feat(phd-ch27): trinity identity — three proofs of φ²+φ⁻²=3

[gHashTag]

Closes #265

feat(phd-ch27): Trinity Identity — Three Proofs of φ²+φ⁻²=3

Chapter Overview

Lane L27 · docs/phd/chapters/27-trinity-identity.tex
Type: SURVEY / THEORY
Lines: 1657 (+1234 from 423-line stub)
Zenodo anchor DOI: 10.5281/zenodo.19227877

This PR delivers the central theoretical chapter of the Trinity S³AI — Flos Aureus v6.2 monograph, proving the Trinity Anchor Theorem:

φ² + φ⁻² = 3 (exact, purely algebraic, not numerical)

Rule of Three — Three Independent Proofs

Strand	Method	Key Identity
Strand I — Algebraic	Minimal polynomial x²=x+1	φ²=φ+1, φ⁻²=2−φ → sum=3
Strand II — Geometric	Regular pentagon cos(36°)/cos(72°)	4cos²(36°)+4cos²(72°)=3
Strand III — Matrix-trace	Fibonacci matrix M, tr(M²)	tr(M²)=3=φ²+φ⁻² by eigenvalue theorem

Requirements Met (R3, R5, R6, R9, R10, R11, R12, R14)

• ≥1500 lines: 1657 lines ✅
• ≥2 Q1/Q2 cites: koshy_fib_lucas (Wiley), vajda_fib_lucas (Dover), olds_pell (MAA), kolmogorov_kar_1957 (Doklady), arnold_kar_1957 (Doklady), liu_kan2_2025 (ICLR Oral) ✅
• ≥1 theorem+proof+qed: Trinity Anchor Theorem (thm:trinity-anchor) with three-way proof and \qed ✅
• Rule of Three: Strand I / Strand II / Strand III section headers ✅
• R6 φ-only: All numeric constants φ-derived (φ⁰=1, φ¹=φ, φ⁻¹=φ−1, φ²=φ+1, φ⁻²=2−φ) ✅
• coqcite: \coqcite{lucas_2_eq_3}{trinity-clara/proofs/lucas_closure_gf16.v}{1--120}{Proven} ✅
• admittedbox: GF16 rounding bound (gf16_precision.v INV-3) honestly admitted (R5) ✅
• Related Work: §10 with Kolmogorov 1957 + Arnold 1957 (KART) + KAN (EPIC 🎯 EPIC L-KAT — Kolmogorov–Arnold Theorem ↔ Trinity GF16 cross-cutting integration #572 L-KAT-RW) ✅

Coq Citation Map

Theorem	File	Lines	Status
lucas_2_eq_3	trinity-clara/proofs/lucas_closure_gf16.v	1–120	Proven

Commits

• 37ec371 feat(phd-ch27): trinity identity — three proofs of φ²+φ⁻²=3 [agent=scarab-l27]

Related Issues

• ONE SHOT #265 — Lane L27
• EPIC #572 — L-KAT-RW Related Work lane