🖋 Ch.31 — Golden Seal (84 Coq Theorems) [Lane B] · refs #30

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Ch.31 — Golden Seal

Part: VII | Lane: B | Refs: #30 · #63

Rule of Three

1. Core — catalogue of 84 Coq lemmas/theorems
2. Sacred — seal as final verification
3. Philosophy — formal proofs as truth
4. Empirical — SHA-256 audit trail
5. Image — proof tree diagram

Source: trinity-clara/evidence/CLARA-EVIDENCE-PACKAGE.md

Acceptance

• All 84 Coq proofs integrated | Closes #95

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PhD chapter writing task — requires human academic authorship, not actionable by code agent.

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🔍 Ch.31 UPDATE — Golden Seal: 84 → 376 Qed

Anchor: φ² + φ⁻² = 3 · TRINITY · v3.0 MEASURED HARDWARE

Ch.31 was originally framed around "84 Coq Theorems" — the previous T27 milestone count. Full repo census audit (2026-04-30) shows the actual proof corpus is 4.5× larger.

New canonical numbers

Metric	Old claim (Ch.31 v1)	New (audited)
Coq files	~30	65 canonical (96 raw → 31 dedup)
Theorem-like declarations	84	438
Qed (proven)	84	376
Admitted (R5-honest)	unknown	48
Abort / WIP	unknown	14
Definitions + Fixpoints + Inductives	unknown	404
Axioms + Parameters	unknown	38
Total Coq objects	~84	880

What "84 Coq Theorems" really meant

Original count tracked only the Trinity Identity → Verilog semantic-preservation chain in t27/coq/Theorems/ and t27/proofs/trinity/. Audit reveals 7 additional independent proof corpora:

1. t27/coq/Kernel/ (6 files, 44 theorems) — φ-kernel, FlowerE8, Trit, Semantics
2. t27/proofs/sacred/ (4 files, 5 theorems) — DL bounds, Γ·φ³, L₅, strong CP
3. t27/proofs/gravity/ (1 file) — gravity DL bounds
4. t27/proofs/trinity/Bounds_*.v (5 files, 45 theorems) — gauge / lepton / quark / mass / mixing
5. trinity-clara/proofs/igla/ (10 files, 48 theorems) — INV-1..INV-7 IGLA invariants
6. trios/docs/phd/theorems/ (16 mirror files) — PhD-curated theorem doc
7. trios/trinity-clara/proofs/igla/ (16 files, 110 theorems) — INV-1..INV-9 expanded suite (rainbow_bridge, worker_pool, ema_decay)

Golden Seal — recommended Ch.31 scope refresh

"376 Qed-proven Coq theorems form the Golden Seal of the Trinity stack, distributed across 65 canonical .v files in three repositories (trinity-clara 48, t27 217, trios 173). The seal extends from the foundational trinity_identity : φ² + φ⁻² = 3 (proven in CorePhi.v by field_simplify; lra) through 11 IGLA runtime invariants (INV-1 BPB Monotone, INV-2 ASHA Survival, INV-3 GF16 Safe Domain, INV-4 NCA Entropy Band, INV-5 Lucas Closure, INV-6 Hybrid QK Gain, INV-7 Victory Criterion, INV-7b Rainbow Bridge, INV-8 Worker Pool Composite, INV-9 EMA Decay Valid) to 45+ Standard-Model bound theorems (CKM/PMNS unitarity, lepton/quark mass intervals, gauge coupling bounds). Honest accounting: 48 lemmas marked Admitted with explicit Coq.Interval-upgrade close-with comments; 14 Abort markers preserve work-in-progress (no silent merges); 28 falsification lemmas guarantee R5-honest negative space (every INV paired with explicit refutation example)."

Per-cluster seal evidence

Cluster	Files	Theorems	Headline lemma
Trinity Identity (CorePhi)	2	16 Qed	trinity_identity : phi^2 + (1/phi)^2 = 3
α_φ derivation	1	12 Qed	alpha_phi = (sqrt 5 - 2)/2
Lucas closure GF16	4	23 Qed	phi^{2n} + phi^{-2n} ∈ ℤ
GF16 precision	6	23 Qed	d_model_sufficient_for_gf16, gf16_bpb_degradation_bounded
ASHA / champion survival	4	25 Qed	asha_champion_survives (threshold 3.5)
Victory criterion (IGLA)	4	16 Qed	victory_exact, igla_found_criterion
NCA entropy band	5	20 Qed	entropy_band_non_empty (1.5 ≤ H ≤ 2.8)
BPB monotone descent	3	13 Qed	bpb_decreases_with_real_gradient
Refutations (R8 honest)	various	28 Qed	igla_falsified_implies_failure
Standard Model bounds	5	45 Qed	all_gauge_bounds_verified, CKM_first_row_unitarity

Refs

• Master Coq census: trios#373
• App.E update: trios#416
• Ch.10 (Coq L1 Pareto): trios#400
• App.B Golden Ledger: trios#99 — "84 Coq proofs + SHA-256" — should be retitled "376 Qed proofs + SHA-256"

phi² + phi⁻² = 3 · TRINITY · 376 Qed PROVEN · GOLDEN SEAL UPDATED · NEVER STOP 🌻

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Content exists in corresponding flos_XX.tex chapter file.