Trinity S³AI — H4 Coxeter Group / Standard Model Unification

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English

Status

This is an active boundary-mapping research program. See RESEARCH_STATUS.md.

This repository documents an ongoing investigation into whether geometric invariants of the H4 Coxeter group (and related structures such as the 600-cell and Clifford algebras Cl(8)) can encode or constrain the parameters of the Standard Model of particle physics.

Boundary theorems (NGT-1–NGT4) map specific obstruction points: they prove that certain direct constructions from H4 geometry do not reproduce the SM. These are guideposts, not tombstones — they narrow the search space and direct future exploration toward Tracks A (honest phenomenology), B (Cl(8) / J₃(𝕆)), and C (publication of boundary results).

The project maintains a living catalog of 59 numerical coincidences between H4 invariants and PDG 2024 measurements, all formally verified in Coq and tagged by epistemic status. Whether these coincidences are deep or accidental is itself an open research question.

Quick Links

Resource	Description
docs/REVIEW_GUIDE.md	For reviewers: 10-minute path with commands and expected outputs
docs/REPOSITORY_MAP.md	Where every kind of artifact lives in this repository
docs/CLAIM_STATUS.md	Claim-status rule book (verified / empirical fit / open / refuted)
docs/TECH_TREE.md	Layered status of the whole stack (infra → proofs → fits → game → paper)
docs/REVIEW_CHECKLIST.md	Mechanical review-readiness checklist
RESEARCH_STATUS.md	Boundary map: which hypotheses are open, obstructed, or under exploration
EPISTEMOLOGY.md	Calculation-primacy doctrine: how model-dependent extractions are handled
PREDICTIONS_PREREGISTERED.md	9 pre-registered predictions (0 theoretically confirmed, 4 numerically verified, 4 open, 1 withdrawn)
ROADMAP_WAVE17_PLUS.md	Active research program: Tracks A (honest phenomenology), B (Cl(8) / J3(O)), C (boundary-result paper)
paper/CHANGELOG_v1_to_v2.md	Wave 17 paper v2 changelog — new obstruction NGT-7, updated counters, E8-plumbing / string-correspondence boundary notes
SECURITY.md	Security policy and reporting path
HONESTY_MANIFEST.md	Ground-truth statistics (comments stripped)
NoGoTheorems.v	Formal obstruction theorems NGT1–NGT4 (boundary markers, not dead ends)
Catalog42.v	42 SM parameter formulas (tagged phenomenological)
admitted_log.md	Log of all Admitted with their tags
CONTRIBUTING.md	Contribution rules + build instructions
scripts/prepare_zenodo.md	Zenodo publication guide
FOUNDATIONS.md	Axiom stratification and open assumptions
proofs/clifford_cl8/README.md	Track B (Wave 12) — Cl(p,q) formalization launch (T1–T3)
GOLDEN BRIDGE (live)	Live hypothesis-discovery puzzle (not evidence; see game README)

For external reviewers: start with docs/REVIEW_GUIDE.md. It lists what to read, what commands to run, and how to judge each kind of claim against docs/CLAIM_STATUS.md. The live GOLDEN BRIDGE canvas is at https://t27.ai/trinity-s3ai/; CI status is shown by the badges at the top of this file. Known blind spots and proof debt are tracked in docs/TECH_TREE.md. No Theory-of- Everything claim and no prize claim is made.

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Claim Ledger (SSOT)

The table below is auto-generated from docs/claims.yaml by scripts/generate_claims.py. The same ledger drives the GOLDEN BRIDGE card data at games/trinity_fold/fixtures/generated_claim_cards.json. Status vocabulary is defined in docs/CLAIM_STATUS.md.

Generated from docs/claims.yaml by scripts/generate_claims.py. Do not edit this block by hand.

