Trinity S³AI — Boundary-Mapping Research + Hardware-Verified Knowledge Chain

Caution

No Theory-of-Everything claim is made. This repository documents an active boundary-mapping research program — we prove what H4 geometry cannot reproduce, narrowing the search space.

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🎯 Start here based on who you are

Role	Start here	Time	What you'll learn
Physicist / Reviewer	docs/REVIEW_GUIDE.md	10 min	Honest path with commands and expected outputs
Formal methods researcher	proofs/trinity/BoundaryTheorems.v	5 min	BT-1..BT-4 Qed; 51 unproven obligations (0 real Admitted + 51 Axiom/Parameter/Conjecture) across 56 Coq files in proofs/trinity/
Curious visitor	🔗 GOLDEN CHAIN live	2 min	Hardware-verified proof chain puzzle
Contributor	CONTRIBUTING.md + good first issue	15 min	Build instructions, phi-loop protocol

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What Makes This Different

Most unification programs publish only successes. We publish the dead ends too — because a formally proven dead end saves the field from years of wasted effort.

Our Strong Side	What It Means For You
1,325 theorems with Qed.	Every positive claim is machine-checked (~210 are trivial/tautological; ~1,100 are substantive).
4 boundary theorems (BT-1..BT-4)	Four direct H4→SM construction paths are formally ruled out. You don't have to try them.
14 refutation theorems	Specific ansätze (orbifold Z₂, E8-plumbing η, δ_CP = 65.66°) are proven dead and documented.
0 real Admitted. + 51 obligations in proofs/trinity/	No cosmetic edits. Every open gap is an Axiom/Parameter with a literature citation. See COQ_HONEST_STATUS.md for full accounting.
0 fake proofs	The honest parser strips comments before counting. Naive grep finds 77; the real count is 0 in proofs/trinity/.

$TRI is not a cryptocurrency. It is a compression score (bits-per-byte) benchmarked on synthetic φ-structured data; silicon validation is NOT pending: the TTSKY26b submission (2026-05-17) was withdrawn and refunded, so no die will be returned. You cannot mine it on a GPU.

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We maintain a catalog of 130 total formulas/invariants (59 SM-sector phenomenological formulas) between H4 Coxeter invariants and PDG 2024 measurements, plus 4 formal boundary theorems proving obstructions. Whether the coincidences are deep or accidental is itself an open research question — tracked via a 7-layer verification stack and a living claim ledger.

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📊 Living Status

Layer	Status	Key Metric	Details
L0 Infrastructure	✅	Live demo	Rust + Coq CI, anti-numerology gate
L1 Claim Ledger	✅	SSOT claims.yaml	5-status vocabulary enforced
L2 Formal Proofs	✅	0 real Admitted in proofs/trinity/	51 unproven obligations (Axiom/Parameter/Conjecture) in proofs/trinity/
L3 Geometry	⚠️	a₄ factor unresolved	3 derivations don't converge
L4 NCG / Lagrangian	🔴	BT-2, BT-6 bound recovery	No σ-field from H4; no string rescue
L5 Numerical Fits	⚠️	59 formulas, 1 withdrawn	δ_CP withdrawn; sin²θ_W = 84σ genuine failure
L6 GOLDEN CHAIN	✅	Rust + wasm deployed	5-ring Cargo workspace
L7 Publication	🔄	Paper v2 in prep	arXiv + Zenodo pipeline

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🏗️ Verification Stack

flowchart BT
    L7["L7 Publication: paper v2 / arXiv / Zenodo"] --> L6
    L6["L6 GOLDEN CHAIN: silicon-targeted chain"] --> L5
    L5["L5 Numerical Fits: 59 formulas, p-values"] --> L4
    L4["L4 NCG / Lagrangian: spectral action"] --> L3
    L3["L3 Geometry: H4 / 600-cell / Cl(8)"] --> L2
    L2["L2 Formal Proofs: Coq / Lean, BT-1..BT-4"] --> L1
    L1["L1 Claim Ledger: 5-status SSOT"] --> L0
    L0["L0 Infrastructure: CI, honest counter, Pages"]

    style L2 fill:#90EE90,stroke:#333
    style L5 fill:#FFD700,stroke:#333
    style L4 fill:#FF6B6B,stroke:#333
    style L3 fill:#FFD700,stroke:#333

Loading

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🏆 Boundary Results (formally verified)

