coupling_law_cert

formal source

 262    ∀ (t : ℝ), t ≠ 0 → 1 < coshEnhancement t
 263
 264/-- The coupling law certificate is inhabited. -/
 265theorem coupling_law_cert : CouplingLawCert where
 266  enhancement_universal := coupling_identity
 267  perturbative_limit := enhancement_at_zero
 268  enhancement_symmetric := enhancement_symmetric
 269  geometric_dominance := enhancement_gt_one
 270
 271/-! ## §8. Physical Interpretation
 272
 273The coupling law resolves the "Missing Something" as follows:
 274
 2751. **What gap(Z) IS**: The exact (non-perturbative) J-cost running in
 276   log-φ units, evaluated at the anchor scale μ⋆.
 277
 2782. **What f_RG IS**: The perturbative (quadratic-approximation) running
 279   from SM β-functions, which captures only the t²/2 term of J_log.
 280
 2813. **What recognition strength IS**: The cosh enhancement factor
 282   S(t) = 2(cosh t − 1)/t², a universal function of the perturbative
 283   running parameter alone.
 284
 2854. **Why it's large for leptons**: The electron has Z = 1332, giving
 286   gap ≈ 13.95. At this scale, cosh is exponentially larger than
 287   quadratic, so S ≫ 1.
 288
 2895. **Why it's universal**: S depends only on t through cosh, which is
 290   uniquely determined by the RCL → J = cosh − 1. Zero free parameters.
 291
 2926. **Where perturbation theory works**: For t → 0 (weak coupling),
 293   S → 1, and geometric = perturbative. The SM is an excellent
 294   approximation at low coupling / high energy.
 295