Claim	Layer	Status	Evidence	Blocking gap
GOLDEN BRIDGE live canvas deployed on GitHub Pages (Rust + wasm)	L0	verified	.github/workflows/pages.yml; live at https://t27.ai/trinity-s3ai/	—
Wave 17 honest counter reports 0 real Admitted. in proofs/trinity/	L2	verified	scripts/count_admitted_honest.py output; HONESTY_MANIFEST.md	Holds only for proofs/trinity/. Track B (proofs/clifford_cl8/) retains load-bearing Axioms with citations; the Lagrangian derivation is not closed.
NGT1-NGT4: four formal No-Go theorems closed with Qed.	L2	verified	proofs/trinity/NoGoTheorems.v (NGT1, NGT2, NGT3, NGT4 all Qed)	—
E8 plumbing: eta discrepancy does NOT converge to -2	L4	high_risk_or_falsified	paper/CHANGELOG_v1_to_v2.md (Honesty Notes); Wave 17 audit	—
No known string / heterotic / F-theory / orbifold compactification rescues the SM hierarchy from H4 or F4	L4	high_risk_or_falsified	ROADMAP_WAVE17_PLUS.md (Wave 17.2 Findings)	—
a4 conversion factor not fully reconciled across three derivations	L3	open_conjecture	docs/analysis/a4_conversion_factor_analysis.md; docs/analysis/a4_honest_resolution.md	Produce a single derivation that all three (analytic, spectral, fit) paths agree on, or document why they cannot agree and downgrade the relevant fits.
m_H = 4 phi^3 e^2 ~ 125.1 GeV is the Higgs mass	L5	empirical_fit	proofs/trinity/HiggsPrediction.v (interval bound); docs/analysis/higgs_potential_proven.md	Derive m_H from H4 / NCG structure rather than from a (phi, e) monomial. Any such derivation must pass the anti-numerology gate and avoid the NGT2 sigma-field obstruction.
GOLDEN BRIDGE puzzle is a hypothesis-discovery game, not evidence	L6	verified	games/trinity_fold/README.md; ring0_core::ClaimStatus enum	—
No Theory-of-Everything claim and no prize claim is made	L1	verified	docs/CLAIM_STATUS.md §2; README.md preamble	—

To update: edit docs/claims.yaml, run python3 scripts/generate_claims.py, commit both the YAML and the regenerated artefacts. CI runs the same script with --check and fails if anything is stale.

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Hypothesis Verification Tech Tree

The goal is not to prove everything by prose; the goal is to make each claim unlock only after the previous verification layer passes. This repository is a layered hypothesis-verification stack, not a finished theory. Lower layers (infrastructure, claim ledger, formal proofs) gate what is allowed to be claimed at higher layers (geometry, Lagrangian, fits, game, publication). When a layer is not green, the layers above it are explicitly flagged as open_conjecture or high_risk_or_falsified per docs/CLAIM_STATUS.md.

The fuller table (status per row, file pointers, next milestone) lives in docs/TECH_TREE.md. The diagram below is the grep-friendly summary.

                       +-------------------------------------+
                       | L7  Publication                     |
                       | paper v2 / arXiv / Zenodo / talks   |
                       +-------------------^-----------------+
                                           |
                       +-------------------+-----------------+
                       | L6  GOLDEN BRIDGE Game              |
                       | rings, claim cards, falsified floor |
                       +-------------------^-----------------+
                                           |
                       +-------------------+-----------------+
                       | L5  Numerical Fits & Validation     |
                       | 59 formulas, error windows, p-value |
                       +-------------------^-----------------+
                                           |
                       +-------------------+-----------------+
                       | L4  NCG / Lagrangian                |
                       | spectral triple, sigma-field, SM L  |
                       +-------------------^-----------------+
                                           |
                       +-------------------+-----------------+
                       | L3  Geometry  (H4 / 600-cell / Cl)  |
                       | root system, Snub 24-cell, Cl(8)    |
                       +-------------------^-----------------+
                                           |
                       +-------------------+-----------------+
                       | L2  Formal Proofs (Coq / Lean)      |
                       | NGT1..NGT4 Qed, refutation theorems |
                       +-------------------^-----------------+
                                           |
                       +-------------------+-----------------+
                       | L1  Claim Ledger                    |
                       | 5-status vocabulary, no-overclaim   |
                       +-------------------^-----------------+
                                           |
                       +-------------------+-----------------+
                       | L0  Infrastructure                  |
                       | CI, anti-numerology gate, Pages,    |
                       | honest counter, Cargo + Coq builds  |
                       +-------------------------------------+

Each layer below states what is currently Checked, what is a known Blind spot, and what the Next unlock is. Status keywords use docs/CLAIM_STATUS.md.

L0 - Infrastructure

• Checked: Rust workspace tests (trinity_rust/, games/trinity_fold/), Pages / wasm deploy of the GOLDEN BRIDGE canvas, anti-numerology gate, Coq build gate, Lean scaffolding gate, honest counter, live game at https://t27.ai/trinity-s3ai/.
• Blind spots: the gates verify build / tag hygiene, not physics; CI is heavy; no single dashboard yet shows proof-debt and claim-counts in one place.
• Next unlock: CI Observatory dashboard, proof-debt counter wired into PRs, Claim Gate (every PR touching public docs must point to a ledger row), Artifact Gate for the arXiv bundle.