Theorem	Statement	What It Closes
BT-1 (Cosmology)	φᵃ πᵇ eᶜ formulas cannot reproduce Λ or Ω_b	Algebraic ansätze for cosmological constants from H4 invariants are dead ends.
BT-2 (σ-field)	No NCG σ-field from H4 root structure alone	Direct NCG spectral-action derivation from H4 geometry alone is impossible; redirects to Cl(8) / J₃(𝕆).
BT-3 (Chirality)	600-cell D_F is vector-like (antipodal symmetry)	The 600-cell cannot source SM chirality; chirality must enter via a different mechanism.
BT-4 (Mass hierarchy)	2I-equivariant D_F cannot reproduce lepton mass ratios	Icosahedral symmetry-breaking ansatz for lepton masses is insufficient; needs stronger breaking.

Coq Stats (Wave 23): 1,325 theorems with Qed. (~210 trivial, ~1,100 substantive) · 0 real Admitted. + 51 unproven obligations (Axiom/Parameter/Conjecture) in proofs/trinity/ · 14 refutation theorems
Honest p-value: p = 0.077 (mean error, not significant) · p < 0.0001 (SG-hit density, significant)

"Not a proof is also a proof." — We share what we tried to prove and could not, because knowing the boundary is as valuable as knowing the path.

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📊 IGLA RACE: Live BPB Benchmarks

Real training results on FineWeb-Edu 10B token subset, 196K-parameter transformer (hidden=384, 2 attention layers, AdamW lr=0.003). Lower BPB = better compression.

Champion (no quantization, seed=43, 81K steps): best = 2.3900 [empirical_fit]

Format	Best BPB @ 10K	Best BPB @ 40–50K	Verdict
bf16	2.7127	—	Reference
fp16	2.7183	3.0383 @ 50K	Reference
f32	2.7375	3.0391 @ 50K	Reference
gf16	2.7186	3.0260 @ 50K	Matches 16-bit baseline
posit8	3.0099	2.9737 @ 40K	Competitive at medium steps
gf8	3.0099	2.9737 @ 40K	Competitive at medium steps
mxfp8	3.3874	~3.5	Marginal
int8	3.6092	~3.9	Marginal
nf4	3.8659	~4.0	Marginal
fp8_e5m2	4.5309	—	Non-viable
int4	7.0005	~7.0	Non-viable

Key finding [empirical_fit]: GF16 matches fp16/f32 compression despite φ-structured mantissa — no accuracy penalty at 16-bit precision on language modeling.

Surprise [empirical_fit]: posit8 and gf8 (8-bit formats) outperform 16-bit fp16/f32 at 40–50K steps, suggesting φ-aware quantization may stabilize optimization beyond naive bit-width expectations.

Dead formats [verified — refutation]: int4 and fp8_e5m2 fail to converge below 4.5 BPB on this architecture — empirically ruled out for the 196K-param regime.

Raw logs: trios-trainer-igla/.trinity/results/ — see trios-trainer-igla.

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🔗 Why $TRI Is Mined Only on TTSKY26b

The Trinity S³AI knowledge chain is designed to anchor in silicon. $TRI (Trinity Index) is the project's native proof-of-work metric — not a cryptocurrency, but a compression score measured in bits-per-byte (BPB) on φ-structured arithmetic. Hardware validation is pending.

The Chain — Three Links

        Coq  Qed.                Silicon  0x47C0                Boundary  Theorem
            |                         |                          |
            v                         v                          v
   +-------------------+   +-----------------------+   +------------------------+
   | Formal proof of   |   | Lucas chain L2 = 3    |   | BT-1..BT-4 prove which   |
   | 1,762 theorems    |   | anchored in GF(16)      |   | H4→SM paths are closed   |
   | (Track A + B)     |   | submitted for TTSKY26b  |   | (saves field from blind  |
   |                   |   |                         |   | alleys)                  |
   +-------------------+   +-----------------------+   +------------------------+
            |                         |                          |
            +------------+------------+------------+-------------+
                         |
                         v
              +---------------------+
              |  GOLDEN CHAIN score  |
              |  = silicon-targeted |
              |  proof-of-work       |
              +---------------------+

Every link is independently auditable: the Coq side builds in CI, the silicon anchor is submitted for TinyTapeout fabrication, and the boundary theorems are in BoundaryTheorems.v with Qed.

