recoveredComplexMellinBridge_of_admissible
plain-language theorem explainer
Any admissible Mellin kernel over the recovered complex analytic substrate carries Mellin reflection symmetry by this definition. Researchers on the RS-native zeta program cite it as the Phase 3 bridge that encodes the transform-level reflection needed before theta instantiation. The definition is a direct structure assembly that injects the supplied kernel and package while pulling the reflection law from mellin_reciprocal_reflection.
Claim. Let $f : {R} {to} {R}$ be a kernel and $P$ a Mellin-admissible package for $f$, so that $P$ supplies a transform $M$ obeying reciprocal symmetry on $f$ together with the substitution law $M(s) = M(1-s)$. The recovered complex Mellin bridge is the structure $(C, f, P, R)$ where $C$ is the logic-complex analytic substrate certificate and $R$ asserts $M(s) = M(1-s)$ for all real $s$.
background
Phase 3 of the RS-native zeta program supplies a Mellin-transform interface whose reflection theorem follows from reciprocal symmetry. The MellinAdmissibleKernel structure records the precise analytic assumptions an integral definition must meet: it contains a transform function $M$, a reciprocal symmetry condition on the kernel, and a substitution law that converts the transform at $s$ into the transform at $1-s$ under the change of variables $x {mapsto} x^{-1}$ (MODULE_DOC). The RecoveredComplexMellinBridge packages the same data for recovered complex inputs while keeping algebraic/RS content (reciprocal symmetry and substitution) separate from analytic content (integral existence and validity of the substitution). Upstream, logicComplexAnalyticSubstrateCert supplies the Phase 2 compatibility certificate that includes carrier equivalence, zeta transport, and the completed functional equation for the logic Riemann zeta.
proof idea
The definition is a direct structure constructor. It populates the analytic_substrate field with logicComplexAnalyticSubstrateCert, assigns the input kernel and MellinAdmissibleKernel package verbatim, and obtains the reflection field by applying mellin_reciprocal_reflection to the admissible package.
why it matters
This definition supplies the Phase 3 bridge in the RS-native zeta program, encoding Mellin reflection symmetry from reciprocal symmetry as the necessary precursor to the functional equation. It feeds forward to Phase 4 instantiations that will supply an explicit theta kernel, advancing the derivation of the zeta functional equation from first principles. The construction rests on the reciprocal automorphism and the logic-complex substrate certificate, preserving the framework separation of algebraic and analytic layers. It touches no open question but closes the Mellin-reflection scaffolding step.
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