LogicThetaZetaBridge
plain-language theorem explainer
LogicThetaZetaBridge packages the recovered-complex analytic substrate certificate with the theta-to-completed-zeta identification required for Phase 4 of the RS zeta program. Number theorists working on the Recognition Mellin transform would cite this structure when invoking the functional equation under the logic-complex equivalence. The definition is a direct record of the two required certificates with no further computation.
Claim. A structure consisting of a certificate that all analytic zeta operations are performed in the complex numbers through the recovered-complex equivalence, together with the analytic bridge identifying the Recognition Theta Mellin transform with the completed Riemann zeta function.
background
The module ZetaFromTheta forms Phase 4 of the RS-native zeta program. It connects the Recognition Theta program to the completed zeta functional equation by isolating a theta-style Mellin transform that is compatible with the completed zeta function. The current unconditional functional equation remains Mathlib's completedRiemannZeta_one_sub; this module records that theorem under the recovered-complex substrate and names the theta/Mellin bridge that would make it RS-native.
proof idea
The declaration is a structure definition that directly assembles the analytic substrate certificate and the theta-zeta bridge structure. No lemmas or tactics are applied.
why it matters
LogicThetaZetaBridge supplies the exact bridge used by the downstream theorem logic_completed_zeta_functional_equation_from_theta to obtain the functional equation for recovered-complex completed zeta. It fills the Phase 4 slot in the RS zeta program by recording the identification between the Recognition Theta Mellin transform and the completed zeta function without claiming the full analytic proof. This keeps the unconditional Mathlib result available while marking the precise point where an RS-native theta would close the argument.
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