L1 - Claim Ledger

• Checked: five-status vocabulary (verified / empirical_fit / open_conjecture / high_risk_or_falsified / unverified), no Theory-of-Everything wording, no prize wording, the game is framed as a hypothesis puzzle and not as evidence.
• Blind spots: older *_v44.md / *_v46.md / IMPACT_* documents are not all linked to ledger rows yet.
• Next unlock: single source of truth (claims.yaml / claims.json) consumed by README, game tile catalogue, and docs generators, so every public claim is traceable to one row.

L2 - Formal Proofs

• Checked: Coq / Rocq build (Wave 17 baseline: 1 790 Qed.+Defined. across the canonical tree; 0 real Admitted. after comment stripping), Lean 4 scaffolding (TrinityLean/), 14 refutation theorems (*_refuted), NGT1..NGT4 closed with Qed..
• Blind spots: the formal layer does not close the Lagrangian derivation; Track B Clifford statements still rest on cited Axioms; Lean side is red on main (H4RootSystem.lean, Snub24Z3.lean).
• Next unlock: machine-readable proof map (which Qed. discharges which claim row), no floating theorem rule, expanded refutation library, bridge theorems between geometry and the spectral side.

L3 - Geometry (H4 / 600-cell / Clifford)

• Checked: H4 root system (order 14 400, 120 roots), 600-cell realisation, Snub 24-cell Z_3 tripartition as a bridge candidate, Dirac spectrum on the 600-cell reported as empirical_fit.
• Blind spots: no derivation of the Standard Model from geometry alone; no end-to-end bridge geometry -> spectral triple -> gauge -> Lagrangian; a4 conversion factor not fully reconciled across the three derivations.
• Next unlock: geometry atlas (one page per geometric fact, with its status), a4 reconciliation quest, explicit boundary doc separating pure-math facts from physics-dependent ones.

L4 - NCG / Lagrangian

• Checked: Connes-Chamseddine spectral-action benchmark as the comparison target; the derivation chain that the project would need to close is enumerated.
• Blind spots: gauge structure, Higgs sector, Yukawa couplings, and the three-generation structure are not derived; NGT2 (no sigma-field from H4) and NGT-6 (no string / orbifold rescue) bound what can be recovered.
• Next unlock: finite-algebra candidate, Dirac-operator construction, inner-fluctuations check, side-by-side NCG comparison table, and a "no derivation, no claim" gate on the Lagrangian docs.

L5 - Numerical Fits and Validation

• Checked: 26 core + 33 extended formulas matching PDG 2024 / NuFit 6.0 inside stated windows (interval-bound Qed. in proofs/trinity/Catalog42.v), falsifiability protocol, RG running scheme.
• Honest p-value (Wave 20, 500k trials): mean rel. error p = 0.077 (not significant); SG-hit density p < 0.0001 (highly significant). Random search matches average precision but rarely achieves the same density of ultra-precise hits.
• σ-ranking: median σ-distance = 0.085σ; 13/26 within 0.1σ; 3 "ultra-precision traps" (G01, Pr, L01) fail on σ only because measurements are 10⁵–10⁶× more precise than the formulas.
• Blind spots: e^2 (electric charge) derivation, full RG closure, numerology / look-elsewhere risk, post-data prediction risk, G03 (sin²θ_W) is a genuine 84σ failure.
• Next unlock: fit registry tied to the claim ledger, systematic ablations, RG quest, pre-registered prediction lock for any new numerical claim.

L6 - GOLDEN BRIDGE Game

• Checked: Rust + wasm GOLDEN BRIDGE canvas (live on Pages), five-ring Cargo workspace with inward-only dependency tests, every tile carries a ClaimStatus, falsification floor (any falsified tile collapses the bridge), framing the game as "data + geometry -> bridge candidate" and not as proof.
• Blind spots: the game is an educational interface, not a proof engine; level catalogue and claim-ledger SSOT are not yet wired into the game's tile data; "the game proves something" must stay an explicit non-claim.
• Next unlock: levels 1..5 of the GOLDEN BRIDGE puzzle, each level unlocked only when its underlying claim row reaches the required status.

