tabularx}{\textwidth}{>{\raggedright\arraybackslash}p{4.8cm} X}
\toprule
\textbf{Level} & \textbf{Meaning}\\
\midrule
Single-channel exclusion & Constrains branching or overlap structure; not yet decisive because \GG{} is non-universal.\\
Two-channel null (e.g. dilepton + diphoton) & Strongly disfavors universal or clean-channel-dominated versions, but still not enough for the full \GG{} benchmark.\\
Hadronic-top + dijet + Higgs-pair null in the first-cluster window & Severely constrains the geometry-induced coupling pattern.\\
Full cluster-aware combined-channel exclusion across the geometry-allowed coupling corridor & Collider-level falsification of the current lock-scale implementation of \GG{}.\\
Discovery of a clean single RS-like pole at an incompatible mass pattern & Would challenge the specific \GG{} spectral grammar even if it supports some extra-dimensional physics more generally.\\
\bottomrule
\end{tabularx}
\end{table}

\begin{table}[H]
\centering
\caption[The discovery calendar.]{The discovery calendar (recap of Part~\ref{part:hllhcsim}; calibrated significances, dedicated matched-template search). A recap rather than a new result: every significance here is computed in the simulation Part and is reproduced only so the schedule can be read in one place. The dates are machine schedules and not this paper's estimates, and the significances are for a dedicated matched-template search rather than for a reinterpretation of a published limit.}
\label{tab:calendar-recap}
\small
\begin{tabular}{lccl}
\toprule
\textbf{Snapshot} & \textbf{1 expt (froz./impr.)} & \textbf{ATLAS+CMS (impr.)} & \textbf{Milestone}\\
\midrule
end-2026 & \(2.7/3.7\sigma\) & \(4.6\)--\(5.2\sigma\) & evidence on tape; \(5\sigma\) a live possibility\\
end-2031 & \(3.1/4.5\sigma\) & \(5.4\)--\(6.3\sigma\) & single-expt \(5\sigma\) luminosity on tape 2032\\
end-2033 & \(3.3/5.4\sigma\) & \(6.1\)--\(7.6\sigma\) & discovery or exclusion decided in practice\\
end-2041 & \(3.5/6.5\sigma\) & \(6.9\)--\(9.1\sigma\) & property program saturated\\
\bottomrule
\end{tabular}
\end{table}

\section{What is the relation between the QCD gluon and the color-octet \texorpdfstring{$g^*$}{g*}?}
\label{app:gluonrelation}

\noindent\textbf{\emph{What is the relation between the gluon of Standard-Model QCD and the color-octet vector \(g^*\) of \GG{}-Theory?}}\medskip

\noindent It is a question with a clean answer, because in \GG{} the relation is as tight as it could possibly be: \textbf{the gluon and \(g^*\) are the same field} --- two standing waves of the single bulk \(\mathrm{SU}(3)_C\) gauge field, the ground note and the first overtone of one instrument.  Everything below is implicit in Theorem T-KK-2 (\S\ref{subsec:tkk2}); this appendix assembles it in one place.

\paragraph{The construction.}  The bulk \(\mathrm{SU}(3)_C\) field lives on the six-dimensional stage and, like every bulk field, decomposes into modes along the scale throat (Appendix~J).  The boundary conditions are fixed by an observed fact --- their zero-mode sector must \emph{be} the massless gauge fields we see --- and that zero mode, with its exactly flat profile along \(r\), \textbf{is the QCD gluon}.  The first radial excitation of the same field --- one node of standing-wave structure along the RG direction, profile concentrated at the infrared tip, mass set by the Bessel condition --- \textbf{is \(g^*\)}.  Same color octet, same spin, same Lorentz structure, one field.

\paragraph{Why one is massless and the other weighs 3.85 TeV.}  The zero mode is flat, so it costs no gradient energy along the throat; its masslessness is exact, protected by the unbroken boundary gauge invariance.  The excited mode pays standing-wave energy in the scale direction, redshifted to the infrared endpoint --- mass from geometry, with \emph{no Higgs mechanism and no new coupling constant anywhere}.  Every \(g^*\) coupling is \(g_s\) times a derived overlap number.

\paragraph{Why they behave so differently despite being one field.}  The profiles.  The gluon's flat profile overlaps every quark \emph{identically} --- which means QCD's flavor blindness, an axiom in the Standard Model, is here a \emph{theorem about the ground state}: universality is a property of flatness.  The overtone has no such protection: its IR-localized profile overlaps the UV-localized light quarks at \(-0.194\,g_s\) and the IR-localized top at \(+2.75\,g_s\) --- the entire top-philic phenomenology of this paper is nothing but the statement that flavor blindness dies with the first node.

\paragraph{The consequence the paper exploits throughout.}  Because \(g^*\) carries exactly the gluon's quantum numbers, it enters \(q\bar q\to\ttbar\) in the \emph{same color channel} as the \(s\)-channel gluon --- which is why the two amplitudes add coherently (Part~\ref{part:ttanatomy}) and why the interference-sign theorem exists at all.  The color-singlet \(Z'_B\), by contrast, cannot interfere with QCD; that yes/no dichotomy inside one peak is a direct fingerprint of the ``same field'' relation (O-KK-28).

\paragraph{In the composite (Scenario B) reading.}  The statement transposes but survives: the gluon is the elementary boundary field, and \(g^*\) is the composite spin-1 resonance of the boundary theory carrying the gluon's quantum numbers --- precisely the relation of the \(\rho\) meson to the photon in hadronic physics (the same analogy carried, for spin 2, in Appendix~G).  The \(\rho\) is not a ``new photon''; it is what the strong dynamics does in the photon's channel.  Likewise \(g^*\) is not a new gauge boson --- no extended color group, no axigluon-style broken \(\mathrm{SU}(3)\times\mathrm{SU}(3)\), no new charge --- it is what the throat does in the gluon's channel.

\paragraph{One instructive contrast within the cluster itself.}  \(g^*\) has a massless sibling (the gluon), because \(\mathrm{SU}(3)_C\) has a zero mode; \(Z'_B\) does \emph{not} --- the \(\UB\) vector is St\"uckelberg-massive with no massless partner, which is why baryon number generates no long-range force and the proton is stable \citep{ArisakaA9SM}.  So the cluster's two vectors, degenerate in mass, have opposite genealogies: one is an excited copy of a field known since 1979; the other is the lowest note of a field the Standard Model never had.

If a seminar audience asks ``is \(g^*\) a heavy gluon?'', the precise answer is: \emph{it is the gluon --- heard one octave up the scale dimension.}


\section{What is the origin of the baryonic vector \texorpdfstring{$Z'_B$}{Z'B}?}
\label{app:zborigin}

\noindent\textbf{\emph{What is the origin of the baryonic vector boson \(Z'_B\)?  Presumably it is the outcome of local \(\mathrm{U}(1)_B\) gauge invariance --- but why must it sit at \(3.85\)~TeV, degenerate with \(g^*\) and the graviton anchor \(G_1\)?  And is it related to the \(Z\) boson of the weak interaction, or to the photon of \(\mathrm{U}(1)_{\rm EM}\)?}}\medskip

\noindent The guess is exactly right, and the interesting physics is in \emph{why} the gauging is forced, and why the mass has nothing to do with the field.

\paragraph{Baryon number must be gauged --- it is not a choice.}  In the Standard Model, baryon number is an \emph{accidental} global symmetry: nothing enforces it, and it is famously fragile against anything one adds.  In \GG{} the situation is inverted.  The \BIsix{} anomaly ledger of \AThree{} \citep{ArisakaA3BI6} closes the six-dimensional tensor/gauge system only if the bulk gauge set includes \(\UB\) alongside color and the electroweak factors --- the anomaly inflow of the 6D theory does not balance without it.  So \(Z'_B\) is indeed the gauge boson of local \(\UB\) invariance, but gauged out of \emph{consistency}, not model-builder's taste.  And the same ledger is why it earns its keep in the electroweak over-constraint: the trace ratio \(2/5\) that forces \(v=\Mlock(2/5)^3\approx246\)~GeV (\S\ref{sec:constitutional-route}) counts \(\UB\) on equal footing with \(\UQi\).  Remove the baryonic factor, and the theory's proudest number breaks.

\paragraph{The obligatory technical question: the mixed anomalies.}  A field theorist will interject here: gauging baryon number in the Standard Model \emph{alone} is famously obstructed, because the mixed triangles do not vanish --- \(\mathcal A[B\text{--}\SU(2)_L^2]=+\tfrac32\) and \(\mathcal A[B\text{--}\U(1)_Y^2]=-\tfrac32\) over three generations (the gravitational, cubic, and \(B^2Y\) triangles are identically zero, since \(B\) is vector-like on the quarks) --- and the conventional price is a set of new chiral ``anomalon'' fermions.  \GG{} pays no such price: the boundary spectrum descends from unsplit \(\mathrm{SU}(5)\) multiplets whose anomaly polynomial factorizes to rank one, and the two surviving triangles are canceled by the inflow of the single Green--Schwarz counterterm \(\int B_2\wedge X_4\) --- the mechanism proved in \AThree{} and spelled out there in four-dimensional language \citep{ArisakaA3BI6}.  The four-dimensional remnant of \(B_2\) is precisely the St\"uckelberg axion of the previous paragraph: its shift cancels the triangles \emph{and} gives \(Z'_B\) its mass --- one coupling, two consequences --- and no anomalon exists anywhere in the spectrum (O-KK-47).

\paragraph{Why 3.85 TeV: the mass belongs to the geometry, not the field.}  This is the cluster theorem's central intuition (\S\ref{sec:t-kk-1}).  The scale throat is one cavity --- a pipe of integer length \(K=35\).  Every infrared-localized standing wave in that pipe, whatever field it belongs to and whatever its spin, sits at the pipe's pitch, \(\Mlock=\tfrac12 M_{\rm Pl}e^{-35}\).  The spin-2 sector puts its first note there (\(G_1\)); \(\mathrm{SU}(3)_C\) puts its first \emph{overtone} there (\(g^*\), Appendix~E); and \(\UB\) puts its first note there (\(Z'_B\)).  The degeneracy is not a coincidence to be explained; it is the statement that one geometry owns all three masses.  Hadronic physics supplies the perfect precedent: the \(\rho(770)\) and \(\omega(782)\) are near-degenerate not because vector mesons conspire, but because \(\Lambda_{\rm QCD}\) sets both.  In the composite (Scenario~B) reading this is literal --- all three cluster members are resonances of one boundary theory at its single infrared scale.

\paragraph{The crucial asymmetry with \texorpdfstring{$g^*$}{g*}: no massless sibling.}  The gluon exists because the boundary conditions of \(\mathrm{SU}(3)_C\) preserve a flat zero mode (Appendix~E).  For \(\UB\) the boundary conditions do the opposite: the would-be zero mode is removed --- the vector is St\"uckelberg-massive, the abelian mechanism that gives a \(\mathrm{U}(1)\) gauge field a mass without breaking the symmetry and without any Higgs \citep{ArisakaA9SM}.  This is not a technicality; it is forced by observation.  A massless ``baryonic photon'' would mediate a long-range force coupled to baryon number, and E\"otv\"os-type equivalence-principle experiments exclude such a force to fantastic precision.  So the theory \emph{must} deliver \(\UB\) gauged (the anomaly ledger demands it) and yet with no long-range force (the torsion balances demand that) --- and the St\"uckelberg construction is precisely the reconciliation.  The price is one massive vector; and since the lightest note the pipe allows it is the pipe's own, that vector lands at \(3.849\)~TeV.  As a bonus, the surviving gauge invariance still enforces baryon-number conservation: the proton stays stable as a matter of \emph{law}, where the Standard Model could only promise it by accident.

\paragraph{Relation to the \texorpdfstring{$Z$}{Z}: essentially none.}  The \(Z\) is the neutral member of \(\mathrm{SU}(2)_L\times\mathrm{U}(1)_Y\); its mass comes from electroweak symmetry breaking --- in \GG{}, the Wilson-line mechanism on \(S^1_\theta\) (Hosotani mechanism; \citealp{Hosotani1983}) --- and it couples to weak isospin and hypercharge, hence generously to leptons.  \(Z'_B\) is \emph{not} a heavier \(Z\): different group, different mass mechanism (geometry versus symmetry breaking), and it couples to baryon number only --- \(1/3\) per quark, universally, and \emph{exactly zero} to leptons.  That is why the leptophobia is a prediction rather than a tuning: \(\mathrm{BR}(\ell\ell)=0\) identically, which is also what has kept it invisible to the powerful dilepton searches (\S\ref{sec:current-status}).  The only genuine link to the \(Z\) is upstream: the same \(2/5\) trace arithmetic that involves \(\UB\) fixes \(v\), and \(v\) fixes \(M_Z\).

\paragraph{Relation to the photon: the closest cousin, with the opposite boundary fate.}  Both are abelian \(\mathrm{U}(1)\) vectors, so as fields they are siblings.  But \(\mathrm{U}(1)_{\rm EM}\) keeps its zero mode --- unbroken, massless, long-range --- while \(\UB\) has its zero mode deleted.  One way to say it: the photon is a fundamental the pipe lets ring; \(Z'_B\) is a field whose fundamental key is blocked, so the first sound it can make \emph{is} the pipe's note.  (Its coupling strength, the discrete \(g_B\) menu, is the subject of L-KK-1.)

The three cluster members therefore carry three distinct genealogies converging on one mass: \(g^*\) is the excited copy of a field known since 1979; \(G_1\) is the excited copy of the metric sector (Appendix~G); and \(Z'_B\) is the lowest note of a field the Standard Model never had --- the only genuinely \emph{new} gauge boson in the cluster, whose existence is an anomaly-ledger necessity and whose silence in dileptons is its signature.

\section{What is the ``graviton'' of GG-Theory?}
\label{app:graviton}

The word ``graviton'' is doing two different jobs in the literature, and \GG{} adds a third, genuinely new reading.  Because the question ``is your \(G_1\) \emph{the} graviton?'' is guaranteed to be asked, this appendix separates the meanings once, cleanly.

\paragraph{Two meanings already in circulation.}  The graviton of quantum gravity --- string theory's graviton --- is the \emph{massless}, helicity-2 quantum of the quantized four-dimensional gravitational field.  Its existence presupposes that gravity itself is a quantum field, and the notorious difficulties (non-renormalizability, ultraviolet completion) live there.  But the ``KK graviton'' of collider phenomenology has never been that object: a massive spin-2 resonance arising from the mode decomposition of a higher-dimensional metric is an \emph{ordinary particle}, and quantizing a massive spin-2 field on a fixed background is standard field theory, no harder in principle than the \(W\) boson.  Every ATLAS and CMS ``graviton search'' ever performed has searched for the second kind, not the first.  Discovering \(G_1\) would therefore not certify that gravity is quantized in the string-theoretic sense; it would certify that the \emph{metric sector has a massive discrete mode}.

\paragraph{What \(G_1\) is, specifically.}  \(G_1\) is the lowest radial standing wave of the spin-2 sector of the bulk --- and in the renormalization-group reading this paper insists on, it is literally a \emph{composite spin-2 resonance of the boundary theory at its infrared scale}, the analogue of a spin-2 hadron.  Its quantum status is exactly as unproblematic as the \(\rho\) meson's.  It is called ``graviton'' by \emph{pedigree}, not by job description: it descends from the bulk metric --- the \(\mathrm{Spin}(1,5)\) block of the composite connection that \citet{ArisakaA2GG6} forces --- which is why its spin-2-ness, its universal coupling structure to \(T^{\mu\nu}\), and its \(W_LW_L\)-rich branching are inherited rather than chosen.  That pedigree is why the paper calls it ``the state that names the geometry.''

\paragraph{The \OGGEP{} inversion.}  \GGEP{} (AX-2) does \emph{not} quantize gravity; it makes a classical, structural statement --- geometry and gauge are connections of the same species, locked by BI-6.  Quantization enters at AX-3: \OGGEP{} makes the bulk-to-boundary projection \emph{be} observation, and \citet{ArisakaA8QM} recovers quantum mechanics \emph{from} that projection rather than postulating it.  \GG{} thereby inverts the fifty-year program: one does not quantize geometry; \emph{quantization is what geometry does when it is projected}.  The mechanism is concrete --- the gauge phase of charged matter is the holonomy of the \(\theta\) fiber, \(\hbar\) is the compact circle's unit (\(M=1\) in the ledger), and discreteness (of charges, of the radial tower in \(n\)) is the character and standing-wave structure of the two non-spatial dimensions.  In this sense the graviton and the photon are quantized \emph{in the same way, by the same projection}: both are modes of one composite connection, and neither requires a separate act of quantization.

\paragraph{The honest fence.}  What \GG{} does \emph{not} thereby deliver is the full quantum dynamics of the massless four-dimensional graviton --- graviton--graviton scattering near the lock scale, the deep-ultraviolet behavior.  That is exactly what \citet{ArisakaA9SM} registers as its open structural item \(O_{14}\): the composite connection resolves the structural unification, but the dynamical quantum-gravity calculation is not performed anywhere in the cohort.  The clean summary: \emph{discovering \(G_1\) at \(3.849\)~TeV would be evidence for the six-dimensional geometric origin of the world --- not a detection of ``the graviton of quantum gravity'' --- and it neither requires nor proves that massless 4D gravity is quantized.}  What it would prove is stronger in one way (it validates the geometry that makes quantization itself an output) and weaker in another (the string-theoretic question remains deferred, and honestly so).

\paragraph{The massive spin-2 field, technically.}  For completeness, the field-theory status of a massive spin-2 particle deserves one careful paragraph, because ``spin-2 is dangerous'' folklore conflates three separate issues.  A \emph{free} massive spin-2 field has a perfectly healthy description (Fierz--Pauli); the notorious pathologies (the vDVZ discontinuity, the Boulware--Deser ghost) afflict attempts to make a massive graviton \emph{replace} the massless one as the carrier of long-range gravity.  \(G_1\) does no such thing: the massless zero mode remains the graviton of gravity, and \(G_1\) is an additional heavy state whose interactions are cut off at the throat scale --- exactly the situation of every RS phenomenology paper, where the KK graviton is treated by ordinary perturbation theory without apology.  \emph{(Physics intuition: nobody worries that the \(\rho\) meson, a massive spin-1 state, endangers the masslessness of the photon; \(G_1\) stands to the graviton as the \(\rho\) stands to the photon --- a heavy resonance in the same channel, not a rival.)}  In the composite reading this is even more manifest: a spin-2 resonance of a strongly coupled boundary theory needs no fundamental spin-2 field at all, any more than the \(f_2(1270)\) does.

\paragraph{What discovery would and would not settle, item by item.}  It is worth tabulating the epistemics, because they are asymmetric.  Discovery of \(G_1\) with the derived pattern \emph{would} establish: a massive spin-2 state exists (from the angular analysis of Part~\ref{part:ttanatomy}); its couplings follow stress-energy weighted by localization (from the \(W_LW_L\)-rich pattern); and the six-dimensional geometric reading has produced a correct, parameter-free postdiction chain.  It would \emph{not} establish: that 4D gravity is quantized (O\(_{14}\) remains open); that the bulk is the unique UV completion; or that the composite versus weakly-coupled readings of the throat are distinguished (that requires the line-shape and recurrence program of Part~\ref{part:beyond}).  Conversely, \emph{non}-discovery at FCC-hh, after an octet discovery at the LHC, would falsify the cluster reading outright --- the division-of-labor clause of F-KK-1.  \emph{(Physics intuition: the graviton is the cluster's birth certificate, not its driver's license; the octet proves something lives at 3.85~TeV, the graviton proves what family it belongs to.)}

\section{Why does \texorpdfstring{$g^*$}{g*} have the largest production rate --- and why is it so broad?}
\label{app:ratewidth}

\noindent\textbf{\emph{Why does \(g^*\) have the largest production cross section of the three cluster members --- and why is its width so large, \(\sim\!15\%\) of its mass?}}\medskip

\noindent The two questions have one answer at their root --- the coupling texture set by where each field's wavefunction lives along the scale throat --- and this appendix tells it in doors, keys, and exits.  (Appendix~I then opens the hood and derives the same facts from the overlap integrals.)

