 125/-- A Mellin reflection package compatible with recovered complex inputs. This
 126does not define zeta; it records the bridge Phase 4 must instantiate with an
 127explicit theta kernel. -/
 128structure RecoveredComplexMellinBridge where
 129  analytic_substrate : Foundation.LogicComplexCompat.LogicComplexAnalyticSubstrateCert
 130  kernel : ℝ → ℝ
 131  mellin_pkg : MellinAdmissibleKernel kernel
 132  reflection : ∀ s : ℝ, mellin_pkg.M s = mellin_pkg.M (mellinReflect s)
 133
 134/-- Any admissible kernel over the recovered complex analytic substrate carries
 135the Mellin reflection symmetry. -/
 136def recoveredComplexMellinBridge_of_admissible
 137    (kernel : ℝ → ℝ) (pkg : MellinAdmissibleKernel kernel) :
 138    RecoveredComplexMellinBridge where
 139  analytic_substrate := Foundation.LogicComplexCompat.logicComplexAnalyticSubstrateCert
 140  kernel := kernel
 141  mellin_pkg := pkg
 142  reflection := mellin_reciprocal_reflection pkg
 143
 144/-! ## 5. Phase 3 certificate -/
 145
 146/-- Phase 3 certificate: reciprocal symmetry plus a valid Mellin substitution
 147law forces `s ↔ 1-s` reflection at the transform level. -/
 148structure MellinPhase3Cert where
 149  J_reciprocal : ReciprocalSymmetric Cost.Jcost
 150  kernel_inversion :
 151    ∀ s x : ℝ, 0 < x → mellinKernel s x⁻¹ = x ^ (mellinReflect s)
 152  reflection_from_admissibility :
 153    ∀ {f : ℝ → ℝ}, (pkg : MellinAdmissibleKernel f) →
 154      ∀ s : ℝ, pkg.M s = pkg.M (mellinReflect s)
 155
 156def mellinPhase3Cert : MellinPhase3Cert where
 157  J_reciprocal := Jcost_mellin_reciprocal
 158  kernel_inversion := mellinKernel_inversion

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