Jcost_mellin_reflection
plain-language theorem explainer
The theorem states that the Mellin transform M attached to a J-cost admissible kernel satisfies M(s) = M(mellinReflect(s)) for every real s. Workers on the Recognition Science zeta program cite it as the algebraic symmetry step that closes Phase 3. The proof is a direct one-line application of the reciprocal reflection lemma to the kernel package inside the supplied bridge.
Claim. Let $M$ be the Mellin transform of an admissible kernel for the recognition cost $J$. Then $M(s) = M(1-s)$ for all real $s$, where the bridge structure encodes the exact admissibility condition that converts $J$'s reciprocal symmetry into this reflection invariance.
background
This module forms Phase 3 of the RS-native zeta program. It supplies a Mellin-transform interface whose reflection property is derived strictly from reciprocal symmetry of the underlying kernel. The central definition is the structure JCostMellinBridge, which contains a field pkg of type MellinAdmissibleKernel Cost.Jcost; the doc-comment states that this admissibility is the precise analytic condition turning J's reciprocal symmetry into an s ↔ 1-s transform symmetry.
proof idea
The proof is a one-line wrapper that applies the sibling lemma mellin_reciprocal_reflection directly to bridge.pkg. That lemma encodes the reciprocal symmetry of the J-cost kernel and substitutes it into the Mellin transform definition to obtain the stated reflection equality.
why it matters
The result supplies the reflection symmetry required before Phase 4 can instantiate an explicit theta kernel. It sits downstream of the J-uniqueness property (T5) and the Recognition Composition Law, and upstream of the full functional equation for the RS-native zeta function. It touches the open question of recovering the analytic continuation from the eight-tick octave structure.
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