ZetaFromThetaPhase4Cert
plain-language theorem explainer
ZetaFromThetaPhase4Cert packages four assertions into a certificate for Phase 4 of the RS zeta program: the theta kernel matches the recognition theta, the completed Riemann zeta obeys its functional equation, the recovered-complex version does likewise via transport maps, and any ThetaCompletedZetaBridge implies the equation. Number theorists formalizing the Recognition Theta to zeta link cite this when assembling the Phase 4 interface. The declaration is a pure structure definition that bundles the statements without internal proof steps.
Claim. A structure consisting of four fields: (i) the theta kernel equals the recognition theta function, (ii) the completed Riemann zeta satisfies $completedRiemannZeta(s) = completedRiemannZeta(1-s)$ for all $s$ in the complex numbers, (iii) the logic-complex completed zeta satisfies $logicCompletedRiemannZeta(s) = logicCompletedRiemannZeta(fromComplex(1-toComplex(s)))$ for all recovered-complex $s$, and (iv) any ThetaCompletedZetaBridge implies the completed zeta functional equation.
background
Phase 4 of the RS-native zeta program connects the Recognition Theta to the completed zeta functional equation through a Mellin bridge without claiming a full analytic formalization. The module records the precise interface so that the standard Mathlib functional equation can be recovered on the logic-complex substrate once the bridge is supplied.
proof idea
The declaration is a structure definition that declares four independent fields. No lemmas or tactics are invoked inside the structure; the fields are later populated by the downstream definition zetaFromThetaPhase4Cert using completed_zeta_functional_equation_mathlib, logic_completed_zeta_functional_equation, and completed_zeta_functional_equation_from_mellin.
why it matters
This structure is the Phase 4 certificate that feeds directly into zetaFromThetaPhase4Cert. It isolates the theta-kernel identification and the recovered-complex functional equation required to make the zeta equation RS-native via the ThetaCompletedZetaBridge, advancing the connection between recognitionTheta and the completed zeta while leaving the analytic Mellin content as an open sub-conjecture.
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