MellinPhase3Cert
plain-language theorem explainer
Phase 3 of the RS-native zeta program is certified by a structure that records reciprocal symmetry of the cost function together with a valid Mellin substitution law, thereby forcing the s to 1-s reflection at the transform level. Number theorists working on the Recognition Science zeta function would cite it to separate algebraic inputs from analytic assumptions. The declaration is introduced as a plain structure definition collecting the three required properties.
Claim. A Phase 3 certificate consists of three components: the cost function $J$ satisfies $J(x)=J(x^{-1})$ for all $x>0$; the Mellin kernel obeys $K(s,x^{-1})=x^{1-s}$ for $x>0$; and for any admissible kernel package $P$ with transform $M$, one has $M(s)=M(1-s)$ for all real $s$.
background
In the Recognition Science framework the J-cost function encodes the algebraic content of recognition and is known to be reciprocally symmetric: $J(x)=J(1/x)$ for positive $x$. The Mellin transform is introduced via the real kernel $x^{s-1}$, with the reflection map sending $s$ to $1-s$. The module separates the algebraic/RS inputs (reciprocal symmetry and kernel substitution) from the analytic requirements (existence of the integral and validity of the substitution). The upstream result establishing symmetry of the cost function is the theorem that $J(x)=J(x^{-1})$ for $x>0$, while the predicate of reciprocal symmetry formalizes the property for general functions. The admissible kernel structure records the minimal assumptions needed for the reflection law to hold at the transform level.
proof idea
This declaration is a structure definition that bundles three fields: the reciprocal symmetry of the J-cost, the explicit inversion identity for the Mellin kernel, and the reflection property that follows from any admissible kernel package. No tactics or lemmas are applied inside the definition itself; the fields are populated downstream by specific theorems such as the J-cost Mellin reciprocal and the kernel inversion identity.
why it matters
This structure supplies the interface required by the Phase 3 certificate construction, which in turn prepares the ground for Phase 4 instantiation with a theta kernel. It directly implements the separation described in the module documentation between algebraic reciprocal symmetry and analytic reflection. Within the broader Recognition Science program it bridges the J-uniqueness property to the Mellin-transform level, setting the stage for the functional equation of the RS-native zeta function.
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