ThetaCompletedZetaBridge
plain-language theorem explainer
The ThetaCompletedZetaBridge structure records the identification of a Recognition Theta Mellin transform with the completed Riemann zeta function together with its reflection symmetry. Number theorists working on the RS-native zeta program cite it as the Phase 4 analytic bridge. It is introduced directly as a four-field record type that packages an admissible theta kernel, the completed Mellin function, equality to completed zeta, and the inherited s to 1-s symmetry.
Claim. A structure consisting of an admissible theta Mellin package, a function $f:ℂ→ℂ$, the equality $f(s)=ζ^*(s)$ for the completed Riemann zeta function $ζ^*$ and all complex $s$, and the reflection property $f(s)=f(1-s)$ for all $s$.
background
The module ZetaFromTheta isolates the exact bridge needed to connect the Recognition Theta program to the completed zeta functional equation in Phase 4 of the RS-native zeta effort. It does not claim a full analytic proof of the theta Mellin transform; instead it records the minimal data that would make the functional equation RS-native once the theta Mellin continuation is supplied. ThetaMellinAdmissible is the structure whose single field is a MellinAdmissibleKernel applied to the Recognition thetaKernel. Upstream results supply the 8-tick Phase (Fin 8), the arithmetic zeta function, the BRF Cayley theta transform, and the Mellin kernel infrastructure.
proof idea
Defined directly as a structure with four fields. The first field pulls in the admissible theta package; the remaining three fields are the completed Mellin function together with the two pointwise equalities that identify it with completedRiemannZeta and enforce reflection symmetry.
why it matters
This declaration supplies the Phase 4 mathematical bite that lets the Recognition Theta Mellin data imply the completed zeta functional equation. It is used to construct StrongRecognitionThetaMellinFactor and the certificate CompletedZetaFunctionalEquationCert_of_thetaBridge. In the broader framework the bridge transports the Mellin reflection symmetry into the zeta functional equation, closing the loop from the eight-tick theta kernel to the analytic continuation of zeta.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.