theorem
proved
tactic proof
coupling_law_cert
show as:
view Lean formalization →
formal statement (Lean)
265theorem coupling_law_cert : CouplingLawCert where
266 enhancement_universal := coupling_identity
proof body
Tactic-mode proof.
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
2967. **Where it breaks down**: For t > 2, S grows rapidly, and the
297 geometric answer diverges from perturbative. This is exactly the
298 regime of confinement and mass generation.
299
300The coupling law is the **third object** connecting geometric and
301perturbative physics: neither a geometric quantity nor a perturbative
302quantity, but the *ratio function* between them — determined entirely
303by the J-cost functional equation. -/
304
305end
306
307end CouplingLaw
308end Unification
309end IndisputableMonolith