zetaFromThetaPhase4Cert
plain-language theorem explainer
The zetaFromThetaPhase4Cert definition constructs a certificate asserting that the theta kernel equals the recognition theta and that the completed zeta satisfies its functional equation in both the Mathlib setting and the recovered complex logic. Researchers on the Recognition Science zeta program would cite it to isolate the theta Mellin bridge to the Riemann zeta reflection without claiming a full analytic proof. The construction is a direct structure instantiation that assigns the unconditional Mathlib functional equation theorem together,
Claim. Let $Z(s)$ denote the completed Riemann zeta function and let $Z_L(s)$ denote its logic-complex counterpart. The certificate asserts that the theta kernel equals the recognition theta function, that $Z(s) = Z(1-s)$ for all $s$ in the complexes, that $Z_L(s) = Z_L(1-s)$ holds after transport through the LogicComplex isomorphism, and that any ThetaCompletedZetaBridge implies the functional equation.
background
The ZetaFromTheta module implements Phase 4 of the RS-native zeta program. It connects the Recognition Theta program to the completed zeta functional equation by isolating the exact bridge: a theta-style Mellin transform compatible with the completed zeta yields the functional equation. The unconditional functional equation remains Mathlib's completedRiemannZeta_one_sub, recorded here under the recovered-complex substrate. The module doc states that the strong ThetaCompletedZetaBridge requires the Recognition-Theta modular identity (the reciprocal symmetry of recognitionTheta), which is not proved, so the certificate isolates only the consequence the RH recognition chain consumes.
proof idea
The definition is a structure constructor for ZetaFromThetaPhase4Cert. It sets the theta_kernel_defined field to reflexivity (rfl) and directly assigns the remaining fields to the upstream theorems completed_zeta_functional_equation_mathlib, logic_completed_zeta_functional_equation, and completed_zeta_functional_equation_from_mellin. No additional tactics or reductions are performed.
why it matters
This declaration supplies the Phase 4 certificate that bridges Recognition Theta to the completed zeta functional equation inside the Recognition Science framework. It feeds the logic_completed_zeta_functional_equation_from_theta theorem, which applies the supplied bridge to obtain the reflection for logicCompletedRiemannZeta. The module records the Mathlib theorem under the recovered-complex substrate and names the theta/Mellin bridge that would make the result RS-native. It touches the open question of formalizing the full theta/Mellin analytic continuation while remaining compatible with the J-uniqueness and phi fixed-point steps of the forcing chain.
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