The Three Crowns

Crown	Chip	Tile	Modules	Anchor	Role
Phi (Nano)	tt_um_trinity_nano	1×1	51	0xCF	Minimal-area witness
Euler (Compact)	tt_um_ghtag_trinity_gf16	8×2	90	0xAE	GF(16) Galois-field carrier
Gamma (Full)	tt_um_trinity_max_true	8×4	105	0x93	Full Trinity-S³AI DNA

All three crowns are designed to carry the canonical anchor 0x47C0 at {uio_out, uo_out} on reset — a planned silicon witness of φ² + φ⁻² = 3 via the Lucas chain L₂ = 3 → GF(16) dot4 trace. Post-silicon logic-analyzer validation is pending (see docs/hardware/silicon_anchor.md §3.3).

Why GF16?

• φ-structured step: each quantization level approximates φ⁻¹ ≈ 0.618, giving 0.694-bit reduction per level (mathematical identity — log₂ φ identity; Prop 1–2, Thm 3 in gf16_mathematics.md)
• 16.2× better than bf16 on uniform data: 9-bit mantissa vs bf16's 7 bits (empirical_fit — zig-golden-float BENCH-010 Uniform [-100,100])
• 65× wider dynamic range than fp16: max normal ~4.29×10⁹ vs fp16's 65504; exact powers-of-two at -120.00 dB NMSE (empirical_fit — benchmark_nmse.py D-5)
• Zero inference accuracy drop: Fashion-MNIST MLP post-training quantization = 0.00% drop vs fp32 (empirical_fit — zig-golden-float BENCH-008)
• Optimal integer φ-split: round((N−1)/φ²) gives exp=6/mant=9 for N=16; phi-distance 0.049 vs fp16 0.118 (proposed — NeurIPS 2026 OPT Prop 2; not yet peer-reviewed)
• Hardware exclusivity: WITHDRAWN — the TTSKY26b submission (Three Crowns: Phi + Euler + Gamma) was withdrawn and refunded; no silicon route is currently selected

Honest note on BPB: GF16 is a 16-bit float, so raw BPB = 0.500 — identical to fp16 and bfloat16 by bit-width. The 0.125 raw BPB claim belongs to the separate phi-4Q 4-bit quantizer (synthetic φ-structured data only). See docs/hardware/bpb_benchmark.py and docs/hardware/gf16_mathematics.md §2.2.

Honest performance

~1 GOPS @ ~50 MHz @ ~1 W ternary (projected on QMTech XC7A100T).

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⚖️ Our Honest Model — Impossibility as Proof

This project is built on a counter-intuitive principle: proving what cannot be done is as valuable as proving what can.

What we proved	Count	Status
Theorems with Qed.	1,325 (~210 trivial, ~1,100 substantive)	Verified
Boundary theorems (impossibility proofs)	4 (BT-1..BT-4)	Verified
Refutation theorems (*_refuted)	14	Verified dead ends
Real Admitted. (outside comments/strings)	0 in proofs/trinity/	Verified by honest parser
Fake proofs	0	—

Boundary theorems save the field from wasted effort:

• BT-1 — no φ^a π^b e^c monomial can reproduce cosmological constants → saves theorists from algebraic dead ends
• BT-2 — no NCG σ-field from H4 alone → redirects toward Cl(8) / J₃(𝕆) constructions
• BT-3 — 600-cell Dirac operator is vector-like → chirality must come from elsewhere
• BT-4 — 2I-equivariant D_F fails for lepton masses → mass hierarchy needs different symmetry breaking

0 real Admitted gaps in proofs/trinity/ — the honest parser strips comments before counting. Naive grep finds 5 mentions in comments; the real count is 0. Total unproven obligations in proofs/trinity/: 51 (Axiom/Parameter/Conjecture).

0 fake proofs. 0 cosmetic edits to hide gaps. Every number is produced by a comment-stripping parser and treated as ground truth.

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🚀 Quick Start

# Docker (zero dependencies)
docker run -it --rm -v $(pwd):/work coqorg/coq:8.20.1 bash -c \
  "cd /work/proofs/trinity && coq_makefile -f _CoqProject -o Makefile.coq && make -f Makefile.coq -j$(nproc)"

# Validators
pip install mpmath numpy
python3 scripts/validators/validate_v4.py
python3 scripts/anti_numerology_gate.py
python3 scripts/count_admitted_honest.py

Full build matrix: docs/BUILD.md · Contributing: CONTRIBUTING.md

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🛡️ Verify in 5 Minutes (Sceptic's Path)

You do not have to trust our numbers. Run these four commands in order:

# 1. Anti-numerology gate — checks that every φ/π/e formula is honesty-tagged (~5 s)
python3 scripts/anti_numerology_gate.py
# Expected: VERDICT: PASS

# 2. Honest Coq counter — strips comments before counting Admitted (~2 s)
python3 scripts/count_admitted_honest.py
# Expected: proofs/trinity/ → 0 real Admitted + 51 Axiom/Parameter/Conjecture; 56 .v files.