\paragraph{The production race: \texorpdfstring{$g^*$}{g*} gets three multiplicative advantages.}
\emph{First, the door it enters through.}  At \(3.849\)~TeV on a 13-TeV machine, each incoming parton must carry \(x\approx0.3\) of its proton's momentum --- and at such high \(x\) the proton is essentially three valence quarks; the gluon density has collapsed.  The largest initial-state luminosity available is therefore \(q\bar q\) annihilation, and \(g^*\) is built to use exactly that door: it couples to the light-quark color current with strength \(\xi_{\rm UV}\,g_s=-0.194\,g_s\).  The overlap \(0.194\) looks modest, but it multiplies the \emph{strong} coupling --- the largest coupling in nature's book.  \emph{Second, color counting.}  \(g^*\) is a color octet entering the same channel as the gluon: eight resonance states, produced with the full color trace of a strong-interaction vertex, plus coherent interference with the QCD continuum (Part~\ref{part:ttanatomy}).  \(Z'_B\) is a color singlet --- one state, no color enhancement, no interference.  \emph{Third, the competition's handicaps.}  \(Z'_B\) enters the same \(q\bar q\) door but with a feeble key: \(g_B\approx0.2\) (the derived menu value of L-KK-1) times baryon charge \(1/3\) per quark --- an effective coupling \(\approx0.067\) against \(g^*\)'s \(\approx0.2\), an order of magnitude down in rate before color and branching factors, which stretch it to the canonical ratio \(8.7\)~fb versus \(0.030\)~fb (\S\ref{sec:operational}).  The graviton is worse off twice over: its natural door is \(gg\) fusion, exactly the luminosity that has died away at \(x\approx0.3\); and its gluon-pair coupling carries the derived suppression of Corollary T-KK-2c, \(\kappa_{gg}=0.47/K\).  The rate goes as the coupling \emph{squared} --- one integer, two powers: \((1/35)^2\approx1/1225\).  That is why the graviton anchor sits at \(0.045\)~fb, invisible to the HL-LHC and waiting for a higher-energy ring (Part~\ref{part:beyond}), while its color-octet sibling at the \emph{same mass} is a per-experiment discovery candidate (Part~\ref{part:hllhcsim}).

\paragraph{The width: the exit door is the top, at composite strength.}
A resonance's width is set by the couplings through which it can \emph{leave}, and \(g^*\)'s dominant exit is \(\ttbar\) with coupling \(+2.75\,g_s\) --- the top quark lives at the infrared tip, directly on top of the \(g^*\) wavefunction, so the overlap is enormous.  Squaring a coupling of that size and dividing by the usual \(8\pi\) of phase space gives \(\Gamma/m\approx14.6\%\), with \(94\%\) of it in \(\ttbar\) --- the top being the \emph{only} fermion at the tip (Appendix~I).  Hadronic physics supplies the calibration: the \(\rho(770)\), a composite vector decaying through the strong interaction, has \(\Gamma/m\approx19\%\).  In the Scenario~B reading \(g^*\) \emph{is} a \(\rho\)-like composite of the boundary theory (Appendix~E), so a width of this order is not merely tolerable --- it is what honesty about the composite reading demands.  By the same logic \(G_1\) is narrower (\(6.3\%\): universal but weaker stress-tensor couplings), and \(Z'_B\) is narrow (\(g_B\) is small).

\paragraph{The point worth savoring.}
Production rate, width, \(\ttbar\) dominance, and the interference sign are all read off the \emph{same two numbers} --- \(0.194\) in, \(2.75\) out --- and those numbers are overlap integrals of fixed wavefunctions, not inputs.  And the broadness is a fingerprint, not a nuisance: it is why published narrow-resonance limits under-constrain this state by a factor \(\sim\!3\) (\S\ref{sec:current-status}), why the interference-aware template of Part~\ref{part:hllhcsim} gains so much sensitivity, and why finding a \emph{narrow} peak at \(3.85\)~TeV would actually falsify this implementation.

\section{One integer, two powers --- the rate hierarchy and the \texorpdfstring{$t\bar t$}{ttbar} dominance}
\label{app:ratehierarchy}

\noindent\textbf{\emph{How can one integer --- the same \(K=35\) that sets the cluster's mass --- also decide how often each member is produced and what each decays to?}}\medskip

\noindent Appendix~H told that story in doors, keys, and exits; this appendix opens the hood.  Two numerical facts organize the paper's entire phenomenology: the graviton is produced \(\sim\!300\) times more weakly than the octet, and both decay overwhelmingly to top quarks.  Both facts have one origin --- wavefunction geometry along the scale throat --- and by the end of this appendix the reader can check every number with a pocket calculator.

\paragraph{First, picture the corridor.}  Everything below is bookkeeping of who stands where.  The throat is a corridor of length \(K=35\) in the scale direction.  The light quarks live at the ultraviolet wall (\(r=0\)); the top quark and the entire lock cluster live at the infrared tip (\(r=K\)); the gluon's zero mode is flat --- spread evenly along the whole corridor.  One fact of normalization drives all the arithmetic: a mode spread over a corridor of length \(K\) has generic amplitude \(\sim\!1/\sqrt K\) at any one point.  Standing \emph{at} a point costs nothing; being spread \emph{over} the corridor costs \(1/\sqrt K\) per field.

\paragraph{The production hierarchy: one power of \(K\) versus two.}  Production at a hadron collider is an overlap question: \emph{how much do the initial partons' profiles intersect the resonance's home?}  The octet's parents are valence quarks, standing at the UV wall, where the \(g^*\) mode retains a finite foothold.  The production vertex is a \emph{single} boundary amplitude --- one factor of \(1/\sqrt K\), with no cancellation, because the quarks' endpoint localization breaks any orthogonality protection maximally.  Squared: \(\xi_{\rm UV}^2\approx1/K\).  One power.

The graviton pays twice, and then pays again.  Its parents are gluons, whose flat mode has amplitude \(1/\sqrt K\) \emph{everywhere} --- so two initial gluons already cost \(1/K\) in the amplitude.  Then the overlap integral itself, \(\int(\text{flat})^2\times(\text{IR-localized }J_2)\), \emph{nearly cancels}: a democratic source barely projects onto a receiver huddled at the tip, and only the last half e-fold of the corridor contributes.  That surviving remnant is the \(0.47\) of Corollary T-KK-2c: \(\kappa_{gg}=2(1+J_2(x_1))/(Kx_1^2J_2(x_1))\approx0.47/K\), hence rate \(\propto0.22/K^2\).  Two powers.  The ratio of rates is one extra power of the \emph{same} integer that sets the mass (\(\times35\)), times the near-orthogonality remnant (\(\times\sim\!6\)): the factor \(\sim\!300\) between \(8.7\)~fb and \(0.045\)~fb.  \emph{The graviton is hard to make for the same reason its mass is impossible to move.}

The same flat-mode geometry works on the decay side too: the photon and the leptons are flat or UV-spread, so the cluster barely speaks to them.  That is why \(G_1\) is lepton-poor, why the cluster is leptophobic in practice, and why the published diphoton null was a retro-diction rather than a tension.

\paragraph{The decay hierarchy: heaviness and cluster-affinity are the same property.}  Why tops?  Not phase space --- at \(3.85\)~TeV, \(\beta_t=0.996\): every channel is wide open.  The answer is localization, used twice.  A fermion's \emph{mass} is set by how much its profile overlaps the Higgs sector at the infrared endpoint; the top is the only fermion with an unsuppressed profile there --- that is \emph{why} \(y_t\sim1\).  But the lock cluster is concentrated at the \emph{same} endpoint, and decay widths are the same overlap integrals as masses.  So the state that is heavy \emph{because} it lives at the tip is automatically the state the cluster talks to most loudly: tell me a fermion's mass, and I will tell you how strongly the cluster couples to it.

Numerically, the octet's width ratio per flavor is \((g_V^{t2}+g_A^{t2})/\xi_{\rm UV}^2\approx10.6/0.038\approx280\), giving the partials \(528:26:7\)~GeV and the falsifiable fingerprint \(\ttbar:b\bar b:jj\approx70:3.5:1\) (the intermediate \(b\bar b\) is itself diagnostic: \(b_L\) shares the top's doublet and inherits partial IR-ness).  The graviton's version couples to stress-energy weighted by the same overlaps, preferring the IR-localized top \emph{and} the Higgs sector --- by the equivalence theorem, \(W_L\) and \(Z_L\) \emph{are} the Higgs sector in disguise --- hence \(\ttbar:W_LW_L:Z_LZ_L:hh=0.69:0.15:0.08:0.08\).

\paragraph{The strategic consequence.}  The LHC's cleanest channels are this signal's quietest, and its loudest channel is among the hardest experimentally --- which is why the published clean-channel exclusions never touched the prediction, why Part~\ref{part:ttanatomy} lives in top-pair anatomy, and why the discovery analysis of Part~\ref{part:hllhcsim} lives in the merged one-lepton \(\ttbar\) topology.  Bulk-RS models share the qualitative top-philic story; the difference is that here the localization band \(r_{t_R}\in[4,5]\) is tied to the same geometry that produces the top mass itself, not dialed.  The throat is a mass-ordering machine, and the cluster sits at the heavy end of the ordering.

\paragraph{The overlap integrals, written out.}  Now the hood comes off; this is the appendix's one technical passage, and a reader content with the corridor picture may skim it without loss.  The text above says ``overlap''; a course should display one.  With the canonical scale profile, the octet's lowest mode function is \(\chi_1(r)\propto e^{r}\,J_1\!\big(x_1e^{r-K}\big)\) (normalized over \([0,K]\)), and the coupling to a fermion localized at radial position \(r_\Psi\) is \(g\cdot\chi_1(r_\Psi)/\overline{\chi}\), with \(\overline{\chi}\) the normalization.  Evaluating at the UV wall gives the small boundary value \(\chi_1(0)\): the mode is exponentially concentrated at \(r=K\), and its residue at \(r=0\) is what the light quarks see --- carried through, this is \(\xi_{\rm UV}=-0.194\), the sign coming from the node structure of \(J_1\) between the wall and the tip.  For the graviton--gluon coupling the integrand is qualitatively different: the gluon zero mode is \(r\)-independent after canonical normalization (\(f_g^2=1/K\)), so
\begin{equation}
\kappa_{gg}\;\propto\;\frac{1}{K}\int_0^{K}dr\;e^{-2(r-K)}\,\frac{J_2\!\big(x_1e^{r-K}\big)}{J_2(x_1)}
\;=\;\frac{2\,[\,1+J_2(x_1)\,]}{K\,x_1^{2}\,J_2(x_1)}\;\approx\;\frac{0.47}{K},
\label{eq:appJ-kappa}
\end{equation}
where the closed form follows from the Bessel identity \(\int z^{2}J_2\,dz \to z^2 J_1 + 2zJ_0 - 2\!\int\!J_0\) machinery collapsed at the endpoints (the reader should verify the elementary antiderivative \(\frac{d}{dz}[z^2J_2(z)]\cdot z^{-1}\) route reproduces it).  The lesson is visible in the integrand: the exponential weight concentrates all support within the last half e-fold below \(r=K\), which is the geometric content of the ``0.47''.  \emph{(Physics intuition: integrate a whisper spread over the whole corridor against a listener standing in the last doorway --- only the final syllable is heard.)}

\paragraph{The width table, assembled.}  The same couplings assemble the octet's widths with no further input: \(\Gamma(g^*\to q\bar q)=\frac{\alpha_s}{6}\,\xi_{\rm UV}^2\,m\) per light flavor, and \(\Gamma(g^*\to\ttbar)=\frac{\alpha_s}{6}\,m\,\beta_t\big[g_V^{t2}(1+\tfrac12\rho)+g_A^{t2}\beta_t^2\big]/g_s^2\) with \(\rho=4m_t^2/m^2\); plugging the derived numbers gives \(528:26:7\)~GeV for \(\ttbar:b\bar b:jj\) and \(\Gamma/m=14.6\%\) --- the reader can confirm each entry with a pocket calculator, which is precisely the point: nothing between the geometry and the width table has a dial in it.  \emph{(Physics intuition: the width IS the coupling table read out loud; a broad resonance is broad here because the top's doorway at the IR tip is wide open, not because anything was tuned.)}


\section{From the 6D action to the KK spectrum --- a course}
\label{app:kkcourse}

\subsection{Step 1: the textbook derivation, from scratch}

\paragraph{The flat circle first.}  Take a massless 5D scalar on \(\mathcal M_4\times S^1\) of radius \(R\), action \(S=\tfrac12\int d^4x\,dy\,\partial_M\Phi\,\partial^M\Phi\).  Because the circle is compact, expand in its complete Fourier basis, \(\Phi(x,y)=\sum_{n=-\infty}^{\infty}\phi_n(x)\,e^{iny/R}/\sqrt{2\pi R}\).  Substitute and integrate over \(y\): the cross terms vanish by orthogonality, \(\int_0^{2\pi R}dy\,e^{i(n-m)y/R}=2\pi R\,\delta_{nm}\), leaving
\begin{equation}
S=\sum_n\tfrac12\int d^4x\left[\partial_\mu\phi_n\partial^\mu\phi_n^* - \frac{n^2}{R^2}\,|\phi_n|^2\right].
\label{eq:appD-flat}
\end{equation}
Each Fourier mode is a 4D particle of mass \(m_n=|n|/R\): the Kaluza--Klein tower.  \emph{(Physics intuition: the extra dimension is a resonating cavity; the tower is its overtone series, and the mass is the price of wrapping \(n\) wavelengths around the circle.  Momentum along \(y\), quantized by periodicity, masquerades as 4D mass because \(p_M p^M=0\) splits into \(p_\mu p^\mu = -p_y^2\).)}  Worked example: \(R^{-1}=1\)~TeV gives states at 1, 2, 3~TeV with \emph{equal} spacing --- remember this picket fence; the warped case will destroy it.

\paragraph{Now the warped interval.}  Replace the circle by an interval \(y\in[0,\pi r_c]\) with metric \(ds^2=e^{-2ky}\eta_{\mu\nu}dx^\mu dx^\nu+dy^2\) (Randall--Sundrum).  For a graviton fluctuation \(h_{\mu\nu}\), the transverse-traceless equation of motion separates with \(h_{\mu\nu}(x,y)=\sum_n h^{(n)}_{\mu\nu}(x)\,\psi_n(y)\), and the radial problem becomes, in the conformal variable \(z=e^{ky}/k\),
\begin{equation}
-\,z^{2}\psi_n'' + 3z\,\psi_n' = m_n^2 z^{2}\,\psi_n
\quad\Longrightarrow\quad
\psi_n(z)\propto z^{2}\left[J_2(m_n z)+b_n\,Y_2(m_n z)\right],
\label{eq:appD-bessel}
\end{equation}
a Bessel equation (for Bessel-function background see the NIST handbook, \citealp{OlverEtAl2010DLMF}).  \emph{(Physics intuition: the warp factor acts like a position-dependent refractive index; waves accumulate near the low-index end, so the mode functions pile up at the infrared boundary --- the origin of every ``IR-localized'' statement in this paper.)}  Neumann conditions at both ends force \(b_n\simeq0\) (for \(kr_c\gg1\)) and quantize the spectrum on the zeros of \(J_1\):
\begin{equation}
m_n \simeq x_n\,k\,e^{-k\pi r_c},\qquad J_1(x_n)=0:\quad x_1=3.8317,\ x_2=7.0156,\ x_3=10.1735.
\label{eq:appD-tower}
\end{equation}
Two features deserve a pencil check.  The spacing is \emph{not} the picket fence of \eqref{eq:appD-flat}: the ratios \(1:1.831:2.655\) are pure Bessel geometry.  And the overall scale is exponentially redshifted: \(e^{-k\pi r_c}\) turns a Planckian input into a TeV output when \(k\pi r_c\approx35\).  Worked example: \(k=M_{\rm Pl}\), \(k\pi r_c=35\) puts \(m_1\) in the few-TeV range --- the reader should verify that reproducing \(3.85\)~TeV requires the normalization discussed in \S\ref{sec:t-kk-1}.

\paragraph{Couplings from overlaps.}  A boundary or bulk fermion couples to mode \(n\) with strength \(g_n = g\int dy\,\sqrt{-g}\;f_\Psi(y)^2\,\psi_n(y)\big/(\text{norms})\): \emph{coupling is geometry} --- the overlap of where the fermion lives with where the mode lives.  This single line is the origin of \(\xi_{\rm UV}\), of \(r_{t_R}\), of \(\kappa_{gg}\), and of every rate in this paper; Appendix~O works its consequences.

\subsection{Step 2: what the textbook cannot fix}

The construction above contains three blanks that no amount of textbook diligence fills.  \emph{The radius is an input}: nothing determines \(R\) or \(kr_c\), so the tower can sit anywhere, and forty years of KK phenomenology has been a scan over this ignorance.  \emph{The length must be stabilized}: \(r_c\) is a dynamical field (the radion), and without an added mechanism \citep{GoldbergerWise1999} the interval collapses or runs away --- an extra sector invented solely to hold the stage still.  \emph{The normalization of the first pole is convention-laden}: what multiplies \(x_1\) depends on boundary details a search cannot check.  Every one of these blanks is a symptom of the same cause: the fifth dimension was postulated, so its size and dynamics are free.

\subsection{Step 3: the \GG{} derivation, and the worked number}

\GG{} inherits the \emph{mathematics} of \eqref{eq:appD-bessel}--\eqref{eq:appD-tower} --- the canonical metric of \citet{ArisakaA2GG6} (Theorem~2) has the warped form with \(A(r)=r\) --- and closes all three blanks at once, because the dimension is derived rather than postulated (\S\ref{subsec:notKKRS}).  The length is the \emph{integer} \(K=35\) from the \OGNfive{} closure (blank one: no scan; blank two: no radion, a count cannot fluctuate); the scale is anchored by the closed form of Theorem T-KK-1 (blank three); and the reduction formulas collected below apply with \(A(r)=r\), \(R_\theta=1\) understood.  The worked number, end to end: \(\Mlock=\tfrac12 M_{\rm Pl}e^{-35}=\tfrac12\times1.2209\times10^{19}\,\mathrm{GeV}\times6.3051\times10^{-16}=3.849\)~TeV, and the tower \(m_n=(x_n/x_1)\Mlock=3.849,\ 7.047,\ 10.219\)~TeV --- the organ pipe of \S\ref{sec:t-kk-1}, now derived by the reader.

\subsection{Reference: the reduction formulas}

For completeness we collect the minimal reduction formulas used implicitly throughout the paper.  As in the mode-structure section of the main text, a general scale profile \(A(r)\) and circle radius \(R_\theta\) are retained purely as bookkeeping symbols, so the formulas can be compared line-by-line with the warped literature; the canonical values \(A(r)=r\), \(R_\theta=1\) of \citet{ArisakaA2GG6} (Theorem~2) are understood throughout, and nothing here is free metric data.  For the scale-profile metric
\begin{equation}
\dd s_6^2=e^{-2A(r)}g_{\mu\nu}(x)\dd x^\mu\dd x^\nu+\dd r^2+R_\theta^2\dd\theta^2,
\end{equation}
one has
\begin{equation}
\sqrt{-g_6}=e^{-4A(r)}R_\theta\sqrt{-g_4}.
\end{equation}
The Einstein--Hilbert term reduces to
\begin{equation}
M_{\mathrm{Pl}}^2 = M_6^4\,2\pi R_\theta\int \dd r\,e^{-2A(r)},
\end{equation}
while gauge zero modes satisfy
\begin{equation}
\frac{1}{g_4^2}=\frac{2\pi R_\theta}{g_6^2}\int \dd r\,|f_0(r)|^2.
\end{equation}
The absence of extra scale-profile factors in the latter relation is the standard cancellation between the determinant and inverse metrics.  Fermion zero-mode normalization carries an extra \(e^{-3A}\) weight, and Yukawa or resonance couplings inherit profile-overlap integrals.  Those basic formulas are the origin of the non-universal coupling pattern discussed in the main text.

