Wave 8.1: A4Bridge.v — a4 bridge analysis, Boundary Theorem BT-8

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Summary

Adds proofs/trinity/A4Bridge.v establishing Boundary Theorem BT-8: the conversion factor between the two competing a_4 definitions cannot be identified with any exact H4/600-cell integer invariant.

Background

The a_4 gap problem (documented in lagrangian_roadmap.md §APPENDIX B, HARSH_REVIEW_v49.md §4, FINAL_STATUS_v44.md) asks:

• Coq_a4 = (5+6*phi)/(16*phi) ≈ 0.568 (heat kernel, SpectralAction600Cell.v)
• Trinity_a4 = 8*phi^3 ≈ 33.889 (H4 invariant fit, HiggsPrediction.v)
• Ratio k ≈ 59.649 — can this be identified with an H4 integer invariant?

A4Conversion.v already proves the algebraic identity Trinity_a4 = k * Coq_a4. This PR answers the geometric question negatively (BT-8).

Numerical Analysis (bc, scale=20)

phi = (1+sqrt(5))/2 ≈ 1.61803398874989484820
Coq_a4  = (5+6*phi)/(16*phi)     ≈ 0.56813562148434214006
Trinity_a4 = 8*phi^3              ≈ 33.88854381999831757096
k_ratio = 128*phi^4/(5+6*phi)    ≈ 59.64868693052419061620

Rationalized: k = (704 + 192*sqrt(5)) / 19

H4 candidates:
  h(H4) = 30         k/30  = 1.988  -- not 30
  2*h   = 60         60-k  = 0.351  -- not 60 (closest integer)
  sum exponents = 60 (1+11+19+29)   -- not 60
  cells_600/10 = 60                  -- not 60
  edges_600/12 = 60                  -- not 60
  vertices_600 = 120                 -- k < 60 < 120

Proof Structure

• k_ratio_gt_59 / k_ratio_lt_60: interval arithmetic (i_prec 30)
• a4_algebraic_bridge: Trinity_a4 = k_ratio * Coq_a4 (field tactic)
• k_ratio_rationalized: k = (704+192*sqrt5)/19 (mirrors A4Conversion.v)
• BT8_no_H4_integer_bridge: main conjunction theorem

Outcome

Boundary Theorem BT-8: No exact H4-integer bridge exists.
The conversion factor k = (704+192*sqrt5)/19 is irrational (element of Q(sqrt5) \ Q).
The ~0.59% deviation from 60 is a genuine mathematical boundary of the H4 framework.

BT-8 joins BT-1..BT-4 (BoundaryTheorems.v) as a Coq-formal impossibility result.

Files Changed

• proofs/trinity/A4Bridge.v (new, 370 lines)
• proofs/trinity/_CoqProject (add A4Bridge.v entry)

R1 Compliance

No Python files. No .py added. All code in English.