# 3. Formula error bounds — verifies numeric claims against PDG 2024 (~10 s)
pip install mpmath numpy
python3 scripts/validators/validate_v4.py
# Expected: error claims match the table below

# 4. GOLDEN CHAIN ring-boundary tests — checks inward-only dependency (~30 s)
cd games/trinity_fold && cargo test --workspace
# Expected: all tests pass, including ring_boundaries.rs

If any step fails, the README statistics are stale — please open an issue.

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📚 Citation

@software{trinity_s3ai_2026,
  title  = {Trinity-s3ai: Active Boundary-Mapping Research Program},
  author = {Dmitrii Vasilev},
  year   = {2026},
  url    = {https://github.com/gHashTag/trinity-s3ai},
  note   = {v1.0-wave23}
}

See CITATION.cff for metadata. Zenodo DOI badge pending deposit.

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🔍 Full Claim Ledger (auto-generated from claims.yaml)

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

Claim	Layer	Status	Evidence	Blocking gap
GOLDEN CHAIN 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.
BT-1–BT-4: four formal Boundary theorems closed with Qed.	L2	verified	proofs/trinity/BoundaryTheorems.v (BT1, BT2, BT3, BT4 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 BT2 sigma-field boundary.
GOLDEN CHAIN puzzle is a silicon-targeted proof chain, 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	—
GF16 (1-6-9, bias 31) phi-anchored float format specified and FPGA-verified	L0	verified	docs/hardware/gf16_spec.md §2–2.6; t27/conformance/FORMAT-SPEC-001.json; NeurIPS 2026 OPT paper (Prop 1–2, Thm 3); IBM DLFloat16 (ARITH 2019, layout identical, 6 differences documented)	—
BPB compression benchmark: phi-4Q quantizer raw BPB 0.125 vs 0.500 for fp16/bf16/GF16	L5	empirical_fit	docs/hardware/bpb_benchmark.py; docs/hardware/bpb_results.json (generated 2026-05-25)	Extend benchmark to real physics datasets (e.g. PDG parameter vectors, MNIST weights) and verify the advantage persists outside synthetic phi-monomial data.
IGLA RACE champion: f32 baseline BPB = 2.3900 on 81K steps (196K params)	L5	empirical_fit	trios-trainer-igla/.trinity/results/champion_81k_seed43.log; seed=43, hidden=384, 2 attn layers, AdamW lr=0.003	Reproduce on independent hardware and confirm the BPB floor is stable across seeds. Run seeds 44–50 to 81K with identical hyperparameters.
IGLA RACE: GF16 matches fp16/f32 BPB on language modeling (empirical parity)	L5	empirical_fit	trios-trainer-igla/.trinity/results/v2_gf16_seed44_81k.log (ema_bpb=3.0260 @ 50K); v2_fp16_seed44_81k.log (ema_bpb=3.0383 @ 50K); v2_f32_seed42_81k.log (ema_bpb=3.0391 @ 50K)	Run GF16 to 81K steps (currently only 50K available) and verify the parity holds at full convergence. Run multiple seeds to bound variance.
IGLA RACE: posit8 and gf8 outperform 16-bit baselines at 50K steps (surprise)	L5	empirical_fit	trios-trainer-igla/.trinity/results/v2_posit8_seed44_50k.log (ema_bpb=2.9737 @ 40K); v2_gf8_seed44_50k.log (ema_bpb=2.9737 @ 40K)	Extend to 81K steps and multiple seeds. Verify the advantage is not a seed artifact. Compare against int8 to isolate phi-aware benefit.
IGLA RACE: int4 and fp8_e5m2 non-viable on 196K-param transformer	L5	high_risk_or_falsified	trios-trainer-igla/.trinity/results/v3_int4_seed46_10k.log (ema_bpb=7.0005); v3_fp8_e5m2_seed47_10k.log (ema_bpb=4.5309)	—
φ-anchored hyperparameters falsified on IGLA RACE: +0.1013 BPB worse than conventional	L5	high_risk_or_falsified	docs/audit/PHI_ABLATION_DESIGN.md; trios-trainer-igla/.trinity/gardener_harvest.log (control best=2.6167, treatment best=2.7180)	—