\section{From parton distributions to cross sections --- a course}
\label{app:pdfcourse}

\subsection{Step 1: the factorized cross section, from scratch}

\paragraph{Factorization.}  A proton is a beam of partons.  The master formula of hadron-collider physics \citep{EllisStirlingWebber1996,HalzenMartin1984} factorizes any hard process into universal parton distribution functions (PDFs) \(f_i(x,\mu_F)\) --- the probability density for parton \(i\) to carry momentum fraction \(x\), measured once and for all in deep-inelastic scattering --- and a computable partonic cross section \(\hat\sigma\):
\begin{equation}
\sigma(pp\to X)=\sum_{i,j}\int_0^1\!dx_1\!\int_0^1\!dx_2\;f_i(x_1,\mu_F)\,f_j(x_2,\mu_F)\;\hat\sigma_{ij\to X}(\hat s=x_1x_2 s).
\label{eq:appE-fact}
\end{equation}
\emph{(Physics intuition: the protons are grocery bags; the PDFs are the shopping lists; the partonic cross section is what happens when two chosen items collide.  Nothing about the bag needs recomputing when the collision changes.)}

\paragraph{The parton luminosity.}  For an \(s\)-channel resonance of mass \(m\), only the combination \(\tau=\hat s/s=m^2/s\) matters, and \eqref{eq:appE-fact} collapses to a single object worth naming, the \emph{parton luminosity}
\begin{equation}
\frac{d\mathcal L_{ij}}{d\tau}=\int_\tau^1\frac{dx}{x}\,f_i(x)\,f_j\!\left(\frac{\tau}{x}\right),
\qquad
\sigma = \sum_{ij}\frac{d\mathcal L_{ij}}{d\tau}\cdot\hat\sigma_{ij}(\hat s = m^2)\cdot\tau .
\label{eq:appE-lumi}
\end{equation}
\emph{(Physics intuition: the luminosity is the number of parton pairs available at the right invariant mass; everything about the machine and the proton is in it, and everything about the new physics is in \(\hat\sigma\).  When this paper says the octet rate grows \(\times330\) from 13 to 85~TeV, it is quoting nothing but the growth of \(d\mathcal L_{q\bar q}/d\tau\) at \(m=3.85\)~TeV.)}

\paragraph{The narrow-resonance master formula.}  For a resonance \(V\) of spin \(J\), color dimension \(N_V\), produced from partons \(ij\) and observed in channel \(f\), integrating the Breit--Wigner gives the textbook result
\begin{equation}
\sigma\times\mathrm{BR}(pp\to V\to f)
=\frac{(2J+1)\,N_V}{N_iN_j}\;\frac{4\pi^2}{m^3}\;\Gamma_{ij}\,\mathrm{BR}_f\;
\frac{d\mathcal L_{ij}}{d\tau}\bigg|_{\tau=m^2/s}\times(1+\delta_{ij}),
\label{eq:appE-nwa}
\end{equation}
with \(\Gamma_{ij}\) the partial width back into the production channel --- the formula every resonance search's sensitivity plot silently uses.  \emph{(Physics intuition: a resonance is a funnel; how much flows through is (spin and color counting) \(\times\) (how wide the funnel's mouth is, \(\Gamma_{ij}\)) \(\times\) (how much beam arrives at the funnel, the luminosity).)}  \textbf{Worked example (the reader should reproduce this).}  For \(g^*\) at \(3.849\)~TeV at \(\sqrt s=13\)~TeV: CTEQ6 gives \(d\mathcal L_{q\bar q}/d\tau\approx\) the archived grid value; the couplings of \S\ref{subsec:tkk2} give \(\Gamma_{q\bar q}\) per light flavor and \(\mathrm{BR}(\ttbar)=0.94\); assembling \eqref{eq:appE-nwa} with the color factors of a spin-1 octet and multiplying by \(K_{q\bar q}=1.3\) returns \(\sigma\times\mathrm{BR}=8.7\)~fb --- the number of Table~\ref{tab:sigmabr}, to be checked against the full line-shape integration (the archived workbook does both; they agree to a few percent, the accuracy of the narrow-width approximation at \(\Gamma/m=15\%\)).

\subsection{Step 2: what the chain assumes but cannot fix}

Four external inputs, each honestly labeled [E] in the ledger: the PDFs themselves (a measured vintage with tails least constrained exactly where high-mass searches live --- the origin of the \([7.0\text{--}12]\)~fb band); the factorization/renormalization scales (formally arbitrary, practically a residual of truncating perturbation theory); the K-factors (someone must compute NLO for your process; we cite \citealp{ChivukulaEtAl2012,MathewsRavindran2005}); and, in every conventional application, \emph{the couplings in \(\Gamma_{ij}\)} --- which textbook phenomenology takes from a model's free parameters, so that the entire chain, however rigorous, ends in a dial.

\subsection{Step 3: the \GG{} chain, closed at both ends}

\GG{} changes exactly one thing, and it is the last one: \(\Gamma_{ij}\) and \(\mathrm{BR}_f\) stop being dials.  The production coupling \(\xi_{\rm UV}=-0.194\) and the decay table descend from T-KK-2, and \(\kappa_{gg}=0.47/K\) from T-KK-2c, so \eqref{eq:appE-nwa} becomes an arithmetic pipeline with nothing to choose: PDFs in, falsifiable femtobarns out --- at 13, 14, and 85~TeV (\S\ref{sec:beyond-fcchh}), with the PDF band carried, not hidden.  The pipeline is archived (parton-luminosity grids and the assembling workbook), and the student who has reproduced the worked example above owns every rate in this paper.

\section{The statistics of discovery --- a worked course}
\label{app:statcourse}

\subsection{Step 1: from the Poisson likelihood to \texorpdfstring{\(Z\)}{Z}, with the arithmetic shown}

Part~\ref{part:hllhcsim} (\S\ref{sec:sim-stats}) derives the formalism --- the likelihood ratio \(q_0\), Wilks' theorem, the Asimov dataset, and the Cowan formula with background uncertainty.  This appendix does what a course must and the body could not: it \emph{shows the arithmetic}, bin by bin, for one real case, so the reader can audit every significance in the paper by extending the same table.  Take end-2026 (0.466~ab\(^{-1}\), 13~TeV), analysis bins of Table~\ref{tab:sim-cases}:

\begin{table}[htbp]
\centering
\small
\caption[The significance arithmetic worked bin by bin for one dataset, so that every other number.]{The significance arithmetic worked bin by bin for one dataset, so that every other number in this Part can be audited by extending the same table. Signal, background, fractional background uncertainty and the resulting per-bin significance are shown in the order the formula uses them. Nothing here is new physics; it is the bookkeeping behind the headline figures, printed because a forecast whose arithmetic is not shown cannot be checked.}
\label{tab:worked-significance}\small
\begin{tabular}{lcccccc}
\toprule
bin [TeV] & \(s=S{+}I\) & \(b\) & \(f_b\) & \(\sqrt{b}\) & \(f_b b\) & \(Z_{\rm bin}\) (Eq.~\ref{eq:sim-zcowan})\\
\midrule
3.0--3.5 & 145 & 178 & 0.069 & 13.3 & 12.2 & 6.8\\
3.5--4.2 & 133 & 75 & 0.132 & 8.7 & 9.9 & 7.4\\
4.2--4.9 & 27 & 19 & 0.226 & 4.3 & 4.3 & 3.5\\
\bottomrule
\end{tabular}
\end{table}

Quadrature over the five analysis bins (the two flanking bins add \(5.2\) and \(0.5\)) gives \(Z=11.9\); dividing by the measured calibration \(F_{\rm cal}=4.33\) gives the \(2.7\sigma\) of every table in Part~\ref{part:hllhcsim}.  \emph{(Physics intuition: each bin is a coin-flip experiment against a bank whose ledger is only good to \(f_b\); the third column pair shows the moment the systematic term overtakes Poisson noise --- already true in every bin at this luminosity, the ``wall'' of \S\ref{sec:sim-wall}.)}  The reader should now recompute the 2041 row of Table~\ref{tab:sim-cases} and watch the frozen-\(f_b\) ceiling emerge from the arithmetic: multiplying \(s\) and \(b\) by seven changes \(Z_{\rm bin}\) hardly at all once \(f_bb\gg\sqrt b\).

\subsection{Step 2: what counting cannot see}

The table above prices two of the three suppressions of \S\ref{sec:sim-ladder}; the third --- the background's freedom to \emph{reshape} --- is invisible to per-bin counting by construction.  The two-parameter demonstration of Figure~\ref{fig:sim-absorb} is the minimal honest exhibit, and the measured \(F_{\rm cal}\) is the full-machinery price.  A student should internalize the moral before running any fit: a smooth broad excess and a background normalization are nearly the same shape, and only correlated, sign-changing structure (the interference pattern) is safe from absorption.

\subsection{Step 3: the pre-registered payoff, worked}

The look-elsewhere arithmetic of \S\ref{sec:sim-lee}, once through with numbers: resolution \(10\%\) gives \(N_{\rm cells}=10\ln(5000/400)=25.3\) independent mass hypotheses; three width templates give \(N_{\rm trials}\approx76\); a local \(p=2.87\times10^{-7}\) (five sigma) becomes \(p_{\rm glob}=76\times2.87\times10^{-7}=2.2\times10^{-5}\), i.e.\ \(Z_{\rm glob}=4.1\) --- the discount a scanned search pays and a pre-registered point does not.  Inverting: global five sigma demands local \(5.8\sigma\).  \emph{(Physics intuition: seventy-six lottery tickets make a win seventy-six times less surprising; the derivation of the mass is what tears up the other seventy-five tickets in advance.)}  Combined with the matched-template factor of \S\ref{sec:sim-matched}, this is the entire statistical content of ``pre-registration'' --- no new mathematics, only the refusal to scan.

\subsection{Reference: the significance-scaling workbook}

Suppose a channel \(f\) has signal efficiency \(\varepsilon_f\), signal cross section \(\sigma_f\), and background yield per unit luminosity \(b_f\).  Then
\begin{equation}
S_f = \mathcal L\,\sigma_f\,\varepsilon_f,
\qquad B_f = \mathcal L\,b_f.
\end{equation}
The simplest background-dominated significance estimator is
\begin{equation}
\mathcal Z_f \approx \frac{S_f}{\sqrt{B_f}} = \sqrt{\mathcal L}\,\frac{\sigma_f\varepsilon_f}{\sqrt{b_f}}.
\end{equation}
If several channels are statistically independent, one may combine them approximately by
\begin{equation}
\mathcal Z_{\mathrm{comb}} \approx \sqrt{\sum_f \mathcal Z_f^2}.
\end{equation}
The key lesson is qualitative: because \GG{} is intrinsically multi-channel and non-universal, combined significance can grow faster than any single clean channel would suggest.  That is why the collider program should be designed as a combined-channel problem from the beginning.

\section{Benchmark / reinterpretation card for experimentalists}

The card below, the template histograms it references, and the full numerical workbook behind them are available at \url{https://arisaka-gg.org/}.

\paragraph*{Search card for the first \GG{} cluster.}
\begin{itemize}
\item \textbf{Primary target:} spin-2-anchored resonance cluster centered at \(3.849\,\mathrm{TeV}\); the discovery rate is carried by the color-octet member (the spin-2 anchor is an FCC-hh measurement at its derived coupling, T-KK-2c).
\item \textbf{Key channels:} \(\ttbar\), boosted dijet, \(hh\), diboson, with clean \(\llbar\) and \(\gamma\gamma\) cross-checks.
\item \textbf{Preferred analysis style:} cluster-aware rather than single-pole-only.
\item \textbf{Most diagnostic pattern:} first cluster + second radial family near \(7\,\mathrm{TeV}\).
\item \textbf{Most useful reduced data:} high-mass jet kinematics, top/Higgs tags, b-tag summaries, timing-assisted pile-up suppression, lepton four-vectors, and channel-combination metadata.
\item \textbf{Rate anchor (model-level, D\(^{*}\)):} color-octet \(\sigma\times\mathrm{BR}(\ttbar)=8.7\,[7.0\text{--}12]\)~fb at 3.85~TeV (13~TeV, LO\(\times\)K); the Scenario-B cluster is octet-dominated (\(G_1\), at its derived \(\kappa_{gg}=0.47/K\), adds only 0.03~fb and is an FCC-hh target: 205~fb at 100~TeV); the free-field 2.46-TeV scenario is already a factor \(\sim\!3.6\) above the observed broad-template limit; \(Z'_B\): 0.012--0.46~fb across the derived-norm L-KK-1 menu, LHC-invisible on every reading.  See Table~\ref{tab:sigmabr} and Fig.~\ref{fig:overlay}.
\item \textbf{Discovery fingerprint (pre-registered, Part~\ref{part:ttanatomy}):} \((1+\cos^{2}\theta^{*})\) angle; \(P_t=+0.91\); resonant \(A_{FB}=0\); constructive-below/destructive-above interference line shape with \(-O(100)\)~GeV centroid pull; nothing in \(\llbar/\gamma\gamma\).
\item \textbf{Collider-level kill criterion:} no combined-channel excess and full-corridor exclusion in the first-cluster window after Run-2 + full Run-3 + early-HL-LHC analyses.
\end{itemize}

\subsection{Implementation walkthrough}
The card above states \emph{what}; this walkthrough states \emph{how}, in the order an analysis team would proceed.  \emph{(1) Template construction.}  Build the truth-level \(d\sigma/dm_{\ttbar}\) from Eq.~\eqref{eq:tt-lineshape} with the fixed inputs (\(\Mlock\), \(\Gamma/m=14.6\%\), \(\xi_{\rm UV}=-0.194\), \(g_V^t=2.75g_s\), \(g_A^t=1.75g_s\)), keeping the Breit--Wigner and interference pieces as \emph{separate} histograms so the fit can float the interference coefficient (sign free --- the data must be allowed to disagree with P-KK-4).  Fold with the experiment's own \(m_{\ttbar}\) response.  \emph{(2) Normalization strategy.}  Anchor the floating \(\ttbar\) normalization at \(m_{\ttbar}\lesssim1\)~TeV or in the dilepton channel; the derived contamination there is \(\lesssim\)~percent (Eq.~\ref{eq:sim-contam}), and \S\ref{sec:sim-contamination} prices what a naive full-range float costs.  \emph{(3) Fit and statistics.}  A two-template profile fit (\(\mu_{\rm BW}\), \(\mu_{\rm int}\)) over the merged-SR spectrum; quote both the pre-registered point (\(\mu_{\rm BW}=\mu_{\rm int}=1\)) and the free fit.  \emph{(4) Validation.}  Closure on background-only pseudo-data; sign-flip injection to confirm the template distinguishes P-KK-4's pattern; and the two calibration masses of \S\ref{sec:sim-calib} as the cross-check that the simplified machinery of this paper reproduces the collaboration's own limits.  \emph{(5) Reporting.}  Whatever the outcome, publish the measured \((\mu_{\rm BW},\mu_{\rm int})\) with its covariance: a null kills F-KK-2B; a signal with the wrong sign kills P-KK-4; a signal with the right sign is the discovery of Part~\ref{part:hllhcsim}.  \emph{(Physics intuition: the entire walkthrough is one sentence --- fit the shape this paper derived, let the sign float, and anchor the bank where the signal cannot reach.)}


\medskip
\noindent\fbox{\parbox{0.965\textwidth}{\textbf{Template T1 --- the deliverable in one box.}\quad
\(T_1(m_{\ttbar}) \;=\; B(m_{\ttbar})\,\big[\,1 + \mu_{\rm BW}\,f_{\rm BW}(m_{\ttbar}) + \mu_{\rm int}\,f_{\rm int}(m_{\ttbar})\,\big]\):
anchored background \(B\) with its normalization fixed at \(m_{\ttbar}\lesssim1\)~TeV (\S\ref{sec:sim-contamination});
Breit--Wigner component at \(m=3.849\)~TeV with \(\Gamma/m=0.146\); interference component with the theorem-fixed
sign (enhancement below the pole, deficit above, crossing near \(4.9\)~TeV) --- both shapes from
Eq.~\eqref{eq:tt-lineshape}.  Validity window \(2.5\)--\(4.9\)~TeV.  Free parameters: the two signal strengths and
the background nuisances, nothing else.  Fit, and publish \((\mu_{\rm BW},\mu_{\rm int})\) with covariance;
interpretation per the walkthrough above.}}
\medskip

\section{Glossary of collider terms}

\begin{description}[style=nextline,leftmargin=3.2cm]
\item[Cluster-aware search] A heavy-resonance search that allows the signal model to contain two or more nearby poles or partially overlapping structures rather than one isolated narrow resonance.
\item[First lock-scale cluster] The first experimentally accessible \GG{} heavy-resonance family, centered near \(3.849\,\mathrm{TeV}\).
\item[Geometry-allowed corridor] The range of effective couplings and branching patterns compatible with the current theorem-level \GG{} geometry, as opposed to a freely scanned benchmark set.
\item[Leptophobic admixture] A contribution from a baryonic or non-universal partner state that enhances hadronic channels relative to dileptons.
\item[Partner pattern] The correlated appearance of the first cluster, KK gluon, and \(\UB\) vector, rather than one isolated signal in one channel.
\item[Reduced stream] A trigger or scouting data format in which only compact high-level or Level-1 information is stored, allowing much higher recording rates than full-event storage.
\item[TLA / trigger-level analysis] The ATLAS approach to saving high-level trigger objects rather than full events for selected searches, especially valuable at high rates.
\item[L1 scouting] The CMS reduced-format strategy in which Level-1 trigger information from essentially all collisions is recorded to disk for dedicated analyses.
\end{description}





\section{Promotion roadmap for the open items}
\label{app:roadmap}

The paper's open items (\S\ref{sec:disc-open}), restated as a roadmap: current grade, the calculation or measurement that closes each, and the impact of closure.

\small
\setlength{\tabcolsep}{4pt}
\begin{longtable}{>{\raggedright\arraybackslash}p{3.1cm} >{\centering\arraybackslash}p{1.4cm} >{\raggedright\arraybackslash}p{4.6cm} >{\raggedright\arraybackslash}p{4.2cm}}
\caption[The open items as a roadmap rather than as a list of regrets.]{The open items as a roadmap rather than as a list of regrets. Each carries its current grade, the specific calculation or measurement that would close it, and what closing it would buy; an item with no closing condition would be an admission rather than a work order, and there are none here. The ordering is by how soon each could be done, which is not the same as by how much it matters.}\label{tab:open-roadmap}\\
\toprule
\textbf{Item} & \textbf{Grade} & \textbf{Closing condition} & \textbf{Impact of closure}\\
\midrule
L-KK-1: the \(g_B\) menu & [M] & The \GDOS{} counting that fixed \(\alpha_s(M_Z)\), applied to the \(\UB\) line & \(Z'_B\) rates single-valued; fifth-force window sharpened\\
NLO interference K-factor & [E] & Dedicated NLO computation of the \(g^*\)--QCD interference & Low-side template firmed; the one theory input the reanalysis needs first\\
\(n\ge2\) overlap couplings & [M] & T-KK-2c machinery evaluated at \(x_2, x_3\) & \(G_2\)/\(G_3\) rates promoted from estimate to derivation (FCC-hh spectroscopy)\\
Composite recurrence spacing & open & Strong-coupling (dual) spectrum calculation & Vector-recurrence search targets fixed\\
Detector-level 2D angular study & [E] & Full simulation with the anchored background cocktail & The bounded \(10\)--\(30\%\) discovery gain banked\\
Composite-scenario width band & [D\(^*\)] & Line-shape fit at HL-LHC statistics (self-measuring) & \(\Gamma/m\) becomes a measured cross-check of the overlap band\\
\bottomrule
\end{longtable}


Closure of any item sharpens the program; closure of none is required for the verdict.  The falsifier stands on the theorem-grade spine alone.

Each row above deserves its sentence of context for the student choosing a project.  \emph{L-KK-1} is the most self-contained: the GDOS counting that produced \(\alpha_s=1/(\pi e)\) is documented in \citet{ArisakaA9SM}, and its \(\UB\) application is a bounded, well-posed exercise with a four-point answer space.  \emph{The NLO interference K-factor} is conventional (if demanding) perturbative QCD --- the one entry a standard SM-theory group could adopt tomorrow, with immediate experimental consumers.  \emph{The \(n\ge2\) overlaps} reuse Eq.~\eqref{eq:appJ-kappa} at higher zeros: an afternoon for the diagonal couplings, a paper for the full mixing structure.  \emph{The recurrence spacing} is the hard one --- genuinely strong-coupling, awaiting either a dual calculation or the data themselves.  \emph{The 2D angular study} belongs inside a collaboration (it needs the real background cocktail), and \emph{the width band} closes itself with data.  A reader looking for where to help should start at the top of the table; a reader looking to falsify should note that none of these items shields the verdict.


The course is complete: a reader who has worked Appendices~J through~L owns every derivation chain in the paper --- spectrum, cross section, and significance.  One service remains, and it is the one a claim of this size owes most: the objections, which the next Part collects and answers, fifty-eight of them, with a pointer into the text or an honest ``not yet.''

\GGPart[Common Objections]{Common Objections and Responses}{The strongest objections a collider experimentalist, phenomenologist, statistician, or string theorist is most likely to raise on first reading --- answered, with pointers into the paper}
\label{part:objections}

\newenvironment{objqabody}%
  {\par\nobreak\begingroup\leftskip=1.8em\parindent=0pt\relax}%
  {\par\endgroup}

A paper that claims a zero-parameter collider target derived from integers will be met --- rightly --- with skepticism from every quarter.  This Part collects fifty-eight objections, grouped by theme, each with an extended response and a pointer to where in the paper the full treatment lives.  The objections are not strawmen; they are the questions the author has been asked, sharpened.  Where the honest answer is ``not yet,'' we say so, and point to the open-item roadmap (Appendix~O) that records exactly which claims are theorems, which are conditional, and which are estimates.  After all, the test of a framework is not that it deflects every question, but that it knows which questions it has answered and which it has not.

