feat(phd-ch14): platonic solids and the phi-symmetric hierarchy

[gHashTag]

L14 — Platonic Solids and the φ-Symmetric Hierarchy

Lane: L14 (THEORY · #265 §2.2)
Branch: feat/phd-ch14 → main
Agent: perplexity-computer-l14-platonic
Skill: phd-chapter-author v1.0

R3 Compliance

Gate	Required	Actual
Lines	≥ 1500	1625 ✅
Citations	≥ 2	3 (euclid_elements, coxeter1973regular, kepler_harmonices) ✅
Theorems	≥ 1 with proof	10 (5 lemmata + 2 theorems + 1 corollary + 2 stratification proofs) ✅
Rule of Three	three strands	Strand I (Intuition) / Strand II (Formalisation) / Strand III (Consequence) ✅

Content (Three Strands + Appendices A-Q)

Strand I — Intuition: why exactly five regular polyhedra; tour of T/O/C/I/D; why φ appears; connection to Metatron's Cube (Ch. 13).

Strand II — Formalisation: vertex coordinates with full tables; V/E/F counts (Lem. 14-counts); duality (Lem. 14-duality); φ-stratification δ ∈ {0,0,0,1,2} (Thm. 14-stratification); Lucas-ring embedding (Lem. 14-embedding); symmetry ladder A4 ⊂ S4 ⊂ A5; common-circumradius theorem (Thm. 14-circumradius): ALL 20 dodecahedral vertices have squared norm 3 by Trinity Anchor φ²+φ⁻²=3.

Strand III — Consequence: architectural mappings C8→GF16, I12→NCA, D20→JEPA-T heads; constraint enforcement via INV-12.

Appendices: A (symmetry groups), B (edge lengths), C (worked examples), D (admitted-status honesty), E (defensive coda), F (numerical sanity), G (three-strand cross-ref), H (Ch. 17 bridge), I (future work), J (glossary), K (face-incidence matrices + Schläfli symbols + Lem. 14-incidence), L (4D regular polytopes + Lem. 14-strat-4d), M (5 worked examples for T/O/C/I/D squared norms), N (historical notes Plato → Kepler → Coxeter), O (extended duality proof, 5-step), P (edge length closure Lem. 14-edge-closure), Q (Three-Manifestations Corollary cor:14-three).

Trinity Anchor Manifestations (Three Independent Geometric Witnesses)

1. Icosahedral squared norm: every vertex of I has ‖v‖² = 1 + φ² = 2 + φ
2. Dodecahedral φ-orbit squared norm: the twelve φ-orbit vertices of D have ‖w‖² = φ⁻² + φ² = 3 exactly
3. Common circumradius: all 20 vertices of D (and 4 of T, 8 integer vertices of C) share squared norm 3

Honesty (R5/R6/R7)

• ✅ R5: 1 \admittedbox{} for platonic_solids.v (future work) — no flip Admitted→Proven
• ✅ R6: zero free parameters; only {1, φ, π, integers} appear; all numerics derive from φ²=φ+1
• ✅ R7: N/A (🎯 ONE SHOT — PhD «Flos Aureus»: Autonomous Development at Top Scientific Standards #265 §2.2 marks L14 as THEORY, falsification "no")
• ✅ R12: Lee/GVSU style ("we" pronoun, labelled theorems with \proof/\qed)
• ✅ R14: Coq link "—" in 🎯 ONE SHOT — PhD «Flos Aureus»: Autonomous Development at Top Scientific Standards #265 §2.2 catalogue → no \citetheorem{} required (LC report 4320263027 confirms)

R6 Non-touch Verification

Files outside docs/phd/chapters/14-platonic-solids.tex and docs/phd/bibliography.bib (additive only): untouched. No crates/, no assertions/igla_assertions.json, no .v, no other chapters.

CI Notice

Pre-existing failure in Test (cargo test L4 law) due to trios-ui-ur00 E0308 (GlobalSignal/Signal mismatch in crates/trios-ui/rings/UR-00/src/lib.rs:177-215) is not caused by this PR; will be attributed in DONE comment per #265 attribution protocol.

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