GF16 silicon tapeout on TinyTapeout TTSKY26a shuttle (submitted, pending silicon)	L0	open_conjecture	docs/hardware/silicon_anchor.md §3.1; PR #322 in TinyTapeout/tinytapeout-sky-26a for gHashTag/tt-trinity-gf16	Await silicon return from Sky130 fab (~Aug 2026). Post-silicon validation requires logic-analyzer reset witness, clock sweep, and conformance run.
0x47C0 reset-time silicon anchor validates Lucas chain L2=3	L0	open_conjecture	docs/hardware/silicon_anchor.md §1–2; no RTL testbench found in t27 repo	Document the anchor in RTL (add reset witness to gf16_mul.v or wrapper), add testbench simulation, or downgrade claim to design-intent only.
phi-structured quantization step gives ~0.694-bit reduction per level	L3	verified	docs/hardware/gf16_mathematics.md §3, §6.1–6.3; log2(phi) = 0.6942419136...; NeurIPS 2026 Prop 1 (Golden Self-Similarity), Prop 2 (Optimal Integer Rounding), Thm 3 (Universal Attractor)	—
Coq/Flocq proof: phi^2 = phi + 1 holds exactly in IEEE binary64	L2	verified	proofs/trinity/PhiFloat.v (adapted from t27/coq/Kernel/PhiFloat.v); phi_sq_f64 = phi_plus_one_f64 by vm_compute; residual 0.0 < tolerance 1e-15	Extend to GF16: prove phi identity holds (or bound error) in the 1-6-9 format. Currently proved only for binary64, not for GF16.
Three Crowns (Phi/Euler/Gamma) submitted to TinyTapeout TTSKY26b	L0	open_conjecture	docs/hardware/silicon_anchor.md §3.2; 21 closed PRs in TinyTapeout/tinytapeout-sky-26b (9 phi + 7 euler + 5 gamma)	Await silicon return from Sky130 fab (~Nov 2026). Post-silicon validation of all three crowns required before upgrade to verified.
The weight alphabet base is forced: any r > 1 with r*r = r + 1 is phi	L1	verified	derivations/golden_alphabet/GoldenAlphabet.v: phi_quadratic, phi_unique (Qed, coqchk clean, negative control run)	—
Multiplying by phi on integer pairs is the Fibonacci step: one addition, no shift	L1	verified	derivations/golden_alphabet/GoldenAlphabet.v: mulphi_correct, addp_correct, negp_correct (Qed)	—
The linear path of a {-phi, 0, +phi} network is exact at any fan-in and depth	L1	verified	derivations/golden_alphabet/GoldenAlphabet.v: dot_exact (Qed); Z[phi] closure	Rounding still enters at the nonlinearity; that is outside this statement.
The gain of k stacked phi-ternary layers is F_(k+1)*phi + F_k, a pair of integers	L1	verified	derivations/golden_alphabet/GoldenAlphabet.v: phi_pow_fib (Qed, by induction)	—

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.

🌊 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	BT-1–BT-4 boundary 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 + |V_ub| + λ added)
23	README redesign: lean structure, audience selector, Mermaid tech tree, collapsible ledger
24	φ falsifiability ablation: φ-anchored hyperparameters falsified (+0.1013 BPB worse than conventional) — see docs/audit/PHI_ABLATION_DESIGN.md
25–28	Queued: Coq gap closure (Wave 25), IGLA Gate-2 push (Wave 26), Lean 4 port (Wave 27), GOLDEN CHAIN integration (Wave 28) — see docs/WAVES_24_28.md

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🔗 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
docs/CLAIM_STATUS.md	Claim-status rule book (5 canonical statuses)
docs/TECH_TREE.md	Layered status of the whole stack
RESEARCH_STATUS.md	Boundary map: open, obstructed, or under exploration
docs/WAVES_24_28.md	Next waves: IGLA RACE, Coq gaps, Lean port, game integration
HONESTY_MANIFEST.md	Ground-truth statistics (comments stripped)
ROADMAP_WAVE17_PLUS.md	Tracks A, B, C of active research
paper/CHANGELOG_v1_to_v2.md	Paper v2 changelog — new boundary notes
SECURITY.md	Security policy and reporting path
docs/BUILD.md	Full build matrix (Coq, Rust, Python)