\bigskip
\section{Methodology and the derivation chain}

\bigskip
\noindent\textbf{Objection 1 (O-KK-1) (the numerology skeptic)}\par\nobreak
\noindent\textbf{``\(\tfrac12 M_{\rm Pl}e^{-35}\) is one formula among millions of integer combinations.  You found one that lands near the current frontier and built a paper on it.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  The charge would stick if \(35\) had been searched for.  It was not: \(K=35\) is the terminal entry of the \OGNfive{} Diophantine cascade \(K=LJ=M^2+N^2+L^2\) proved in \citet{ArisakaA3BI6}, fixed before any collider comparison, and the \emph{same} integer is spent again on a number measured decades ago --- \(v=\Mlock(2/5)^3=246\)~GeV.  A formula-mining exercise gets one landing; it does not get two landings from one integer, and it cannot survive pre-registration, which converts the coincidence charge into a scheduled experiment (\S\ref{sec:sim-lee}).  The paper's posture is forcing, not searching: every step of the chain from the axioms to \(3.849\)~TeV is written down (\S\ref{sec:t-kk-1}) precisely so that the reader can locate the step at which, on the skeptical reading, the cheat must have occurred --- and find that there is no free step available.\end{objqabody}

\bigskip
\noindent\textbf{Objection 2 (O-KK-2) (the factor skeptic)}\par\nobreak
\noindent\textbf{``The \(\tfrac12\) in \(\tfrac12 M_{\rm Pl}e^{-K}\) looks like a fitted fudge dressed as a constant.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  The \(\tfrac12\) is the normalization of the lowest physical mode on the canonical metric of \citet{ArisakaA2GG6} (its KK-spectrum appendix), not an adjustable prefactor: the anchor paragraph of \S\ref{sec:t-kk-1} traces it, and the same appendix records why the naive Bessel prefactor \(2x_1\approx7.66\) is \emph{not} an independent correction.  The operational point is sharper than the aesthetic one: whatever one thinks of the \(\tfrac12\), it was fixed before the comparison, it cannot be revised without breaking the \(v\) over-constraint, and the experimental window \(3.85\pm0.10\)~TeV is set by resolution and breadth, not by prefactor anxiety.\end{objqabody}

\bigskip
\noindent\textbf{Objection 3 (O-KK-3) (the post-hoc skeptic)}\par\nobreak
\noindent\textbf{``How convenient that the prediction sits exactly at today's exclusion frontier --- neither excluded nor safely far away.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  Invert the logic.  A prediction fixed by integers has to land \emph{somewhere}; landing at the live frontier is simply what a claim looks like in the years when it is about to be decided --- had the LHC run five years longer before this paper appeared, the same number would sit just behind the frontier, and five years less, just ahead of it.  The suspicious pattern would be a prediction that \emph{tracks} the frontier across revisions; \S\ref{sec:disc-significance} documents the opposite (the excluded Scenario~A was not moved but retired, \S\ref{subsec:tkk2}).  Meanwhile the same chain fixes \(v=246\)~GeV, which sits at no frontier at all.\end{objqabody}

\bigskip
\noindent\textbf{Objection 4 (O-KK-4) (the imports skeptic)}\par\nobreak
\noindent\textbf{``The physics is `frozen' upstream in the cohort.  This paper cannot claim an independent test of anything.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  Correct --- and embraced.  Every import is a named theorem with a named owner (\S\ref{sec:t-kk-1}; the References Part records each), so the collider verdict tests the conjunction: a null at \(\lesssim2\)~ab\(^{-1}\) retires not a detachable module but the lock-scale implementation, taking the electroweak derivation of \citet{ArisakaA9SM} with it.  Dependence is what makes the falsifier valuable; an ``independent'' collider module that could fail without consequence upstream would be exactly the escape-hatch structure \S1 criticizes.\end{objqabody}

\bigskip
\noindent\textbf{Objection 5 (O-KK-5) (the hidden-choices skeptic)}\par\nobreak
\noindent\textbf{``No free parameters?  You chose Bessel \(J_1\), you chose Neumann boundary conditions, you chose the orbifold.  Choices all the way down.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  Each item on that list is fixed by an observed fact, not selected: the boundary conditions are the ones whose zero-mode sector \emph{is} the observed massless Standard-Model gauge fields (Theorem T-KK-2's premise, \S\ref{subsec:tkk2}); given them, the radial equation and its \(J_1\) zeros follow with no residual freedom; and the metric on which all of this is posed is itself Theorem~2 of \citet{ArisakaA2GG6}, not an ansatz.  The honest residue of choice is recorded where it lives: the discrete \(g_B\) menu (L-KK-1) and the graded estimates of Appendix~O --- a short list, published as such.\end{objqabody}

\bigskip
\noindent\textbf{Objection 6 (O-KK-6) (the self-grading skeptic)}\par\nobreak
\noindent\textbf{``Your [T]/[D\(^*\)]/[M]/[E] grades are self-assigned.  Every crackpot marks his own homework.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  The grades are not verdicts but \emph{addresses}: each graded claim carries the location of its derivation (the ledger of \S\ref{subsec:provenance}, the symbol table of Appendix~B), so a reader can regrade any entry by checking the cited step.  The discipline's value is asymmetric: self-assigned grades that flatter would be exposed by the first hostile reader; self-assigned grades that are honest --- including [M] for a coupling menu and estimate labels on the \(n=2\) rates --- cost the paper rhetorical force and buy it checkability.  The reader is invited to audit exactly that trade.\end{objqabody}

\bigskip
\noindent\textbf{Objection 7 (O-KK-7) (the K-factor skeptic)}\par\nobreak
\noindent\textbf{``You quote LO\(\times\)K rates and call the result `derived.'  NLO could move everything.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  The paper is careful about which claims are shapes and which are rates.  The theorem-grade content --- masses, degeneracy, sign, angular forms, leptophobia, \(A_{FB}=0\) --- survives any revision of K-factors or PDFs (Part~\ref{part:ttanatomy}).  The rates carry published K-factors with citations and quoted bands (\([7.0\text{--}12]\)~fb), and the single most K-sensitive object, the interference template, is flagged as the first theory task of the program (\S\ref{sec:sim-contamination}, Appendix~O).  Nothing in the falsification logic requires the central rate to be exact: the kill window is stated against the band.\end{objqabody}

\bigskip
\noindent\textbf{Objection 8 (O-KK-8) (the metaphysics skeptic)}\par\nobreak
\noindent\textbf{``An `RG-scale dimension' is philosophy, not physics.  You cannot test what a coordinate \emph{means}.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  The meaning is tested through the consequences that differ between the readings.  A spatial fifth dimension of the same metric drags a radion (testable: Higgs-precision null, P-KK-9), permits a tunable length (testable: the mass could not be an integer output), and makes the \(\theta\) tower's scale a dial (testable: angular satellites, P-KK-7).  The non-spatial reading forbids all three.  \S\ref{subsec:notKKRS} and Table~\ref{tab:kkrsgg} convert the ontology into rows of falsifiable differences --- which is precisely what rescues it from philosophy.\end{objqabody}

\bigskip
\noindent\textbf{Objection 9 (O-KK-9) (the sociology skeptic)}\par\nobreak
\noindent\textbf{``A single-author program of this scope cannot be trusted; big claims need big collaborations.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  Trust is the wrong currency; checkability is the right one, and the paper is engineered for it: line-by-line derivations at analysis-team depth (Part~\ref{part:ttanatomy}), an anchored and calibrated simulation whose every number is reproducible from stated inputs (\S\ref{sec:sim-method}), a reinterpretation card (Appendix~M), and a falsifier that hands the verdict to two thousand-person collaborations.  The claim's fate does not rest on the author's authority --- by construction, it rests on ATLAS and CMS.\end{objqabody}

\bigskip
\noindent\textbf{Objection 10 (O-KK-10) (the Popperian bookkeeper)}\par\nobreak
\noindent\textbf{``Scenario A died and the theory survived.  That is one falsification absorbed already --- classic degenerating research program.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  A degenerating program absorbs falsifications by \emph{adding} freedom; this one \emph{spent} freedom.  The dichotomy was a pre-stated, discrete either/or with both masses and rates published before the comparison (\S\ref{subsec:tkk2}); excluding one horn answered the pre-registered question (the throat vectors are composite) and left the survivor \emph{more} exposed, not less --- one scenario, one mass, one rate, no third option.  Lakatos's test is whether the adjustment predicted anything new: here the surviving scenario's degenerate cluster and its \(8.7\)~fb rate were on the table from the start.\end{objqabody}

\bigskip
\section{The mass and the fine structure}

\bigskip
\noindent\textbf{Objection 11 (O-KK-11) (the error-bar skeptic)}\par\nobreak
\noindent\textbf{``What is the theory uncertainty on 3.849 TeV?  Every prediction needs an error bar.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  The centroid is a closed form in CODATA constants and an integer, so its \emph{theory} error is the within-scenario interaction correction, \(\mathcal O(1/K)\lesssim3\%\) (\S\ref{subsec:tkk2}); the quoted window \(\pm0.10\)~TeV is \emph{experimental} (resolution and breadth), chosen so that the search, not the prediction, sets the tolerance.  Contrast the usual situation, in which the theory band is the widest one in the plot: here the prediction is the sharpest object in the analysis, and \S\ref{sec:t-kk-1}'s anchor paragraph states exactly which constants it consumes.\end{objqabody}

\bigskip
\noindent\textbf{Objection 12 (O-KK-12) (the neighboring-integer skeptic)}\par\nobreak
\noindent\textbf{``If 3.85 TeV is excluded, you will discover that \(K\) was really 34 or 36.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  No such move exists.  \(K=35\) is not a fitted exponent but the closure of \(K=LJ=M^2+N^2+L^2\) with \((M,N,L;J)=(1,3,5;7)\): neighboring integers break the Diophantine identity itself, and even granting the vandalism, \(e^{\pm1}\) moves the mass by a factor \(2.7\) --- to \(1.4\) or \(10.5\)~TeV --- and destroys the already-verified \(v=\Mlock(2/5)^3=246\)~GeV by the same factor.  The integer is load-bearing across the cohort precisely so that it cannot be quietly renegotiated here.\end{objqabody}

\bigskip
\noindent\textbf{Objection 13 (O-KK-13) (the degeneracy skeptic)}\par\nobreak
\noindent\textbf{``Exact mass degeneracy \(m_{g^*}=m_{Z'_B}\) is untestable underneath 15\% widths --- a free miracle.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  The degeneracy is a theorem (same boundary-value problem, \S\ref{subsec:tkk2}), and it is testable in two ways: any resolved splitting beyond the width scale falsifies it outright, and even unresolved, the two states separate by \emph{channel decomposition} --- color-octet versus singlet production, dijet-to-\(\ttbar\) ratios, and the interference dichotomy of O-KK-28 --- the FCC-hh unfolding program of Part~\ref{part:beyond}.  Far from free, the degeneracy is one of the few claims that a single clean measurement could kill.\end{objqabody}

\bigskip
\noindent\textbf{Objection 14 (O-KK-14) (the hardening skeptic)}\par\nobreak
\noindent\textbf{``The `composite-hardened' scenario reads like an escape hatch drilled after 2.46 TeV got into trouble.  And say it plainly: a theory with a discrete menu of \(N\) readings has \(N\) lives.  You had two; you have spent one.  Why should anyone believe the second death would be accepted?''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  The record shows the opposite order: the caution that the free-field root ratio ``need not survive in the non-spatial spectrum'' is intrinsic to the construction (\S\ref{subsec:tkk2}, Step 6), the dichotomy was published \emph{with both rates} --- including Scenario A's unmissable \(224\)~fb, hardly the number one quotes when building an escape --- and the exclusion was then carried out in this same paper against the horn that data could reach (\S\ref{sec:operational}).  An escape hatch is something you exit through quietly; this was a doorway walked through in public, with the door then bolted.  As for the stronger form of the charge: the menu had exactly two entries, both stated in advance with their rates, and no third reading exists --- Scenario~B is not one refuge among many but the sole survivor of a pre-declared either/or.  The paper's own falsifier prices this in: the kill window of \S\ref{subsec:killwindow} is written against Scenario~B, and a null there is \emph{terminal} --- there is no Scenario~C to retreat to, and the requalification that retired Scenario~A in public (\S\ref{sec:operational}) is the standing demonstration that this program accepts its deaths in print.\end{objqabody}

\bigskip
\noindent\textbf{Objection 15 (O-KK-15) (the borrowed-math skeptic)}\par\nobreak
\noindent\textbf{``You disown Randall--Sundrum on every page and then solve its exact Bessel problem.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  Deliberately: the eigenvalue problem is shared because the metric is (Theorem 2), and shared mathematics is what makes the derivation checkable by anyone fluent in the warped literature.  What differs is the ontology, and \S\ref{subsec:notKKRS} itemizes where the difference pays observable rent: no radion, integer length, composite reading of the resonances (which is what made the Scenario dichotomy a physical question), multi-field clustering, and a GUT-scale angular tower.  Borrowing RS's Bessel functions while rejecting its spatial fifth dimension is no more inconsistent than borrowing Fourier analysis from heat conduction.\end{objqabody}

\bigskip
\noindent\textbf{Objection 16 (O-KK-16) (the band-is-a-dial skeptic)}\par\nobreak
\noindent\textbf{``\(\Gamma/m=14.6\%\) rests on \(r_{t_R}\in[4,5]\).  A band is a dial with a euphemism.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  A dial is continuous freedom exercised \emph{after} seeing data; the band is a derived-conditional bracket stated \emph{before}, with its central value, its endpoints, and its downstream consequences (width, \(\mathrm{BR}\) fingerprint, \(P_t\) band) all published together (Part~\ref{part:ttanatomy}).  Crucially, the band is self-closing: the HL-LHC line-shape fit measures \(\Gamma/m\) to a few percent and \(P_t\) to \(\pm0.08\), turning the bracket into a cross-check rather than a refuge (Appendix~O).  A dial that publishes its own execution date is not much of a dial.\end{objqabody}

\bigskip
\noindent\textbf{Objection 17 (O-KK-17) (the absence skeptic)}\par\nobreak
\noindent\textbf{``Predicting the \emph{absence} of angular satellites is a cheap unfalsifiable negative.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  It is exactly as falsifiable as any positive claim: a spatial sixth dimension of accessible radius \emph{requires} \(\ell=\pm1\) partners decorating the cluster, so observing satellites falsifies the non-spatial reading at a stroke (P-KK-7), while their absence discriminates \GG{} from the entire spatial-compactification class.  Negatives are cheap only when nothing predicts the positive; here the alternative does, loudly, which is what gives the silence its evidential weight --- the same logic as the muon-collider null of \S\ref{sec:beyond-muc}.\end{objqabody}

\bigskip
\noindent\textbf{Objection 18 (O-KK-18) (the overtone skeptic)}\par\nobreak
\noindent\textbf{``If FCC-hh finds nothing at 7.047 TeV you will blame the estimated coupling, not the theory.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  The grades pre-empt the dodge: the \emph{position} is theorem-grade (\(x_2/x_1\), no freedom) while only the \emph{rate} is an estimate, and Part~\ref{part:beyond} states the consequence in advance --- at FCC-hh statistics, a same-coupling rate of order \(10\)~fb means even an order-of-magnitude coupling suppression leaves thousands of events, so a genuine position-null at 7.047~TeV kills the radial tower, coupling excuses notwithstanding.  The commitment is written where it can be enforced: an overtone series that misses its Bessel ratio is not this geometry.\end{objqabody}

\bigskip
\section{Couplings, rates, and suppressions}

\bigskip
\noindent\textbf{Objection 19 (O-KK-19) (the convenient-suppression skeptic)}\par\nobreak
\noindent\textbf{``\(\kappa_{gg}=0.47/K\) is suspiciously tiny --- engineered so the graviton hides from every LHC null.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  The order of events refutes the engineering charge: the falsifier was staked \emph{before} the coupling was computed, and the computation (Corollary T-KK-2c, a closed Bessel form with no inputs to tune) then \emph{retro-dicted} the published diphoton null by a factor of several thousand --- a result that would have been an embarrassing tension had the overlap come out large.  Nor is the smallness an escape: it converts into a positive, dated commitment (\(\sigma(G_1)\approx0.13\)~pb at FCC-hh, discovery in the first weeks or falsification of the cluster reading, \S\ref{sec:beyond-fcchh}).  A suppression that hides a state forever would be convenient; one that reschedules it, with a rate attached, is just a derivation.\end{objqabody}

\bigskip
\noindent\textbf{Objection 20 (O-KK-20) (the exact-zero skeptic)}\par\nobreak
\noindent\textbf{``\(\mathrm{BR}(Z'_B\to\ell\ell)=0\) \emph{exactly} is impossible in quantum field theory: kinetic mixing and loops always generate something.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  Stated precisely: the zero is exact at the level of the \(\UB\) charge assignment --- leptons carry no baryon number, so there is no tree coupling to cancel, which is already stronger than the tuned leptophobia of model-built \(Z'\)s.  Radiative kinetic mixing is an honest caveat, and it is small, ultraviolet-calculable, and most importantly \emph{bounded by the same measurement that tests the zero}: the FCC-hh dilepton null at the \(10^{-6}\) level (\S\ref{sec:beyond-fcchh}).  The operational form of the claim, immune to loop pedantry, is the one the paper bets on: any \emph{observable} dilepton line at the cluster mass falsifies the assignment.\end{objqabody}

\bigskip
\noindent\textbf{Objection 21 (O-KK-21) (the nomenclature skeptic)}\par\nobreak
\noindent\textbf{``You call \(G_1\) a graviton, but \GG{} never quantizes gravity.  Which is it?''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  Appendix~G answers at length; the short form: ``graviton'' names \(G_1\) by \emph{pedigree} (it is a mode of the bulk metric sector, whence its spin-2-ness and stress-tensor couplings), not by the quantum-gravity job description.  As a massive spin-2 resonance --- in the RG reading, a composite state of the boundary theory --- its quantum status is as unproblematic as a \(\rho\) meson's, and its discovery would evidence the six-dimensional geometry while neither requiring nor proving that massless 4D gravity is quantized.  The genuinely deep statement runs the other way: in \GG{}, quantization itself is an output of the \OGGEP{} projection, for graviton and photon alike.\end{objqabody}

\bigskip
\noindent\textbf{Objection 22 (O-KK-22) (the why-tops skeptic)}\par\nobreak
\noindent\textbf{``Why should everything decay to top quarks?  Top-philia smells like a choice made to dodge the dilepton limits.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  It is the same geometry that makes the top heavy, used twice: a fermion's mass and its coupling to the infrared-localized cluster are the \emph{same} overlap integral, so the state that is heavy because it lives at the throat's tip is automatically the state the cluster talks to (Appendix~I).  The numbers are not adjustable --- \((g_V^{t2}+g_A^{t2})/\xi_{\rm UV}^2\approx280\) per flavor, giving \(\ttbar:b\bar b:jj\approx70:3.5:1\) --- and the alleged convenience cuts both ways: top-philia moves the signal into the experimentally \emph{hardest} clean channel, which no one dodging limits would choose.\end{objqabody}

\bigskip
\noindent\textbf{Objection 23 (O-KK-23) (the menu skeptic)}\par\nobreak
\noindent\textbf{``A `discrete menu' for \(g_B\) is a dial with extra steps.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  A dial is a continuum; the menu is four values, each produced by a named counting rule, published with its consequences (\(Z'_B\) rates spanning 0.2--6.8\% of the cluster sum) and registered as an open lemma with its closing condition stated --- the same \GDOS{} counting that fixed \(\alpha_s(M_Z)\) (L-KK-1; Appendix~O).  Nothing downstream hides behind the menu: the falsifier is carried by the octet, whose coupling is not on any menu, and every menu point is separately falsifiable through the fifth-force window it also feeds \citep{ArisakaA9SM}.\end{objqabody}

\bigskip
\noindent\textbf{Objection 24 (O-KK-24) (the sign skeptic)}\par\nobreak
\noindent\textbf{``A \emph{sign} prediction, \(\xi_{\rm UV}<0\)?  Cross sections are sign-blind.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  Squares are sign-blind; interference is not.  The octet shares the gluon's color channel, so the sign enters linearly through the interference term and imprints a parameter-free \emph{shape}: enhancement below the pole, deficit above, the summed spectrum crossing \emph{below} the Standard Model near \(4.9\)~TeV (Part~\ref{part:ttanatomy}, P-KK-4).  Flip the sign and the shape mirrors --- a distinction a template fit measures directly, and the quantitative Figure~\ref{fig:lhc-signature} draws.  Few BSM frameworks are falsifiable at the level of a relative sign; this one is.\end{objqabody}

