MellinAdmissibleKernel
plain-language theorem explainer
MellinAdmissibleKernel records the minimal analytic assumptions on a Mellin transform M of a kernel f so that reciprocal symmetry of f yields the reflection M(s) = M(1-s). Researchers formalizing the Recognition Science zeta program cite it as the interface that isolates algebraic symmetry from the substitution law. The declaration is a bare structure with three fields that directly encode the required properties.
Claim. Let $f : {R} {to} {R}$. A Mellin-admissible kernel for $f$ consists of a function $M : {R} {to} {R}$ together with a proof that $f$ is reciprocally symmetric and that $M(s) = M(1-s)$ for all real $s$, where the equality follows from the validity of the substitution $x {mapsto} x^{-1}$ in the underlying integral.
background
The MellinTransform module implements Phase 3 of the RS-native zeta program. It deliberately separates algebraic/RS content (reciprocal symmetry of the kernel) from analytic content (existence of the transform and validity of the $x {mapsto} x^{-1}$ change of variables). The structure bundles a candidate transform $M$ with the hypothesis that $f$ satisfies ReciprocalSymmetric (drawn from the reciprocal automorphism in CostAlgebra and the reciprocal event in LedgerForcing) and the substitution invariance $M(s) = M(mellinReflect s)$. Upstream results on cost functions induced by multiplicative recognizers and observer forcing supply the concrete kernels that later instantiate this interface.
proof idea
As a structure definition there is no proof body. The three fields are declared directly: the transform function $M$, the reciprocal symmetry hypothesis on $f$, and the substitution law that encodes the change-of-variables invariance.
why it matters
This supplies the parent structure for JCostMellinBridge (which applies it to Cost.Jcost) and for MellinPhase3Cert (which derives the $s {leftrightarrow} 1-s$ reflection from reciprocal symmetry plus admissibility). It fills the transform-level bridge in the RS zeta program before Phase 4 instantiates an explicit theta kernel. The reflection law here is the first step toward the functional equation that will rest on the Recognition Composition Law.
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