 250/-! ## §7. The Coupling Certificate -/
 251
 252/-- The coupling law certificate: packages the full bridge. -/
 253structure CouplingLawCert where
 254  enhancement_universal :
 255    ∀ (t : ℝ), t ≠ 0 →
 256      exactCost t = coshEnhancement t * perturbativeCost t
 257  perturbative_limit :
 258    coshEnhancement 0 = 1
 259  enhancement_symmetric :
 260    ∀ (t : ℝ), coshEnhancement (-t) = coshEnhancement t
 261  geometric_dominance :
 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.

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