\bigskip
\noindent\textbf{Objection 25 (O-KK-25) (the precision pedant)}\par\nobreak
\noindent\textbf{``You write \(|\xi_{\rm UV}|\approx1/\sqrt K=0.169\) in one place and \(0.194\) in another.  Which is the prediction?''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  Both, at their stated grades: \(1/\sqrt K\) is the leading-order scaling that exhibits the \(K\)-dependence, and \(0.194\) is the exact mode value including the derived warped correction (\S\ref{subsec:tkk2}), the number every rate in the paper uses.  The refinement went in the derived direction before any comparison --- the correction is not a knob but a computed \(+15\%\) --- and the practical spread between the two is far inside the quoted rate band.  When a scaling and its exact completion are both published with their derivations, the right description is convergence, not equivocation.\end{objqabody}

\bigskip
\noindent\textbf{Objection 26 (O-KK-26) (the hidden-channel skeptic)}\par\nobreak
\noindent\textbf{``The 70:3.5:1 fingerprint assumes no exotic decay channels.  A single hidden state ruins it.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  The lock cluster is the \emph{lowest-lying} new structure of the derived spectrum, so within \GG{} there is nothing below it for it to decay into --- the fingerprint is closed by construction, not by assumption.  If nature nevertheless supplies an extra light state, then \GG{}'s spectrum is wrong and the paper is falsified on grounds far more direct than a branching-ratio distortion.  Either way the fingerprint does its job: it is exact within the theory and diagnostic against alternatives (\S\ref{sec:disc-comparison}).\end{objqabody}

\bigskip
\noindent\textbf{Objection 27 (O-KK-27) (the washout skeptic)}\par\nobreak
\noindent\textbf{``\(P_t=+0.91\) will never survive parton shower, hadronization, and reconstruction.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  Top polarization is special precisely because the top decays before hadronizing: the charged lepton is a maximal-power spin analyzer whose angular distribution survives to reconstruction level, which is why ATLAS and CMS already publish top-polarization measurements today.  Part~\ref{part:ttanatomy} carries the lepton-analyzer derivation, and Part~\ref{part:hllhcsim} propagates purity-corrected statistics to \(\delta P_t=0.14\) by 2033 and \(0.08\) by 2041 --- enough to resolve the \emph{interior} of the derived band \([+0.88,+0.92]\), not merely its existence.\end{objqabody}

\bigskip
\noindent\textbf{Objection 28 (O-KK-28) (the interference-consistency skeptic)}\par\nobreak
\noindent\textbf{``Interference with QCD requires identical color flow.  Did you actually check which states interfere?''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  Yes, and the answer is itself a diagnostic.  The color octet enters the \emph{same} color channel as the \(s\)-channel gluon, so it interferes coherently (the amplitude algebra is written out in Part~\ref{part:ttanatomy}); the color-singlet \(Z'_B\) does not interfere with QCD at all --- nor with the octet, since the same color trace kills the \(g^*\)--\(Z'_B\) cross-term, which is why the cluster rates may be summed incoherently.  The cluster therefore predicts a two-component line shape --- an interfering octet plus a non-interfering singlet at the same mass --- and that yes/no dichotomy within one peak is one more handle the FCC-hh unfolding uses to separate the degenerate pair (O-KK-13, \S\ref{sec:beyond-fcchh}).\end{objqabody}

\bigskip
\section{The simulation and its statistics}

\bigskip
\noindent\textbf{Objection 29 (O-KK-29) (the fudge-factor skeptic)}\par\nobreak
\noindent\textbf{``\(F_{\rm cal}=4.33\) is a fudge factor by another name.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  A fudge factor is chosen to produce a desired answer; \(F_{\rm cal}\) is \emph{measured} against answers already published: narrow test signals injected at two masses reproduce ATLAS's expected limits with mutually consistent factors (4.40 and 4.27), and the calibrated framework then \emph{predicts} the 15\%-width limit at 8.4~fb, correctly ordered between ATLAS's own narrow and broad templates --- a validation with no adjustable input (\S\ref{sec:sim-calib}).  Its application is conservative by construction (exclusion-derived, applied to discovery), and the one methodological temptation it forbids --- double-profiling on top of post-fit uncertainties --- is documented in the same section.\end{objqabody}

\bigskip
\noindent\textbf{Objection 30 (O-KK-30) (the systematics-optimism skeptic)}\par\nobreak
\noindent\textbf{``Halving the background systematics is a wish, not a scenario.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  The halving is the standard HL-LHC projection convention \citep{CERNYellowReport2019}, and \S\ref{sec:sim-impsyst} decomposes it: the control-region and template-statistics components shrink as \(1/\sqrt L\) by arithmetic alone (\(\times2.8\)--\(4.9\) over the program), object calibrations improve data-driven toward known floors, and only the theory-modeling class requires actual work.  The paper also quotes the frozen scenario --- no improvement of anything for fifteen years --- as the pessimistic bound, and notes that at end-2026 roughly half the ``improvement'' is automatic with the Run-3 dataset itself.  Reality lives between two published curves; neither is a wish.\end{objqabody}

\bigskip
\noindent\textbf{Objection 31 (O-KK-31) (the where-is-the-hint skeptic)}\par\nobreak
\noindent\textbf{``You claim \(2.7\)--\(3.7\sigma\) sits on tape.  ATLAS analyzed that region and saw nothing.  One of you is wrong.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  Neither is --- because ``analyzed that region'' and ``ran this analysis'' differ by the two factors this paper quantifies.  The published search used templates mismatched to the coherent 15\%-width shape (a factor \(\sim\!3\) dilution in rate sensitivity, \S\ref{sec:sim-matched}) and coarse-binned counting that a floating background absorbs (\S\ref{sec:sim-ladder}); under those conditions the predicted signal produces \emph{sub-2\(\sigma\) local} fluctuations in their own quotes --- fully consistent with ``saw nothing.''  Indeed \S\ref{sec:operational} shows the framework reproduces ATLAS's observed limit with the signal \emph{present}: the expected excess sits just below their published exclusion.  The claim is not that ATLAS missed a bump; it is that a differently designed fit extracts what their design integrates over.\end{objqabody}

\bigskip
\noindent\textbf{Objection 32 (O-KK-32) (the global-trials skeptic)}\par\nobreak
\noindent\textbf{``No look-elsewhere?  Your cohort makes dozens of predictions.  Across twenty-two papers, something was bound to sit near a frontier.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  The look-elsewhere effect is a property of a \emph{decision rule}, not of an author's productivity: the discovery claim at issue is a single pre-registered mass point with a single pre-stated test, and its local \(p\)-value is its global one (\S\ref{sec:sim-lee}).  Cross-paper trials would matter if the program's survival required only \emph{one} of many predictions to succeed; the cohort's structure is the opposite --- conjunctive, with shared integers, so that failures propagate (O-KK-4).  A framework whose predictions must \emph{all} hold pays no trials discount on any one of them; it pays the far steeper price that any one can kill it.\end{objqabody}

\bigskip
\noindent\textbf{Objection 33 (O-KK-33) (the asymptotics skeptic)}\par\nobreak
\noindent\textbf{``Asimov formulas fail at small counts, and your tail bins are small.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  The analysis bins carrying the significance have \(b\gtrsim19\) even at the smallest snapshot and grow from there (Table~\ref{tab:sim-cases}), inside the validated Asimov regime; the truly sparse display bins beyond 5.6~TeV contribute nothing to the quoted \(Z\).  Residual asymptotic error is second-order against the calibration factor, which was measured against ATLAS's full toy-based machinery and therefore absorbs precisely such effects (\S\ref{sec:sim-calib}).\end{objqabody}

\bigskip
\noindent\textbf{Objection 34 (O-KK-34) (the correlations skeptic)}\par\nobreak
\noindent\textbf{``Treating per-bin systematics as uncorrelated is wrong, and everyone knows correlated backgrounds absorb broad signals better.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  Agreed on the physics --- which is why the paper does not rely on the uncorrelated model alone.  The correlation-driven absorption is demonstrated explicitly (the two-parameter profile of \S\ref{sec:sim-ladder}, which eats 75\% of the window signal), its full-machinery magnitude is what \(F_{\rm cal}\) was measured to capture, and the caveat ledger records the treatment (item 5, \S\ref{sec:sim-meaning}).  The quoted significances are thus calibrated against a fit that \emph{does} exploit correlations; the per-bin model is the bookkeeping, not the claim.\end{objqabody}

\bigskip
\noindent\textbf{Objection 35 (O-KK-35) (the own-goal skeptic)}\par\nobreak
\noindent\textbf{``Your sideband-contamination warning undermines your own sensitivity numbers.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  The numbers already stand on ground the warning does not touch: they use ATLAS's post-fit uncertainties, which encode the real fit's freedom, calibrated by \(F_{\rm cal}\) (\S\ref{sec:sim-calib}).  The warning (\S\ref{sec:sim-contamination}) is design guidance for the \emph{future dedicated fit} --- anchor the normalization below 1~TeV, where the derived contamination falls to the percent level --- and the two-parameter demo quantifies what ignoring it costs.  A paper that flags the trap and prices it is not undermined by the trap; it is the reason the trap will be stepped around.\end{objqabody}

\bigskip
\noindent\textbf{Objection 36 (O-KK-36) (the combination skeptic)}\par\nobreak
\noindent\textbf{``Your ATLAS+CMS \(\sqrt2\) assumes uncorrelated experiments.  Theory systematics are common to both.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  Which is why \(\sqrt2\) is never quoted alone: Table~\ref{tab:sim-combo} brackets both extremes --- fully correlated (end-2026 improved: \(4.6\sigma\)) and fully uncorrelated (\(5.2\sigma\)) --- with the statement that reality lies between, and the physical sorting is given (modeling: correlated; detector calibrations and control-region statistics: not).  Every headline in the paper carries the bracket, not the optimistic edge.\end{objqabody}

\bigskip
\noindent\textbf{Objection 37 (O-KK-37) (the seed skeptic)}\par\nobreak
\noindent\textbf{``One random seed for the pseudo-data.  Cherry-picked fluctuations?''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  Every quoted significance is an Asimov (median-expected) quantity, computed from expectations and independent of any seed; the single fixed seed generates only the \emph{display} points of the spectra figures, stated as such in the caveat ledger (item 8) and reproducible from the archived workbook.  Change the seed and the dots move within their error bars; not one number in any table moves.\end{objqabody}

\bigskip
\noindent\textbf{Objection 38 (O-KK-38) (the PDF-vintage skeptic)}\par\nobreak
\noindent\textbf{``CTEQ6 is two decades old.  Modern PDFs could shift your rates.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  They could, and the quoted bands already say by how much: the \([7.0\text{--}12]\)~fb spread on the octet rate is dominated by exactly this class of uncertainty, and every conclusion --- the frontier position, the kill window, the calendar --- is stated against the band, not the central value.  Shapes, which carry the theorem-grade content, are far more stable across PDF generations than tails of absolute rates; and the paper names the permanent fix (LHeC-class high-\(x\) input, \S\ref{sec:beyond-linear}) rather than pretending the issue away.\end{objqabody}

\bigskip
\section{Reducibility, non-spatial dimensions, and neighboring frameworks}

\bigskip
\noindent\textbf{Objection 39 (O-KK-39) (the category skeptic)}\par\nobreak
\noindent\textbf{``A dimension is something you can move through.  A `scale dimension' is a category error.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  By that criterion time is a category error too --- one cannot ``move through'' time in the spatial sense, yet Minkowski's fusion \citep{Minkowski1908} is not up for debate.  The disciplined criterion classifies a dimension by its symmetry generator, and \S\ref{subsec:notKKRS} applies it: translations generate space and time; a dilatation generates \(I_r\); a compact charge generates \(S^1_\theta\).  ``Motion'' along \(r\) is transmutation --- change of renormalization scale --- which is as physical as anything in field theory; what it is not, is walking.\end{objqabody}

\bigskip
\noindent\textbf{Objection 40 (O-KK-40) (the holography skeptic)}\par\nobreak
\noindent\textbf{``This is AdS/CFT rebranded: everyone knows the radial coordinate is the RG scale.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  In holography that identification is a \emph{duality} between two complete descriptions, either of which may be taken as fundamental; in \GG{} it is a single literal construction --- there is no spatial-bulk description to dualize to, no radion, no tunable length, no landscape of compactifications (Table~\ref{tab:kkrsgg}).  The distinction pays observable rent: a genuine AdS\(_5\) dual would carry the usual RS phenomenology (radion, adjustable \(kr_c\), single pole); \GG{} forbids each item, and the forbiddances are on the prediction list (P-KK-7, P-KK-9).  Rebranding changes names; this changes what experiments should see.\end{objqabody}

\bigskip
\noindent\textbf{Objection 41 (O-KK-41) (the SUSY skeptic)}\par\nobreak
\noindent\textbf{``No supersymmetry anywhere?  Serious UV physics requires it.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  In \GG{} the absence is a theorem, not a taste: supercharges must square-root translations, and the two extra directions have none to offer --- a dilatation and a compact charge admit no super-partner extension (\S\ref{subsec:notKKRS}; the algebraic statement is Theorem-level in \citealp{ArisakaA2GG6}).  The framework therefore \emph{predicts} the null that fifty years of superpartner searches have delivered, rather than perpetually postponing their discovery --- one of the few readings under which the LHC's SUSY silence counts as evidence \emph{for} something.  (The careful reader will note the loophole class: exotic superalgebras that do not square-root translations.  They exist as mathematics; what they cannot deliver is the phenomenological content --- superpartners protecting scalar masses --- that made supersymmetry physics rather than algebra, so the claim stands where it matters.)\end{objqabody}

\bigskip
\noindent\textbf{Objection 42 (O-KK-42) (the sixth-dimension skeptic)}\par\nobreak
\noindent\textbf{``Any honest 6D theory has a second KK tower.  Where are the \(\theta\) modes?''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  At \(\sim\EGUT\), and immovably so: the phase circle is topological, with no radius modulus to dial its tower downward --- \(R_\theta=1\) is a convention on a dimensionless angle, not a size (\S\ref{subsec:notKKRS}).  The observable content is the pair of statements the paper registers: the radial tower at derived TeV positions (P-KK-8) and \emph{no} angular satellites decorating it (P-KK-7).  A spatial 6D construction generically fails the second; that is the discrimination, available at the same experiments.\end{objqabody}

\bigskip
\noindent\textbf{Objection 43 (O-KK-43) (the stabilization skeptic)}\par\nobreak
\noindent\textbf{``Without a radion, what keeps the throat from collapsing or running away?''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  Collapse and runaway are fates of \emph{sizes}; the throat's length is a \emph{count} --- the integer \(K=35\) of the cascade --- and a count has no continuous deformation to run along (\S\ref{subsec:notKKRS}; \citealp{ArisakaA4Euclid}).  Asking what stabilizes it is asking what stabilizes the number of colors in QCD.  The claim is radical but testable: a stabilized modulus always leaves a scalar in the spectrum and fingerprints in Higgs couplings, so the Higgs-precision null (P-KK-9) is exactly the experiment that distinguishes ``no modulus'' from ``hidden modulus.''\end{objqabody}

\bigskip
\noindent\textbf{Objection 44 (O-KK-44) (the missing-partners skeptic)}\par\nobreak
\noindent\textbf{``Warped models come with KK fermions, KK \(W'/Z'\), custodians.  Your spectrum looks suspiciously sparse.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  Sparseness is forced, not curated: the low-lying content is the lowest radial modes of the fields actually present in the bulk group of \citet{ArisakaA2GG6}, and that group was derived, not assembled to populate a spectrum.  There are no custodial partners because there is no custodial problem to solve --- the electroweak sector is not protected by tuning but derived (\citealp{ArisakaA9SM}).  Heavier structure exists (radial recurrences, Part~\ref{part:beyond}) at derived or dual-calculable positions; what does not exist is the model-builder's freedom to add a state whenever a search wants one.\end{objqabody}

\bigskip
\noindent\textbf{Objection 45 (O-KK-45) (the no-free-lunch skeptic)}\par\nobreak
\noindent\textbf{``No landscape, no moduli, no tuning --- because you hid all the arbitrariness in the `constitution.'  The lunch is not free.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  Correct, and the paper's ancestors say so themselves: the price of forced low-energy structure is the strength of the constitutional premises, stated as such in \citet{ArisakaA2GG6} (Theorem~0's assumptions are listed, not smuggled).  The relevant comparison is where each program places its irreducible assumptions --- at inaccessible trans-Planckian scales, or in axioms whose consequences land at \(3.85\)~TeV within one PhD of publication.  Lunches are never free; this one at least comes with a bill the LHC can audit.\end{objqabody}

\bigskip
\noindent\textbf{Objection 46 (O-KK-46) (the convenient-split skeptic)}\par\nobreak
\noindent\textbf{``Radial tower testable, angular tower at the GUT scale --- the split conveniently exposes only what you can survive.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  The split runs the other way from convenience: it \emph{exposes} the radial tower at exactly the energy the LHC reaches now, with a rate the current run can kill (F-KK-2B), while the ``safe'' angular sector still makes a falsifiable claim through its absence (P-KK-7) and through its lightest member elsewhere in the cohort --- the \(8.1\,\mu\)eV kaxion of \citet{ArisakaC1Kaxion}, under active haloscope test.  A construction optimized for survival would have put the radial tower at 40~TeV; this one put it under the beam.\end{objqabody}

\bigskip
\bigskip
\noindent\textbf{Objection 47 (O-KK-47) (the anomaly skeptic)}\par\nobreak
\noindent\textbf{``You cannot simply gauge baryon number.  With Standard-Model fermion content the mixed triangles \(\U(1)_B\)--\(\SU(2)_L^2\) and \(\U(1)_B\)--\(\U(1)_Y^2\) are non-zero (\(\pm3/2\) over three generations); every honest \(Z'_B\) model buys them off with new chiral `anomalon' fermions, which then have masses, signatures, and problems of their own.  Where are yours?''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  Nowhere --- because none exist, and their absence is a theorem, not an oversight.  The premise is correct: with the Standard Model alone, \(\mathcal A[B\text{--}\SU(2)_L^2]=+\tfrac32\) and \(\mathcal A[B\text{--}Y^2]=-\tfrac32\) (the gravitational, cubic, and \(B^2Y\) triangles vanish identically), and the conventional remedy is indeed anomalon matter.  But in \GG{} the boundary spectrum is not the Standard Model with baryon charges appended by hand: it descends from unsplit \(\mathrm{SU}(5)\) multiplets on the \OGNfive{}-locked charge lattice, and the microscopic anomaly polynomial of that spectrum factorizes to \emph{rank one} (\AThree{}, Theorem T-BI6-6.1).  The two surviving mixed triangles are the boundary projections of the cross-blocks of the \BIsix{} anomaly matrix, and the gauge variation of the single Green--Schwarz counterterm \(\int B_2\wedge X_4\) cancels them by inflow, with the compensating couplings splitting across \(\SU(2)_L\times\U(1)_Y\subset\mathrm{SU}(5)\) in exactly the ratios required (\citealp{ArisakaA3BI6}, the boundary-mixed-anomalies subsection).  The four-dimensional remnant of \(B_2\) is a St\"uckelberg axion whose shift performs the cancellation \emph{and} is eaten by the baryonic vector: the \(Z'_B\) mass of Appendix~F and the anomaly bookkeeping are one coupling seen twice.  Note the collider corollary, which is a falsifier of its own: anomalon models necessarily populate the accessible range with new charged chiral states, while \GG{} \emph{forbids} them --- discovering anomaly-canceling fermions accompanying a \(3.85\)-TeV \(Z'\) would count against this implementation as surely as a clean null.\end{objqabody}

\section{Cohort dependence and standing}

\bigskip
\noindent\textbf{Objection 48 (O-KK-48) (the house-of-cards skeptic)}\par\nobreak
\noindent\textbf{``If A9's electroweak derivation fails, this paper falls with it.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  Yes --- deliberately.  The over-constraint \(v=\Mlock(2/5)^3\) means the two papers share one integer and one fate, so a collider null at \(3.85\)~TeV wounds A9 and a failure of A9's chain wounds this paper; the exposure is stated in \S\ref{sec:disc-cohort} in both directions.  A house of cards is fragile \emph{and unfalsifiable-by-parts}; this structure is fragile \emph{because} falsifiable-as-a-whole --- the property the whole program regards as the point.\end{objqabody}