L7 - Publication

• Checked: TRIOS PhD-style pipeline (paper v2, seminar talk v2, arXiv tarball, Zenodo deposit plan, citation metadata), all public documents written in English.
• Blind spots: drift risk between README, paper PDF, and game tile text; possible Cyrillic / non-English leakage in older artefacts; secrets in the tree.
• Next unlock: generated publication artefacts (README / paper / game catalogue all driven from claims.yaml), strict no-Russian gate on public docs, automated secret-scan gate.

The diagram and the per-layer bullets above are intentionally a summary. Status per row, file pointers, and next milestones live in docs/TECH_TREE.md; when the two disagree, the TECH_TREE file is the source of truth and this section is updated.

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Track B — Cl(p,q) formalization (Wave 12 launch)

Following the no-go results on the H4/600-cell side, Wave 12 launches a parallel Track B investigating Cl(8) / J3(O) / triality as an alternative basis for the three-generations question (per outputs/B_program_T1_T12.md).

The new directory proofs/clifford_cl8/ introduces:

• T1 — definition of Cl(p,q) via universal property (CliffordAlgebra.v) — mostly Qed
• T2 — statement of Cl(0,6) ≅ M_8(R) ⊕ M_8(R) (Cl6_iso_M8R.v) — stated, Admitted with citation (Lounesto 2001 Table 16.3, Wieser-Song 2022 §6, arXiv:2110.03551)
• T3 — statement of Bott 8-periodicity Cl(n+8) ≅ Cl(n) ⊗ Cl(8) (Cl8_periodicity.v) — stated, Admitted with citation (Atiyah-Bott-Shapiro 1964 Table 3, Lawson-Michelsohn 1989 Prop. I.4.1)

Track B is intentionally decoupled from the H4 core (no Trinity imports); its Admitted set is logged separately as TRACK_B_CLIFFORD in admitted_log.md. The launch PR provides the scaffolding and statement-level theorems; T4–T12 and the Admitted discharges are follow-up work.

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Trinity Fold — hypothesis-discovery puzzle prototype

games/trinity_fold/ is a small, opt-in puzzle prototype inspired by FoldIt, AlphaFold's Evoformer two-tower split, and CASP-style held-out benchmarking. It scores candidate boards built from theory tiles (constants, symmetries, geometry blocks, fields, constraints, observables) against a fixed catalog and surfaces the worst claim status across all active tiles. It does not prove or test any Theory of Everything claim. Every tile carries a ClaimStatus (verified / empirical_fit / open_conjecture / high_risk_or_falsified / unverified), and boards containing falsified tiles are floored at a negative total. See games/trinity_fold/README.md for the honesty policy and scoring spec.

The prototype is fully decoupled from the Coq proofs and from the anti_numerology_check CI gate; it lives in its own Rust Cargo workspace (five ring crates: ring0_core ← ring1_constraints ← ring2_search ← ring3_adapters ← app) so a broken game build never blocks proof work and so each layer can be audited independently. The inward-only dependency direction is enforced by integration tests in games/trinity_fold/crates/app/tests/ring_boundaries.rs.

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The Four Obstruction Theorems

Theorem	Statement	File
NGT1 (Cosmology)	φ^a π^b e^c formulas cannot reproduce Λ or Ω_b	NoGoTheorems.v
NGT2 (σ-field)	No NCG σ-field from H4 root structure alone	NoGoTheorems.v
NGT3 (Chirality)	600-cell D_F is vector-like (antipodal symmetry)	NoGoTheorems.v
NGT4 (Mass hierarchy)	2I-equivariant D_F cannot reproduce lepton mass ratios	NoGoTheorems.v

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Coq Proof Statistics (Wave 15.1 — Honest Audit)

Metric	Value
Coq version	8.20.1
Coq .v files (proofs + derivations)	79
Qed. theorems	1 762
Real Admitted. (outside comments/strings)	5
Axiom + Conjecture + Parameter	85
Refutation theorems	14
SG-class formulas (error < 0.01 %)	11
V-class formulas (error 0.01–0.3 %)	12
Honest p-value (500k trials)	p = 0.077 (mean error); p < 0.0001 (SG-hit density)
Anti-numerology gate	✓ PASS (59 formulas tagged)
Honest-count script	scripts/count_admitted_honest.py

Audit note (Wave 15.1): Prior counts mixed comment text with real proof obligations. A naive grep -c "Admitted" finds 77 matches in proofs/trinity/, but all 77 are inside historical comments (tags, TODOs, cross-references). The honest parser strips comments and strings before counting. The result: proofs/trinity/ has 0 real Admitted., proofs/clifford_cl8/ has 4, and derivations/chirality/ has 1. Full methodology and commitment are in HONESTY_MANIFEST.md. All 5 open gaps are documented with citations or [OPEN_PROBLEM] labels.

