📖 Ch.3 — Trinity Identity (φ-squared plus phi-minus-squared = 3) (700w · 🟢 P2)

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Ch.3 — Trinity Identity (φ-squared plus phi-minus-squared = 3)

Master: trios#380 GOLDEN SUNFLOWERS Book
Priority: 🟢 P2 · Word target: 700 · Owner: GoldenSunWeaver

🎯 Scope (FROZEN)

Statement, Coq proof sketch, multiple proofs (algebraic, geometric, recursive). Tie to T1 (root theorem). Why φ²+φ⁻²=3 is the 'blooming axiom' of the field.

🔑 Key claims / formulas

• T1 statement: φ²+φ⁻²=3 ↔ φ²−φ−1=0 (defining equation)
• Coq formalization: coq/T1_trinity.v (existing in t27)
• Geometric proof via Lucas L₂=3
• Recursive proof: φⁿ + φ⁻ⁿ = Lⁿ for Lucas Lₙ

📦 Deliverables

• paper/sections/3_trinity_identity_squared_plus_phiminussquared__3.tex
• Reference t27#319 Ring 051 Phi-Split Optimality (cited)
• Coq lemma already exists — link, don't redo

🔗 Dependencies

• Ch.1, Ch.2

📝 Draft (Markdown — paste here when ready)

Operator types наполни Ch.3 → General fills this section with research-backed Markdown.

✅ Definition of Done (NeurIPS checklist aligned)

• Word count 700 ±10%
• All claims cited (primary literature, not just our repos)
• No AI as author (Acknowledgments only)
• Compiles in main.tex via tectonic, no warnings
• Limitations stated (where applicable)
• Reproducibility info present (where applicable)
• PR with Closes #<this> + green CI
• L1 (no .sh) · L3 (clippy zero) · L4 (tests pass) · L8 (PUSH FIRST)

🤖 ONE-SHOT directive (когда operator typed ONE SHOT Ch.3)

A CHAPTER-AGENT: take Markdown draft from this issue body §"Draft", convert to LaTeX,
place in paper/sections/3_<slug>.tex, ensure compile via tectonic with main.tex,
open PR linked to #, run cargo clippy/test, push, request review.
Hard deadline per priority (P0 → T-3h, P1 → T-1h, P2 → defer).

phi^2 + phi^-2 = 3 · TRINITY · NEVER STOP