\bigskip
\noindent\textbf{Objection 49 (O-KK-49) (the analysis-cost skeptic)}\par\nobreak
\noindent\textbf{``Why should a collaboration spend person-years on an arXiv-only, single-author benchmark?''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  Because the request is priced and the payoff is asymmetric.  The ask is one template added to an existing merged-SR analysis on data already recorded --- thesis-scale, per the blueprint of Part~\ref{part:hllhcsim} --- and the deliverable is valuable under \emph{every} outcome: the world's best limit on any broad top-philic resonance, a reusable interference-aware template, and, in the tail of the distribution, the first particle beyond the Standard Model in collider history.  Collaborations reinterpret benchmarks of far weaker epistemic standing (fixed-coupling scans with no derivation) as routine practice; a zero-parameter, pre-registered target with a published discovery calendar clears that bar with room to spare.\end{objqabody}

\bigskip
\noindent\textbf{Objection 50 (O-KK-50) (the closed-loop skeptic)}\par\nobreak
\noindent\textbf{``The cohort cites itself.  Twenty-two papers in a loop is not external validation.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  The internal citations carry the \emph{derivations}; the \emph{validations} are external by construction, and Section~B of the References Part lists them: ATLAS and CMS measurements, HEPData records, published K-factors, the PDG ledger, the strategy documents.  Every contact between the cohort and reality happens on instruments the cohort does not own --- most decisively the falsifier itself, whose verdict will be signed by three thousand people with no stake in the program.\end{objqabody}

\bigskip
\noindent\textbf{Objection 51 (O-KK-51) (the pace skeptic)}\par\nobreak
\noindent\textbf{``Twenty-two papers in roughly a year.  Depth takes decades; this is speed-writing.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  Pace is evidence about process, not about correctness, and this paper is built so the two can be separated: its chain is locally checkable given the cited theorems --- a reader can verify T-KK-1 in an afternoon, T-KK-2's boundary-value problems in a day, the simulation from the archived workbook --- without adjudicating the cohort's production history.  Where depth is genuinely unfinished, the paper says so in its own voice (the open lemma, the estimates, Appendix~O).  Judge the derivation on the page; the calendar behind it is biography.\end{objqabody}

\bigskip
\noindent\textbf{Objection 52 (O-KK-52) (the schedule skeptic)}\par\nobreak
\noindent\textbf{``Your calendar leans on CERN schedules that slip as a law of nature.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  Every decision point in the paper is stated in integrated luminosity, with calendar years attached only through the official planning documents quoted at their source (\S\ref{sec:sim-equiv}, Part~\ref{part:beyond}).  If LS3 stretches, the \(0.80\)~ab\(^{-1}\) five-sigma crossing arrives later in wall-clock time and at exactly the same point in data; F-KK-2B fires at \(\lesssim2\)~ab\(^{-1}\) whenever those inverse attobarns exist.  Slippage delays the verdict; it cannot dilute it.\end{objqabody}

\bigskip
\noindent\textbf{Objection 53 (O-KK-53) (the attribution skeptic)}\par\nobreak
\noindent\textbf{``If ATLAS finds \emph{some} 3.9-TeV top-philic octet, every model-builder will claim it.  Discovery would prove nothing about \GG{}.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  The mass alone, granted --- which is why the paper never bets on the mass alone.  The registered fingerprint is conjunctive: exact vector degeneracy at one mass, exact leptophobia, the derived interference sign with its below-SM crossing, \(P_t\) in \([+0.88,+0.92]\), \emph{no} angular satellites, and an overtone at \(1.831\times\) the mass (P-KK-2 through P-KK-8).  \S\ref{sec:disc-comparison} walks the neighboring frameworks through that list; each fails multiple clauses, and none predicts the ratio-locked second family.  Imitators can gate-crash a bump; they cannot gate-crash a spectrum.\end{objqabody}

\bigskip
\section{Experimental practice}

\bigskip
\noindent\textbf{Objection 54 (O-KK-54) (the practitioner)}\par\nobreak
\noindent\textbf{``Broad \(\Gamma/m=15\%\) resonances at \(S/B\sim1\) are exactly what tt-resonance analyses are worst at.  `Just reanalyze' is glib.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  Agreed that broad-plus-interfering is the hard case --- that is why the paper does the analysis design work instead of gesturing at it: the anchor analysis is read cut-by-cut (\S\ref{sec:sim-anchor}), the template is fully specified with its sign structure (Part~\ref{part:ttanatomy}), the fit-design trap is located and priced (\S\ref{sec:sim-contamination}), the statistics are calibrated to the published machinery (\S\ref{sec:sim-calib}), and the whole chain is packaged in the reinterpretation card (Appendix~M).  ``Just reanalyze'' would be glib; ``reanalyze along this worked blueprint'' is a costed proposal.\end{objqabody}

\bigskip
\noindent\textbf{Objection 55 (O-KK-55) (the template-validation skeptic)}\par\nobreak
\noindent\textbf{``Interference-aware templates are notoriously hard to validate.  How would we know the template is right?''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  Three independent handles.  The template's \emph{shape} is falsifiable within the fit itself --- the sign is left floating and the data choose (a wrong-sign preference falsifies P-KK-4, not the analysis); its \emph{normalization region} is validated where the derived contamination is provably negligible (below \(\sim\!1\)~TeV, \S\ref{sec:sim-contamination}); and its \emph{theory} systematic is the named NLO task of Appendix~O, commissioned before unblinding rather than argued after.  This is a stronger validation position than the generic templates now in use, whose widths and shapes answer to no derivation at all.\end{objqabody}

\bigskip
\noindent\textbf{Objection 56 (O-KK-56) (the trigger skeptic)}\par\nobreak
\noindent\textbf{``The 2015--2026 data were triggered without your signal in mind.  Whatever is on tape may not contain it.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  The kinematic chain settles this (\S\ref{sec:sim-trigger}): a 3.85-TeV decay delivers \(\sim\!500\)~GeV leptons to single-lepton triggers thresholded at 26--28~GeV --- the signal has been sailing onto tape at essentially 100\% trigger efficiency since 2015, in the same streams the published \(\ttbar\) searches already analyze.  The scouting and reduced-stream proposals of Table~\ref{tab:trigger} are future insurance for \emph{other} corners of the program, not a precondition for this one.\end{objqabody}

\bigskip
\noindent\textbf{Objection 57 (O-KK-57) (the governance skeptic)}\par\nobreak
\noindent\textbf{``Who `owns' a joint pre-registered search?  ATLAS and CMS do not co-analyze.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  Nothing here requires co-analysis: the card (Appendix~M) is collaboration-neutral, each experiment runs its own fit on its own data, and the combination follows the established precedent of the Higgs-discovery-era and \(W\)-mass combinations --- independent analyses, joint statistical combination afterward, with the correlation brackets of Table~\ref{tab:sim-combo} spanning exactly the choices a combination committee would face.  The pre-registration lives in the public template, not in anyone's governance chart.\end{objqabody}

\bigskip
\noindent\textbf{Objection 58 (O-KK-58) (the last skeptic standing)}\par\nobreak
\noindent\textbf{``If your framework is right, the evidence has been sitting on tape for years while everyone looked past it.  That strains belief more than any equation.''}\par\nobreak\smallskip
\begin{objqabody}\noindent\emph{Response.}  It should strain belief less than its precedents.  The \(\Omega^-\) sat in bubble-chamber film until someone knew what to scan for; the CMB was in Bell Labs' horn as noise; pentaquarks hid in LHCb's own recorded data for years of running.  Recorded is not the same as extracted: extraction requires a template, and \S\ref{sec:sim-matched} quantifies exactly how far the published templates are from this signal (a factor \(\sim\!3\) in rate, plus the absorption of \S\ref{sec:sim-ladder}).  The claim is not that thousands of physicists were careless; it is that no one runs an analysis for a signal no theory had specified --- and that, precisely, is what this paper exists to change.\end{objqabody}


Fifty-eight objections, answered where an answer exists and conceded where it does not: that is as close to referee-proofing as a single document can come, and the reader who has found their own doubt missing from the list is invited to regard it as the fifty-ninth and write.  All that remains is the record itself --- the works this paper stands on, each with a note explaining why.

\GGPart[References]{References}{Cited works in two groups --- the \GGTheory{} cohort and the external literature --- each entry carrying a relevance note and its first citation point}
\label{part:references}

The reference list is organized in two sections.  \textbf{Section A} lists the eighteen sister papers of the \GGTheory{} cohort, in the canonical order of the program (the constitutional root, the GG6 Trio, the boundary law and its pillars, the three textbook readouts, the capstone, the two phenomenology branches, and the Foundations Branch); every cohort paper is cited in the body of this one, because the lock cluster sits inside a single deductive structure and its verdict propagates through all of it.  \textbf{Section B} lists the external literature --- the theoretical ancestors, the experimental analyses this paper is anchored to, the statistical methods, and the strategy documents of the post-LHC landscape --- alphabetically by first author.

Every entry carries a \textbf{Relevance} paragraph stating what the work contributes to this paper and where it enters, followed by the location of its first citation.  The list contains no entry that is not cited in the text.

All cohort preprints are available at \url{https://arisaka-gg.org/}.  External references are cited under their published or preprint identifiers as given.

\renewcommand{\bibsection}{\vspace{1.2em}}

\section*{A.\ Papers by Katsushi Arisaka (the \GGTheory{} cohort)}

\begin{thebibliography}{99}

\bibitem[Arisaka(2026, A1-Constitution)]{ArisakaA1Constitution}

\textbf{Arisaka, K.\ (2026).}
\textit{The \GGConstitution{}: Four Axioms and Seven Postulates of the \GGTheory{} Program, From Which \GG{}, \BIsix{}, \OGNfive{}, and \GDOS{} Descend as Theorems}.
Arisaka-GG-A1-Constitution preprint.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The foundational charter from which the present paper inherits its rules of admissibility. The lock cluster is presented here as a downstream consequence of the four axioms and seven postulates --- in particular \MSMF{} (no continuous dials) and \POS{}-MF/POS-7 (falsifiability) --- rather than as an independent collider ansatz. We cite it whenever the constitutional provenance of a claim, the naming convention, or the pre-declared-falsifier discipline is invoked.  First cited in \S\ref{sec:central-claim}.

\bibitem[Arisaka(2026, A2-GG6)]{ArisakaA2GG6}

\textbf{Arisaka, K.\ (2026).}
\textit{\GGTheory{} with \GG{}: Geometry--Gauge Locking at the 6D Bulk (\GG{}) at the Heart of Grand Geometric Theory (\GGTheory{})}.
Arisaka-GG-A2-GG6 preprint.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The geometric stage of this paper. We cite A2-GG6 for the canonical scale-profile metric of Theorem~2, the forced bulk of Theorem~0, the non-spatial scale ($I_r$) and phase ($S^1_\theta$) directions, the two-tower convention (angular tower at $\EGUT$, observable radial lock cluster at $\Mlock$), and the Randall--Sundrum reading of the radial spectrum that underlies the entire collider analysis here.  First cited in the Executive Summary.

\bibitem[Arisaka(2026, A3-BI6)]{ArisakaA3BI6}

\textbf{Arisaka, K.\ (2026).}
\textit{Anomaly Freedom of \BIsix{}: The Modified Bianchi Identity (\BIsix{}) for the Six-Force, Six-Dimensional Bulk (\GG{})}.
Arisaka-GG-A3-BI6 preprint.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} Supplies the rigorous proof of the \OGNfive{} Diophantine cascade $(M,N,L;J,K)=(1,3,5;7,35)$ from the single-tensor anomaly identity. The lock integer $K=LJ=35$ that fixes the scale depth --- and hence the derived mass $\Mlock=\tfrac12 M_{\mathrm{Pl}}e^{-35}$ of Theorem~T-KK-1 --- is established there; we cite it wherever the integer ledger is the source of a parameter-free number.  First cited in \S\ref{sec:plainenglish}.

\bibitem[Arisaka(2026, A4-Euclid)]{ArisakaA4Euclid}

\textbf{Arisaka, K.\ (2026).}
\textit{From Euclid to \GG{}: The Realization of Euclid's Elements in Six Dimensions --- \GUfive{}, \OGNfive{}, and a Parameter-Free Reconstruction of Modern Physics}.
Arisaka-GG-A4-Euclid preprint.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} Owns the 6D Einstein--Hilbert reduction and the derivation of the universal constants \GUfive{} as bulk conversion factors.  We cite it specifically for the bulk-to-Planck relation $M_6=\sqrt{2/\pi}\,M_{\mathrm{Pl}}$ (Theorem~T-Euclid-6) and the radial scale-profile integral $\tfrac12(1-e^{-2K})\to\tfrac12$ that together supply the factor $\tfrac12$ in $\Mlock=\tfrac12 M_{\mathrm{Pl}}e^{-K}$, ensuring the centerpiece mass carries no fitted normalization.  First cited in \S\ref{subsec:notKKRS}.

\bibitem[Arisaka(2026, A5-GDOS)]{ArisakaA5GDOS}

\textbf{Arisaka, K.\ (2026).}
\textit{\GDOS{}: Geometric Dynamics of Open Systems --- 6D \GG{} Bulk Projected to 4D Boundary for Quantum Measurement, General Relativity, and QFT}.
Arisaka-GG-A5-GDOS preprint.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} Supplies the open-system (\GDOS) framework on which \S\ref{sec:gdos-section} rests: a collider event is not a closed-system observable but a 6D bulk excitation registered only after projection to the 4D boundary through a dissipative detector and a statistical reconstruction pipeline. We cite it for the open-system framing of detection (\OGGEP{}/\GDOS), not for any collider number.  First cited in \S\ref{sec:central-claim}.

\bibitem[Arisaka(2026, A6-GRIP)]{ArisakaA6Gi4}

\textbf{Arisaka, K.\ (2026).}
\textit{The \GRIP{} on \GDOS{} --- \GRIP{}: the Geometric Engine of Symmetry Breaking, from the Big Bang to Mind}.
Arisaka-GG-A6-GRIP preprint (the runtime definitions of \textsf{GGA}, \textsf{GRIP-In}, and \textsf{GRIP-Out}, and the two-regime figure).
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} Provides the \GRIP{} (Gi-4) four-pillar cascade language used in the open-system reading of trigger, reconstruction, and decision-making. Cited alongside \AFive{} for the detection-as-open-system view of \S\ref{sec:gdos-section}; it contributes no quantitative input to the lock-cluster prediction.  First cited in \S\ref{sec:central-claim}.

\bibitem[Arisaka(2026, A7-Qi4)]{ArisakaA7Qi4}

\textbf{Arisaka, K.\ (2026).}
\textit{The \QiFour{} Pillars --- \QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}: The Quantum-Information Backbone of \GDOS{}}.
Arisaka-GG-A7-Qi4 preprint.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The quantum-information pillars that, with the \GiFour{} pillars of \citet{ArisakaA6Gi4}, force the boundary law \GDOS{}; cited for the origin story of the program (Part~\ref{part:conclusions}) and the information charge \(\UQi\) that partners \(\UB\) in the BI-6 identity.  First cited in \S\ref{sec:central-claim}.

\bibitem[Arisaka(2026, A8-QM)]{ArisakaA8QM}
\textbf{Arisaka, K.} (2026).
\textit{Quantum Mechanics Derived from \GGTheory{} as Projection from 6D Bulk to 4D Boundary}.
Arisaka-GG-A8-QM preprint.
\par\noindent\textbf{Relevance.} Derives measurement as a complete runtime event (apparatus arming = endogenous \GRIPout{}; registration = \GRIPin{} descent; the record = the halt) --- the anatomy applied in \S\ref{sec:gdos-section} to the trigger and reconstruction chain.  First cited in \S\ref{sec:central-claim}.

\bibitem[Arisaka(2026, A9-SM)]{ArisakaA9SM}

\textbf{Arisaka, K.\ (2026).}
\textit{The Standard Model as Theorem of \GGTheory{}: Its 28 Parameters as Observations of One Six-Dimensional Geometry}.
Arisaka-GG-A9-SM preprint.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} Owns the scale chain $\EPl\to\EGUT\to\Mlock\to v$ and the $\tfrac12$ normalization of the lock scale used in the centerpiece. Crucially, it derives the electroweak vacuum expectation value $v=\Mlock(2/5)^3\approx 246$\,GeV from the same $\Mlock$, which is the over-constraint that gives the collider test its unusual falsifying power: a wrong lock cluster would also unseat the electroweak derivation. First cited in \S\ref{sec:plainenglish}.

%% ===== ATLAS =====.  First cited in \S\ref{sec:plainenglish}.

\bibitem[Arisaka(2026, A10-Cosmos)]{ArisakaA10Cosmos}

\textbf{Arisaka, K.\ (2026).}
\textit{The Standard Cosmology as Theorem of \GGTheory{}: The Cosmic Inventory as Observations of One Six-Dimensional Geometry}.
Arisaka-GG-A10-Cosmos preprint.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The cosmological member of the Consequence Trio: the same six-dimensional geometry whose radial lock spectrum this paper sends to the LHC fixes, in A10, the cosmic inventory (\(\Omega_m\approx1/\pi\)) and the derived \(H_0\).  Cited in the cohort map as the third textbook read-out alongside \citet{ArisakaA8QM} and \citet{ArisakaA9SM}; a collider verdict on the lock cluster propagates to A10's geometry exactly as it propagates to the electroweak ledger.  First cited in \S\ref{sec:central-claim}.

\bibitem[Arisaka(2026, A11-Origin)]{ArisakaA11Origin}

\textbf{Arisaka, K.\ (2026).}
\textit{The Geometric Inevitability of Cosmos, Life, and Mind: The Self-Referential Closure of the \GGTheory{} Program}.
Arisaka-GG-A11-Origin preprint.
\par\noindent In preparation. To be posted at \url{https://arisaka-gg.org/} on completion; no result of the
present paper depends on it.
\par\noindent\textbf{Relevance.} The capstone of the A-series: the paper that closes the constitutional arc which the present paper's falsifier exposes to data.  Cited in the cohort map; the shared deductive funnel of \S\ref{sec:central-claim} is carried, with different payloads, by A9, A10, and A11, so the octet verdict at \(3.85\)~TeV is also a stress test of the closure A11 completes.  First cited in \S\ref{sec:central-claim}.

\bibitem[Arisaka(2026, P2-N3)]{ArisakaP2N3}

\textbf{Arisaka, K.\ (2026).}
\textit{Why Three Generations: A Three-Leg Derivation of $N_g=3$ from \BIsix{}, \MSMF{}, and \GDOS{} Morse Stability}.
Arisaka-GG-P2-N3 preprint.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The second Predictions-Branch paper: the three-leg derivation of the generation count from the same \BIsix{} anomaly structure that locks this paper's gauge content.  Cited in the cohort map; the top-quark localization band \(r_{t_R}\in[4,5]\) that fixes the octet's width and polarization here presupposes the three-family structure P2 derives.  First cited in \S\ref{sec:cohort-p1}.

\bibitem[Arisaka(2026, P3-Electron)]{ArisakaP3Electron}

\textbf{Arisaka, K.\ (2026).}
\textit{The Electron as the Ground State of Nature: The Six Oldest Questions about the Electron, Closed and Fenced within \GGTheory{}}.
Arisaka-GG-P3-Electron preprint.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The third Predictions-Branch paper: the electron as the geometric ground state.  Cited in the cohort map; its UV-localized lepton profiles are the same localization data that make this paper's cluster leptophobic in practice --- the two papers read one geometry from opposite ends of the mass ladder.  First cited in \S\ref{sec:cohort-p1}.

\bibitem[Arisaka(2026, P4-Real)]{ArisakaP4Real}

\textbf{Arisaka, K.\ (2026).}
\textit{The Real Geometry Beneath the Complex Numbers of Physics: Positive Frequency is Holomorphy --- from Cardano's Cubics to Twistor Space, via the Sixth Dimension of \GGTheory{}}.
Arisaka-GG-P4-Real preprint.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The origin-of-\(i\) member paper of the Foundations Branch: from the two real structures of the arena this paper's spectrum lives on --- the compact phase circle and the causal clock --- it derives the single imaginary unit of boundary physics, so the interference phases that this paper's amplitudes and line shapes take for granted are outputs of the geometry rather than inputs.  First cited in \S\ref{sec:cohort-p1}.

\bibitem[Arisaka(2026, P5-CPT)]{ArisakaP5CPT}

\textbf{Arisaka, K.\ (2026).}
\textit{The Mirrors of Matter: Why Every Particle Has a Twin, Why Nature's Mirrors Break, and Why the Universe Is Made of Matter}.
Arisaka-GG-P5-CPT preprint.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The mirrors-and-matter member paper of the Foundations Branch: C, P, and T as orientation reversals of the three real bulk factors, with the matter--antimatter asymmetry at the end of the chain.  Its dated kill list names this paper's HL-LHC lock-cluster verdict as a companion channel of the same shared geometry --- the collider face of a battery whose other faces are decided elsewhere.  First cited in \S\ref{sec:cohort-p1}.