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Build Instructions

Prerequisites

• Coq 8.20.1
• opam (OCaml package manager)
• coq-interval, coq-coquelicot (Coq libraries)

Local build

# Install dependencies (first time only)
opam repo add coq-released https://coq.inria.fr/opam/released
opam install coq.8.20.1 coq-interval coq-coquelicot

# Build
cd proofs/trinity
coq_makefile -f _CoqProject -o Makefile.coq
make -f Makefile.coq -j$(nproc)

Docker

docker run -it --rm -v $(pwd):/work coqorg/coq:8.20.1-ocaml-4.14-flambda bash
cd /work/proofs/trinity
coq_makefile -f _CoqProject -o Makefile.coq
make -f Makefile.coq -j4

Python validators

pip install mpmath numpy
python3 scripts/validators/validate_v4.py     # formula error bounds
python3 scripts/anti_numerology_gate.py       # honesty check

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Repository Structure

proofs/trinity/              — Coq .v source files (50 files)
  CorePhi.v                  — Golden ratio φ and its algebra
  Catalog42.v                — 42 SM parameter formulas [phenomenological_fit]
  NoGoTheorems.v             — NGT1–NGT4 formal obstruction theorems
  HiggsPrediction.v          — Higgs mass formulas [phenomenological_fit]
  CosmologyOrigins.v         — Cosmological formulas [HONEST: ...]
  ...
scripts/
  count_admitted_honest.py   — Honest parser: strips comments before counting
  anti_numerology_gate.py    — CI gate: detect untagged φ/π/e formulas
  README_anti_numerology.md  — Gate documentation
  prepare_zenodo.md          — Zenodo publication guide
.github/
  workflows/ci.yml           — CI: anti_numerology_check → build
  workflows/release.yml      — Release: coq bundle + PDF
  PULL_REQUEST_TEMPLATE.md   — Enforces "verified/open" PR section
RESEARCH_STATUS.md                   — Honest summary of results
CITATION.cff                 — Citation metadata (Zenodo)
CONTRIBUTING.md              — Contribution rules (Russian + English)

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Honesty Statement

The 59 formulas in Catalog42.v and related files match Standard Model parameters to <0.01% precision using combinations of φ, π, and e. This is a catalog of numerical coincidences, not a physical theory. The No-Go theorems prove that these coincidences cannot be elevated to a consistent NCG model of the Standard Model.

Every formula is tagged with [phenomenological_fit] and is automatically checked by the anti-numerology CI gate (scripts/anti_numerology_gate.py). Any new formula without an approved honesty tag will be rejected by CI.

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Citation

See CITATION.cff for citation metadata.

After Zenodo deposit, the DOI badge above will be updated. For now, cite the GitHub repository:

@software{trinity_s3ai_2026,
  title  = {Trinity-s3ai: A Constructive Negative Result on H4-Based Standard Model Unification},
  author = {[Author Name]},
  year   = {2026},
  url    = {https://github.com/gHashTag/trinity-s3ai},
  note   = {Version v1.0-wave10}
}

Also see CITATION.bib for formatted references.

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Wave History

Wave	Key addition
1–3	Initial φ/π/e formula catalog
4.1	Honesty tags: [phenomenological_fit], [NUMERICAL_FIT] added
5–8	NCG derivations, spectral action
9.6	NGT1–NGT4 obstruction theorems formalized
10.5	Anti-numerology CI gate; CITATION.cff; CONTRIBUTING.md
12	Track B launch: Cl(p,q) formalization (T1–T3)
15.1	Honest counting system: comments stripped before statistics
17	Formal retraction of δ_CP = 65.66° (5.6σ excluded); 0 real Admitted. in proofs/trinity/
18	Merge of honesty-pass PRs #21–#23 (Lagrangian status, δ_CP withdrawal, Coq audit)
19	Merge of honesty-pass PRs #29, #31, #32 (calculation-primacy, N_gen=3 withdrawn, Strong CP withdrawn); all old PRs merged
20	Honest phenomenology refresh: 500k-trial p-value, σ-ranking updated (26 obs, δ_CP withdrawn, sin²θ_13/23/W +