\bibitem[Arisaka(2026, P6-Proton)]{ArisakaP6Proton}

\textbf{Arisaka, K.\ (2026).}
\textit{The Proton as the Ground Geometric Attractor of Matter: The Eleven Questions of the Proton, Closed and Fenced within \GGTheory{}}.
Arisaka-GG-P6-Proton preprint.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The sibling standing falsifier, and the honest description of the relation is that it is a sibling rather than an input. Nothing in the lock-cluster derivation uses the proton and nothing in this paper's physics turns on it. What the two share is the ledger: the same integers that place a resonance at \(3.85\)~TeV also forbid the proton from decaying, so the framework is exposed at both ends of the energy axis at once --- a resonance that must appear where it is predicted, and a decay that must never appear at all. A reader weighing how much this framework has staked should count both. Cited at roster level in the cohort map. First cited in \S\ref{sec:cohort-p1}.

\bibitem[Arisaka(2026, W1-Symmetry)]{ArisakaW1Symmetry}

\textbf{Arisaka, K.\ (2026).}
\textit{Symmetry, Conservation, and Geometry: Seven Steps from Noether, Bianchi, to Hopkins--Singer, and the Ledger the Open-System Turn Pays Out}.
Arisaka-GG-W1-Symmetry preprint.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The background this paper's central negative claim needs and does not supply. The structural results here are absences --- no radius modulus, no radion, no landscape, no supersymmetry requirement --- and each is derived rather than assumed; but an absence reads as a result only to someone who already accepts that a conservation statement need not descend from a symmetry. W1 is the cohort's survey of exactly that century-long shift, from Noether to cohomology. It supplies no number and no theorem the lock-cluster chain uses. First cited in \S\ref{subsec:notKKRS}.

\bibitem[Arisaka(2026, W2-Holonomy)]{ArisakaW2Holonomy}

\textbf{Arisaka, K.\ (2026).}
\textit{The Circle That Remembers, the Interval That Forgets: \GGTheory{} and the Sixteen Problems of Particle Physics and Cosmology}.
Arisaka-GG-W2-Holonomy preprint.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The classification the mechanism of this paper's phase circle belongs to. The lock spectrum is shifted by a Wilson line: a constant potential around a compact angle, locally pure gauge, changing no field strength, and moving every charged mode anyway. That is protection and displacement by holonomy rather than by symmetry, which is W2's subject, and the absence of a radius modulus here is an instance of it rather than a coincidence. Cited at roster level in the cohort map, and named at the Wilson-line shift because that is where the mechanism is used. First cited in \S\ref{subsec:nonspatial-convention}.

\bibitem[Arisaka(2026, C1-Kaxion)]{ArisakaC1Kaxion}

\textbf{Arisaka, K.\ (2026).}
\textit{The Kaxion: Cold Dark Matter as the Non-Spatial Compact-Angle Pseudoscalar of \GG{}}.
Arisaka-GG-C1-Kaxion preprint.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The dark-matter companion: the \(8.1\,\mu\)eV kaxion of the same compact \(\theta\) direction whose angular tower this paper predicts \emph{absent} at colliders.  Cited in the cohort map; the two papers are the complementary read-outs of \(S^1_\theta\) --- the haloscope listens to its lightest mode while the LHC confirms the silence of its heavy ones.  First cited in \S\ref{sec:cohort-p1}.

\bibitem[Arisaka(2026, C2-Hubble)]{ArisakaC2Hubble}

\textbf{Arisaka, K.\ (2026).}
\textit{The Hubble Tension in the Light of \GGTheory{}: The Hubble Constant as One Reading of a Derived Expansion History $H(t)$}.
Arisaka-GG-C2-Hubble preprint.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The second Consequences-Branch paper: the derived-\(H_0\) reading of the Hubble tension within the same geometry.  Cited in the cohort map as one of the two phenomenology branches' astrophysical arms that a collider discovery at \(3.85\)~TeV would indirectly fortify.  First cited in \S\ref{sec:cohort-p1}.

\bibitem[Arisaka(2026, C3-BH)]{ArisakaC3BH}

\textbf{Arisaka, K.\ (2026).}
\textit{Black-Hole Entropy under \GGTheory{}: The \GDOS{} Reading of the Horizon as an \OGGEP{} Port --- with the Exact Bekenstein--Hawking $A/4G_N$ Derived from Induced Gravity}.
Arisaka-GG-C3-BH preprint.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The third Consequences-Branch paper: horizon thermodynamics under the same open-system projection whose collider face is this paper's SIP reading of detection (\S\ref{sec:gdos-section}).  Cited in the cohort map; together with \citet{ArisakaC1Kaxion} and \citet{ArisakaC2Hubble} it carries the framework into the arenas a collider cannot reach.  First cited in \S\ref{sec:cohort-p1}.
\end{thebibliography}

\section*{B.\ Other references}

\begin{thebibliography}{99}

\bibitem[Agashe et al.(2008)]{AgasheEtAl2007}

\textbf{Agashe, K., Belyaev, A., Krupovnickas, T., Perez, G., \& Virzi, J. (2008).}
LHC signals from warped extra dimensions.
\textit{Physical Review D}, \textit{77}, 015003. arXiv:hep-ph/0612015.
\par\noindent\textbf{Relevance.} Establishes the top- and longitudinal-boson-rich signatures of IR-localized warped resonances and the UV/IR coupling hierarchy. Cited in \emph{Read-out~II only} (assumptions B2, B4) for the empirical localization--coupling correspondence used to estimate $R_f$.  First cited in \S\ref{subsec:kappagg}.

\bibitem[Allanach et al.(2000)]{AllanachEtAl2000}

\textbf{Allanach, B.~C., Odagiri, K., Parker, M.~A., \& Webber, B.~R. (2000).}
Searching for narrow graviton resonances with the ATLAS detector at the Large Hadron Collider.
\textit{Journal of High Energy Physics}, \textit{0009}, 019. arXiv:hep-ph/0006114.
\par\noindent\textbf{Relevance.} Supplies the corrected massive-vector partial width for a spin-2 resonance (fixing a typographical error in \citealp{HanLykkenZhang1999}) and the benchmark $\sigma\times\mathrm{BR}(e^+e^-)$ table against which the workbook of \S\ref{sec:operational} is validated.  First cited in \S\ref{subsec:workbook}.

\bibitem[ATLAS Collaboration(2012)]{ATLAS2012}

\textbf{ATLAS Collaboration (Aad, G. et al.) (2012).} Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC. \textit{Physics Letters~B}, \textit{716}, 1--29. arXiv:1207.7214.
\par\noindent\textbf{Relevance.} The ATLAS observation of the \(125\)~GeV Higgs boson: the last unchecked entry of the Standard Model's list, completed.  Cited in \S1 as the closing bookend of the confirmation era --- the discovery that finished a structure written down in 1964 and left the list, for the first time in half a century, without a next item.  First cited in Part~I.

\bibitem[ATLAS Collaboration(2019)]{ATLAS2019Dilepton}

\textbf{ATLAS Collaboration (2019).}
Search for high-mass dilepton resonances using 139~fb$^{-1}$ of $pp$ collision data at $\sqrt{s}=13$~TeV with the ATLAS detector.
\textit{Physics Letters B}, \textit{796}, 68--87. arXiv:1903.06248 [hep-ex].
\par\noindent\textbf{Relevance.} The strongest public clean-channel constraint on high-mass spin-1/spin-2 resonances. We cite it as the limit that most directly pressures a universally-coupled lock cluster, and to make the point that the \GG{} prediction evades it for a structural reason --- the partners are leptophobic and top-rich, with $\mathrm{BR}(Z'_B\to\ell^+\ell^-)=0$ exactly --- not because the dilepton limit is weak.  First cited in \S\ref{sec:current-status}.

\bibitem[ATLAS Collaboration(2024)]{ATLAS2024HLUpgrade}

\textbf{ATLAS Collaboration (2024).}
A new ATLAS for the high-luminosity era.
\par\noindent Official ATLAS upgrade overview. \url{https://atlas.cern/}.
\par\noindent\textbf{Relevance.} Documents the Phase-2 tracker, timing, and trigger upgrades that the lock-cluster search depends on. We cite it where the HL-LHC reach and the feasibility of cluster-aware, boosted-top, reduced-stream analyses are argued.  First cited in \S\ref{sec:near-term}.

\bibitem[ATLAS Collaboration(2025a)]{ATLAS2025ttbar}

\textbf{ATLAS Collaboration (2025).}
Search for $t\bar t$ resonances in final states with one or two leptons using 140~fb$^{-1}$ of $pp$ collision data at $\sqrt{s}=13$~TeV.
\par\noindent arXiv:2512.17856 [hep-ex].
\par\noindent\textbf{Relevance.} The most relevant existing search for the priority discovery channel of the lock cluster. We cite it as the template that a dedicated, cluster-aware $t\bar t$-resonance reinterpretation should extend, since the \GG{} graviton and its color-octet/baryonic partners are dominantly top-rich.  First cited in \S\ref{thm:tkk2}.

\bibitem[ATLAS Collaboration(2025b)]{ATLAS2025HighRate}

\textbf{ATLAS Collaboration (2025).}
ATLAS expands its reach into the high-rate frontier.
\par\noindent Official ATLAS briefing on trigger-level analysis. \url{https://atlas.cern/}.
\par\noindent\textbf{Relevance.} Establishes trigger-level (data-scouting) analysis as an available tool. We cite it in the trigger-roadmap section, where a reduced-format stream dedicated to the $3$--$6$\,TeV region is proposed as a high-value, low-cost addition for the clustered signal. First cited in \S\ref{sec:trigger-roadmap}.

%% ===== CERN facility / programme =====.  First cited in \S\ref{sec:trigger-roadmap}.

\bibitem[CEPC Study Group(2025)]{CEPC2025TDR}

\textbf{CEPC Study Group (2025).}
CEPC Technical Design Report --- Accelerator; and CEPC Technical Design Report --- Reference Detector.
IHEP, Beijing, completed October 2025.
\par\noindent\textbf{Relevance.} The mature Chinese Higgs/\(Z\)-factory design.  Not selected for China's 15th Five-Year Plan (March 2026); resubmission intended around 2030.  Status input to the facility landscape of Part~\ref{part:beyond}.  First cited in \S\ref{sec:beyond}.

\bibitem[Cepeda et al.(2019)]{CERNYellowReport2019}

\textbf{Cepeda, M., et al. (2019).}
Report from Working Group 1: Standard Model physics at the HL-LHC and HE-LHC.
\textit{CERN Yellow Reports: Monographs}, CERN-2019-007. arXiv:1902.00134.
\par\noindent\textbf{Relevance.} The canonical HL-LHC projection conventions, including the halved-systematics scenario adopted (and decomposed into its statistical, calibration, and theory classes) in \S\ref{sec:sim-impsyst}.  First cited in \S\ref{sec:sim-impsyst}.

\bibitem[CERN(2023)]{CERN2023HLLHC}

\textbf{CERN (2023).}
The High-Luminosity LHC: a new horizon for science and technology.
\par\noindent Official CERN HL-LHC flyer. \url{https://home.cern/}.
\par\noindent\textbf{Relevance.} The official statement of the $3000$\,fb$^{-1}$/experiment HL-LHC target at $14$\,TeV. We cite it as the dataset that decides the collider falsifier: the spin-1 cluster is either discovered or the current implementation is excluded within this program (the spin-2 anchor, at its derived coupling, awaits a 100-TeV machine; Corollary T-KK-2c).  First cited in \S\ref{sec:near-term}.

\bibitem[CERN(2024a)]{CERN2024CLIC}

\textbf{CERN (2024).}
The Compact Linear Collider (CLIC).
\par\noindent Official CERN accelerator overview. \url{https://home.cern/}.
\par\noindent\textbf{Relevance.} Cited in the future-collider outlook for the indirect electroweak/Higgs-coupling leg of the pattern test, which remains informative even where direct production of the cluster is inaccessible at staged lepton-collider energies.  First cited in \S\ref{sec:longterm}.

\bibitem[CERN(2024b)]{CERN2024Muon}

\textbf{CERN (2024).}
Muon Collider.
\par\noindent Official CERN accelerator overview. \url{https://home.cern/}.
\par\noindent\textbf{Relevance.} Cited for the line-shape and spin diagnostics a multi-TeV muon collider would provide --- precisely the deconvolution of the near-degenerate multi-field cluster that a hadron collider cannot fully resolve.  First cited in \S\ref{sec:longterm}.

\bibitem[CERN(2025)]{CERN2025Luminosity}

\textbf{CERN (2025).}
LHC delivers a record number of particle collisions in 2025.
\par\noindent Official CERN news item, 9 December 2025. \url{https://home.cern/}.
\par\noindent\textbf{Relevance.} The integrated-luminosity backdrop for the ``could this year already matter?'' question. We cite it for the Run-3 dataset size that places the lock scale inside the currently testable range.  First cited in \S\ref{sec:current-status}.

\bibitem[CERN(2026)]{CERN2026Run}

\textbf{CERN (2026).}
Accelerator Report: the 2026 run will be short but intense.
\par\noindent Official CERN news item, 12 March 2026. \url{https://home.cern/}.
\par\noindent\textbf{Relevance.} Establishes the compressed 2026 run window before LS3. We cite it where the paper argues there is genuine, if limited, discovery leverage for the lock cluster in the current year. First cited in \S\ref{sec:current-status}.

%% ===== CMS =====.  First cited in \S\ref{sec:current-status}.

\bibitem[CERN(2026b)]{CERN2026LS3}

\textbf{CERN (2026).}
CERN bids farewell to the LHC and enters Long Shutdown 3.
\par\noindent Official CERN press release, 29 June 2026. \url{https://home.cern/}.
\par\noindent\textbf{Relevance.} The official close-out of Run 3 (last collisions 14 June 2026; $\approx$540~fb$^{-1}$ delivered since startup) and the LS3-to-HL-LHC schedule used for the luminosity-on-tape model of Table~\ref{tab:timeline-inputs} and Figure~\ref{fig:timeline}.  First cited in \S\ref{subsec:timeline}.

\bibitem[Chivukula et al.(2012)]{ChivukulaEtAl2012}

\textbf{Chivukula, R.~S., Farzinnia, A., Foadi, R., \& Simmons, E.~H. (2012).}
Production of massive color-octet vector bosons at next-to-leading order.
\textit{Physical Review D}, \textit{85}, 054005. arXiv:1111.7261.
\par\noindent\textbf{Relevance.} The complete NLO computation for massive color-octet vector production (corrections up to $\sim$30\%, scale dependence reduced to per-cent level); source of the $K_{q\bar q}\simeq1.3$ used for the $g^{*}$ rates in Part~\ref{part:ttanatomy} and \S\ref{sec:operational}.  First cited in \S\ref{sec:ttanatomy-intro}.

\bibitem[CMS Collaboration(2012)]{CMS2012}

\textbf{CMS Collaboration (Chatrchyan, S. et al.) (2012).} Observation of a new boson at a mass of 125~GeV with the CMS experiment at the LHC. \textit{Physics Letters~B}, \textit{716}, 30--61. arXiv:1207.7235.
\par\noindent\textbf{Relevance.} The simultaneous CMS observation of the Higgs boson.  Cited in \S1 alongside the ATLAS paper; the author served on CMS during this era (Part~\ref{part:conclusions}), watching the Higgs \emph{complete} the Standard Model rather than break it --- the experience from which this paper's insistence on pre-registered falsifiability descends.  First cited in Part~I.

\bibitem[CMS Collaboration(2019)]{CMS2019MTD}

\textbf{CMS Collaboration (2019).}
A new precision timing detector for the CMS HL-LHC upgrade (MTD).
\par\noindent Official CMS news item. \url{https://cms.cern/}.
\par\noindent\textbf{Relevance.} Cited where precision timing is promoted from a pile-up tool to a resonance-quality variable for high-mass boosted-hadronic final states --- a concrete way to sharpen the cluster line shape.  First cited in \S\ref{sec:longterm}.

\bibitem[CMS Collaboration(2020)]{CMS2020DileptonPublic}

\textbf{CMS Collaboration (2020).}
High-mass dilepton resonances in $pp$ collisions at $\sqrt{s}=13$~TeV (CMS-EXO-19-019).
\par\noindent Official CMS public-results summary. \url{https://cms.cern/}.
\par\noindent\textbf{Relevance.} The CMS counterpart to the ATLAS dilepton limit; cited together with it as the clean-channel constraint that the leptophobic, non-universal lock cluster evades while a generic universal resonance would not.  First cited in \S\ref{sec:current-status}.

\bibitem[CMS Collaboration(2020b)]{CMS2020Dijet}

\textbf{CMS Collaboration (2020).}
Search for high mass dijet resonances with a new background prediction method in proton--proton collisions at $\sqrt{s}=13$~TeV.
\textit{Journal of High Energy Physics}, \textit{05}, 033. arXiv:1911.03947.
\par\noindent\textbf{Relevance.} The 137~fb$^{-1}$ narrow-dijet limit ($\sigma B A$) whose observed curve appears in Fig.~\ref{fig:overlay}; the reference dijet constraint against which the octet and $Z'_B$ hadronic rates are compared in \S\ref{sec:operational}.  First cited in \S\ref{subsec:sigmabr-results}.

\bibitem[CMS Collaboration(2024a)]{CMS2024Diphoton}

\textbf{CMS Collaboration (2024).}
Search for new physics in high-mass diphoton events at $\sqrt{s}=13$~TeV (CMS-EXO-22-024).
\par\noindent arXiv:2405.09320 [hep-ex].
\par\noindent\textbf{Relevance.} The diphoton spin-2 search; cited as a clean but, for \GG{}, sub-dominant channel: the geometry suppresses light-state couplings, so diphoton is a corroborating rather than a discovery channel for the cluster.  First cited in \S\ref{subsec:kappagg}.

\bibitem[CMS Collaboration(2024b)]{CMS2024DiphotonPublic}

\textbf{CMS Collaboration (2024).}
High-mass diphoton search, public-results summary (CMS-EXO-22-024).
\par\noindent Official CMS public-results summary. \url{https://cms.cern/}.
\par\noindent\textbf{Relevance.} The public-results companion to the diphoton paper, cited in the channel-by-channel exclusion ledger as the traceable diphoton input for a geometry-aware reinterpretation.  First cited in \S\ref{sec:current-status}.

\bibitem[CMS Collaboration(2024c)]{CMS2024HHbbbb}

\textbf{CMS Collaboration (2024).}
Search for resonant Higgs-boson pair production in the $b\bar b b\bar b$ final state at $\sqrt{s}=13$~TeV.
\par\noindent arXiv:2407.13872 [hep-ex].
\par\noindent\textbf{Relevance.} The $hh$ channel that probes the spin-2 graviton's $hh$ decay and the IR-sensitive Higgs sector of the cluster. We cite it as the best scalar-corroborator search and as a template for the boosted-$hh$ leg of a combined-channel \GG{} analysis.  First cited in \S\ref{sec:current-status}.

\bibitem[CMS Collaboration(2024d)]{CMS2024DijetAnomalous}

\textbf{CMS Collaboration (2024).}
Model-agnostic search for dijet resonances with anomalous jet substructure at $\sqrt{s}=13$~TeV.
\par\noindent arXiv:2412.03747 [hep-ex].
\par\noindent\textbf{Relevance.} The boosted-dijet / substructure search most sensitive to the leptophobic, color-octet side of the cluster. We cite it as the channel where the $\UB$ admixture and KK-gluon partner become visible, and as a natural host for a cluster-aware (multi-pole) template.  First cited in \S\ref{thm:tkk2}.

\bibitem[CMS Collaboration(2024e)]{CMS2024UpgradeWeek}

\textbf{CMS Collaboration (2024).}
The CMS Upgrade Week: knowledge sharing within the collaboration.
\par\noindent Official CMS news item. \url{https://cms.cern/}.
\par\noindent\textbf{Relevance.} The CMS Phase-2 upgrade program --- new tracker, MIP timing detector, high-granularity calorimetry, and the Level-1 scouting stream --- as presented in the collaboration's upgrade-week material.  Cited for the detector capabilities that the trigger and reduced-stream proposals of this paper (Table~\ref{tab:trigger}) build on, and that keep the boosted-top performance assumptions of Part~\ref{part:hllhcsim} conservative.  First cited in \S\ref{sec:near-term}.

\bibitem[CMS Collaboration(2025)]{CMS2025Scouting}

\textbf{CMS Collaboration (2025).}
How CMS records all 40 million events per second (Level-1 scouting).
\par\noindent Official CMS news item. \url{https://cms.cern/}.
\par\noindent\textbf{Relevance.} The CMS data-scouting capability cited in the trigger roadmap as the mechanism that would let a dedicated reduced-format stream capture the clustered high-mass signal without saturating the standard trigger budget. First cited in \S\ref{sec:trigger-roadmap}.

%% ===== FCC =====.  First cited in \S\ref{sec:trigger-roadmap}.

\bibitem[Connes \& Lott(1991)]{ConnesLott1991}

\textbf{Connes, A., \& Lott, J. (1991).} Particle models and noncommutative geometry. \textit{Nuclear Physics B (Proceedings Supplements)}, \textit{18}(2), 29--47.
\par\noindent\textbf{Relevance.} The noncommutative-geometry Standard Model: gauge and Higgs structure encoded in a genuinely non-spatial internal geometry.  Cited in the genealogy of \S\ref{subsec:notKKRS} as the furthest prior step toward non-spatial internal dimensions --- \GG{} differs by making its two internal directions literal bulk coordinates, forced by Theorem~0 of \citet{ArisakaA2GG6}.  First cited in \S\ref{subsec:notKKRS}.

\bibitem[Cowan(1998)]{Cowan1998}

\textbf{Cowan, G. (1998).} \textit{Statistical Data Analysis}. Oxford University Press.
\par\noindent\textbf{Relevance.} The standard first statistics text for particle physicists; the recommended companion to the discovery-statistics course of Appendix~L and the formalism of Part~\ref{part:hllhcsim}.  First cited in Part~\ref{part:appendices}.

\bibitem[Cowan et al.(2011)]{CowanEtAl2011}

\textbf{Cowan, G., Cranmer, K., Gross, E., \& Vitells, O. (2011).}
Asymptotic formulae for likelihood-based tests of new physics.
\textit{European Physical Journal C}, \textit{71}, 1554. arXiv:1007.1727.
\par\noindent\textbf{Relevance.} The standard machinery of discovery statistics --- profile-likelihood test statistics, Wilks asymptotics, the Asimov dataset, and the significance formula with background uncertainty --- derived step by step and used for every projection in Part~\ref{part:hllhcsim}.  First cited in \S\ref{sec:sim-stats}.

\bibitem[Davoudiasl, Hewett \& Rizzo(2000)]{DavoudiaslHewettRizzo2000}

\textbf{Davoudiasl, H., Hewett, J.~L., \& Rizzo, T.~G. (2000).}
Phenomenology of the Randall--Sundrum gauge hierarchy model.
\textit{Physical Review Letters}, \textit{84}, 2080--2083. arXiv:hep-ph/9909255.
\par\noindent\textbf{Relevance.} The standard normalization of warped-graviton couplings and production cross sections in terms of $k/\mplred$. We use it in \emph{Read-out~II only} (\S\ref{sec:scenarioB}, assumption B1) as the bulk-graviton template against which $\kappaeff\approx1/\sqrt{K}$ is calibrated --- an external, model-level input, not a \GG{}-A result.  First cited in \S\ref{sec:scenarioB}.

\bibitem[Dobrescu \& Yu(2013)]{DobrescuYu2013}

\textbf{Dobrescu, B.~A., \& Yu, F. (2013).}
Coupling--mass mapping of dijet peak searches.
\textit{Physical Review D}, \textit{88}, 035021. arXiv:1306.2629.
\par\noindent\textbf{Relevance.} The standard widths and coupling--mass mapping for leptophobic baryonic $Z'$ states; \S\ref{sec:operational} uses its conventions for the $g_B^2$ scaling of the $Z'_B$ rates and for the octet width cross-check.  First cited in \S\ref{subsec:workbook}.

\bibitem[Ellis, Stirling \& Webber(1996)]{EllisStirlingWebber1996}

\textbf{Ellis, R.~K., Stirling, W.~J., \& Webber, B.~R. (1996).} \textit{QCD and Collider Physics}. Cambridge University Press.
\par\noindent\textbf{Relevance.} The definitive treatment of factorization, parton luminosities, and resonance production at hadron colliders; the primary companion to Appendix~K, whose master formulas it derives at full depth.  First cited in Part~\ref{part:appendices}.

\bibitem[European Strategy Group(2025)]{ESPP2025}

\textbf{European Strategy Group (2025).}
European Strategy for Particle Physics: 2026 update milestone --- FCC-ee recommended as the preferred next flagship collider.
\par\noindent Official CERN press release, December 2025. \url{https://home.cern/}.
\par\noindent\textbf{Relevance.} Fixes the post-LHC calendar assumed in \S\ref{subsec:timeline}: FCC-ee as the next machine (indirect at the lock scale), with the first direct probe of $3.85$~TeV deferred to FCC-hh.  First cited in \S\ref{subsec:timeline}.

\bibitem[European Strategy Group(2026)]{ESPP2026Adoption}

\textbf{European Strategy Group (2026).}
The European Strategy for Particle Physics --- 2026 Update.
Adopted unanimously by the CERN Council, Budapest, 22 May 2026.
\par\noindent\textbf{Relevance.} Names the FCC-ee the preferred option for the next flagship collider at CERN, with a Council decision expected by 2028, and prioritizes muon-collider R\&D.  The post-LHC planning baseline of Part~\ref{part:beyond}.  First cited in \S\ref{sec:beyond}.

\bibitem[FCC Collaboration(2018)]{FCC2018}

\textbf{FCC Collaboration (2018).}
FCC-hh: the Hadron Collider --- Future Circular Collider Conceptual Design Report.
\par\noindent CERN. \url{https://fcc.web.cern.ch/}.
\par\noindent\textbf{Relevance.} The $100$\,TeV machine that would turn the lock-cluster corridor into a fully mappable sector: the first cluster, the radial $n=2$ ($\sim7$\,TeV) and $n=3$ ($\sim10$\,TeV) families, and the colored partners would all be in reach. Cited in the long-term-prospects section. First cited in \S\ref{subsec:kappagg}.

%% ===== Standard literature =====.  First cited in \S\ref{subsec:kappagg}.

\bibitem[FCC Collaboration(2025)]{FCC2025FSR}

\textbf{FCC Collaboration (2025).}
Future Circular Collider Feasibility Study Report, Volume 1.
\par\noindent arXiv:2505.00272 [physics.acc-ph].
\par\noindent\textbf{Relevance.} The staged FCC scenario (FCC-ee physics $\sim$2047+, FCC-hh $\sim$2074) that sets the right edge of Figure~\ref{fig:timeline} and the date at which $\sigma(G_1)\approx205$~fb becomes directly testable.  First cited in \S\ref{subsec:timeline}.

\bibitem[Franceschini et al.(2016)]{FranceschiniEtAl2016}

\textbf{Franceschini, R., et al. (2016).}
What is the $\gamma\gamma$ resonance at 750 GeV?
\textit{Journal of High Energy Physics}, \textit{03}, 144. arXiv:1512.04933.
\par\noindent\textbf{Relevance.} The reference narrow-width master formula and parton-luminosity coefficients used to validate the workbook of \S\ref{sec:operational} (reproduced there to 2--5\%), together with its quoted NLO K-factors.  First cited in \S\ref{subsec:workbook}.

\bibitem[Fukuda et al.(1998)]{Fukuda1998}

\textbf{Fukuda, Y. et al. (Super-Kamiokande Collaboration) (1998).} Evidence for oscillation of atmospheric neutrinos. \textit{Physical Review Letters}, \textit{81}, 1562--1567.
\par\noindent\textbf{Relevance.} The discovery of atmospheric-neutrino oscillation --- the lone established crack in the Standard-Model edifice of the past fifty years, and pointedly \emph{not} a collider result.  Cited in \S1 as the exception that sharpens the rule: no collider has ever added a particle to the Standard Model's list, and the one confirmed beyond-Standard-Model fact prescribes no collider scale --- the vacuum this paper's derived target fills.  First cited in Part~I.

\bibitem[Goldberger \& Wise(1999)]{GoldbergerWise1999}

\textbf{Goldberger, W.~D., \& Wise, M.~B. (1999).} Modulus stabilization with bulk fields. \textit{Physical Review Letters}, \textit{83}, 4922--4925. arXiv:hep-ph/9907447.
\par\noindent\textbf{Relevance.} The canonical stabilization mechanism for the Randall--Sundrum radion: a bulk scalar whose potential fixes the brane separation.  Cited in Table~\ref{tab:kkrsgg} as the emblem of what every \emph{spatial} extra dimension owes and what \GG{} does not: the scale direction's length is the integer \(K=35\), so there is no modulus to stabilize and no radion --- an absence this paper promotes to a falsifiable Higgs-precision prediction (Part~\ref{part:beyond}).  First cited in \S\ref{subsec:notKKRS}.

\bibitem[Gross \& Vitells(2010)]{GrossVitells2010}

\textbf{Gross, E., \& Vitells, O. (2010).}
Trial factors for the look elsewhere effect in high energy physics.
\textit{European Physical Journal C}, \textit{70}, 525--530. arXiv:1005.1891.
\par\noindent\textbf{Relevance.} The rigorous (up-crossings) treatment of the look-elsewhere effect; \S\ref{sec:sim-lee} uses the resolution-cell estimate it validates to quantify the \(\sim\!75\)-fold trials factor that the pre-registered \GG{} search does not pay.  First cited in \S\ref{sec:sim-lee}.

\bibitem[Halzen \& Martin(1984)]{HalzenMartin1984}

\textbf{Halzen, F., \& Martin, A.~D. (1984).} \textit{Quarks and Leptons: An Introductory Course in Modern Particle Physics}. Wiley.
\par\noindent\textbf{Relevance.} The classic accessible source for cross-section and decay estimates; the gentlest complete route into the worked examples of Appendix~K.  First cited in Part~\ref{part:appendices}.

\bibitem[Han, Lykken \& Zhang(1999)]{HanLykkenZhang1999}

\textbf{Han, T., Lykken, J.~D., \& Zhang, R.-J. (1999).}
On Kaluza--Klein states from large extra dimensions.
\textit{Physical Review D}, \textit{59}, 105006. arXiv:hep-ph/9811350.
\par\noindent\textbf{Relevance.} The canonical spin-2 partial-width formulas used for the $G_1$ rates in \S\ref{sec:operational} (with the massive-vector polynomial taken from \citealp{AllanachEtAl2000}).  First cited in \S\ref{subsec:workbook}.

\bibitem[Hosotani(1983)]{Hosotani1983}

\textbf{Hosotani, Y. (1983).} Dynamical mass generation by compact extra dimensions. \textit{Physics Letters~B}, \textit{126}(5), 309--313.
\par\noindent\textbf{Relevance.} The Hosotani mechanism: gauge-symmetry breaking by the Wilson-line holonomy of a compact direction.  In \GG{} this is not one model-building option among many but the \emph{operative} electroweak mechanism --- the phase direction \(S^1_\theta\) supplies the Wilson line \(\theta_H\) whose selected value gives \(v=\Mlock(2/5)^3\) in \citet{ArisakaA9SM} --- so the comparison row of Table~\ref{tab:kkrsgg} credits the mechanism at its source.  First cited in \S\ref{subsec:notKKRS}.

\bibitem[International Muon Collider Collaboration(2024)]{IMCC2024Interim}

\textbf{Accettura, C., et al.\ (International Muon Collider Collaboration) (2024).}
Interim report for the International Muon Collider Collaboration.
arXiv:2407.12450.
\par\noindent\textbf{Relevance.} The staged \(3\to10\)~TeV design, the ionization-cooling demonstrator roadmap, and the reference schedule (stage-1 near 2050, 10~TeV near 2070) assumed in Part~\ref{part:beyond}; the leptophobia null test of \S\ref{sec:beyond-muc} is defined against this machine.  First cited in \S\ref{sec:beyond}.

\bibitem[Kaluza(1921)]{Kaluza1921}

\textbf{Kaluza, T. (1921).}
Zum Unit{\"a}tsproblem der Physik.
\textit{Sitzungsberichte der Preu{\ss}ischen Akademie der Wissenschaften zu Berlin}, 966--972.
\par\noindent\textbf{Relevance.} The historical origin of the idea that gauge structure can be read geometrically from an extra coordinate. We cite it to situate the \emph{non-spatial} GG-6 reading: the lineage of ``KK'' is retained for discoverability even though the GG-6 directions are scale and phase, not space.  First cited in \S\ref{sec:geometry-origin}.

\bibitem[Klein(1926)]{Klein1926}

\textbf{Klein, O. (1926).}
Quantentheorie und f{\"u}nfdimensionale Relativit{\"a}tstheorie.
\textit{Zeitschrift f{\"u}r Physik}, \textit{37}, 895--906.
\par\noindent\textbf{Relevance.} Introduced compactification and the discrete momentum tower that gives ``Kaluza--Klein'' its spectral meaning. We cite it for the standard mechanism by which a compact direction yields a discrete spectrum --- the precedent we deliberately contrast with the RG-scale-throat origin of the GG-6 radial cluster.  First cited in \S\ref{sec:geometry-origin}.

\bibitem[Lillie, Randall \& Wang(2007)]{LillieRandallWang2007}

\textbf{Lillie, B., Randall, L., \& Wang, L.-T. (2007).}
The bulk RS KK-gluon at the LHC.
\textit{Journal of High Energy Physics}, \textit{0709}, 074. arXiv:hep-ph/0701166.
\par\noindent\textbf{Relevance.} The reference computation of the warped KK-gluon: its top-rich branching pattern, broad width, and light-quark-suppressed production. We borrow its coupling ratios in \emph{Read-out~II only} (assumption B4) to convert the GG-6 color-octet structure into the $\sigma\times\mathrm{BR}(\ttbar)$ ballpark; the result is model-level.  First cited in \S\ref{sec:scenarioB}.

\bibitem[Maldacena(1997/1998)]{Maldacena1997}

\textbf{Maldacena, J. (1998).} The large-\(N\) limit of superconformal field theories and supergravity. \textit{Advances in Theoretical and Mathematical Physics}, \textit{2}, 231--252. arXiv:hep-th/9711200.
\par\noindent\textbf{Relevance.} The AdS/CFT correspondence: the renormalization-group scale of a boundary theory realized as a bulk radial coordinate --- \emph{as a duality}.  Cited in \S\ref{subsec:notKKRS} because \GG{} takes the same identification literally rather than dually: the throat resonances of this paper \emph{are} the composite states of the boundary theory at its infrared scale, which is why the holographic language of ``composite resonances'' describes the lock cluster exactly.  First cited in \S\ref{subsec:notKKRS}.

\bibitem[Mathews \& Ravindran(2005)]{MathewsRavindran2005}

\textbf{Mathews, P., \& Ravindran, V. (2005).}
NLO-QCD corrections to dilepton production in the Randall--Sundrum model.
arXiv:hep-ph/0506158.
\par\noindent\textbf{Relevance.} NLO K-factors for spin-2 (RS graviton) production at the LHC ($K\simeq1.5$--$1.6$); source of the $K_{gg}\simeq1.6$ applied to the $G_1$ rates in Part~\ref{part:ttanatomy}.  First cited in \S\ref{sec:ttanatomy-intro}.

\bibitem[Minkowski(1908)]{Minkowski1908}

\textbf{Minkowski, H. (1908).} Raum und Zeit. \textit{Physikalische Zeitschrift}, \textit{10}, 75--88.
\par\noindent\textbf{Relevance.} The first non-spatial geometric dimension: time, fused into spacetime.  Cited in \S\ref{subsec:notKKRS} as the precedent that legitimizes classifying dimensions by their symmetry generators rather than by spatial intuition --- the classification under which \GG{}'s scale and phase directions are dimensions in exactly Minkowski's sense, and spatial in no sense.  First cited in \S\ref{subsec:notKKRS}.

\bibitem[Olver et al.(2010)]{OlverEtAl2010DLMF}
\textbf{F.~W.~J.~Olver, D.~W.~Lozier, R.~F.~Boisvert, and C.~W.~Clark (eds.).}
\emph{NIST Handbook of Mathematical Functions.}
Cambridge University Press, 2010.  Companion online release: the NIST Digital Library of Mathematical Functions, \url{https://dlmf.nist.gov/}.
\par\noindent\textbf{Relevance.} The standard modern reference for the Bessel functions \(J_n\), \(Y_n\) and their zeros \(x_n\), which carry the entire radial spectrum of this paper --- the lock mass, the tower ratios \(x_2/x_1\) and \(x_3/x_1\), and the mode profiles of the overlap integrals; cited at the boundary-value course of Appendix~\ref{app:kkcourse} as the recommended companion for a reader meeting the machinery for the first time.  First cited in \S\ref{app:kkcourse}.

\bibitem[Particle Data Group(2024)]{PDG2024}

\textbf{Particle Data Group (Navas, S., et al.) (2024).}
Review of Particle Physics.
\textit{Physical Review D}, \textit{110}, 030001.
\par\noindent\textbf{Relevance.} The canonical compilation of measured Standard-Model parameters.  Cited for the world-average strong coupling \(\alpha_s(M_Z)=0.1180\pm0.0009\) against which the \GG{} structural value \(1/(\pi e)=0.11710\) is compared in Part~\ref{part:beyond} --- the single number FCC-ee will turn into a decisive test --- and as the source of the top mass and electroweak inputs used throughout.  First cited in \S\ref{sec:beyond-fccee}.

\bibitem[Peskin \& Schroeder(1995)]{PeskinSchroeder1995}

\textbf{Peskin, M.~E., \& Schroeder, D.~V. (1995).} \textit{An Introduction to Quantum Field Theory}. Westview Press.
\par\noindent\textbf{Relevance.} The standard graduate field-theory text; supplies the mode-decomposition and effective-Lagrangian machinery assumed in the KK course of Appendix~J.  First cited in Part~\ref{part:appendices}.

\bibitem[Pumplin et al.(2002)]{Pumplin2002CTEQ6}

\textbf{Pumplin, J., Stump, D.~R., Huston, J., Lai, H.-L., Nadolsky, P., \& Tung, W.-K. (2002).}
New generation of parton distributions with uncertainties from global QCD analysis.
\textit{Journal of High Energy Physics}, \textit{0207}, 012. arXiv:hep-ph/0201195.
\par\noindent\textbf{Relevance.} The CTEQ6 parton distributions used for the parton luminosities of \S\ref{sec:operational}; the CTEQ6M/CTEQ6L1 spread is part of the quoted uncertainty band.  First cited in \S\ref{subsec:workbook}.

\bibitem[Randall \& Sundrum(1999)]{RandallSundrum1999}

\textbf{Randall, L., \& Sundrum, R. (1999).}
A large mass hierarchy from a small extra dimension.
\textit{Physical Review Letters}, \textit{83}, 3370--3373. arXiv:hep-ph/9905221.
\par\noindent\textbf{Relevance.} The warped two-brane construction whose Bessel-root spectrum and TeV-scale resonances are the closest phenomenological analogue of the GG-6 lock cluster. We cite it throughout the scorecard: GG-6 borrows the warped-throat discreteness but differs decisively in that its infrared scale is \emph{derived} (the integer $K=35$) rather than a fitted curvature, and its cluster is multi-field rather than a single graviton pole.  First cited in \S\ref{sec:geometry-origin}.

\bibitem[Thomson(2013)]{Thomson2013}

\textbf{Thomson, M. (2013).} \textit{Modern Particle Physics}. Cambridge University Press.
\par\noindent\textbf{Relevance.} The experiment-aware modern survey; the recommended first pass over every sector touched by the appendix course, written from the detector side of the field --- the perspective this paper's Preface addresses.  First cited in Part~\ref{part:appendices}.

\bibitem[Weyl(1918)]{Weyl1918}

\textbf{Weyl, H. (1918).} Gravitation und Elektrizit\"at. \textit{Sitzungsberichte der Preussischen Akademie der Wissenschaften}, 465--480.
\par\noindent\textbf{Relevance.} Weyl's original \emph{Eichinvarianz}: the first geometrization of \emph{scale} as a connection.  It failed as a theory of electromagnetism (Einstein's second-clock objection), but it is the ur-precedent for the \GG{} scale dimension \(I_r\) --- the 1918 half of the genealogy traced in \S\ref{subsec:notKKRS}, completed when \GG{} promotes scale to a forced, non-spatial dimension of the bulk.  First cited in \S\ref{subsec:notKKRS}.

\bibitem[Weyl(1929)]{Weyl1929}

\textbf{Weyl, H. (1929).} Elektron und Gravitation.~I. \textit{Zeitschrift f\"ur Physik}, \textit{56}, 330--352.
\par\noindent\textbf{Relevance.} The birth of modern gauge theory: Weyl's reinterpretation of his 1918 idea as a local \emph{phase} (\(U(1)\)) invariance.  The \GG{} phase dimension \(S^1_\theta\) is this insight taken literally --- the \(U(1)\) fiber promoted to a coordinate --- making Weyl the single ancestor of both non-spatial directions (\S\ref{subsec:notKKRS}).  First cited in \S\ref{subsec:notKKRS}.

\end{thebibliography}

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