IndisputableMonolith.NumberTheory.ZetaFromTheta
ZetaFromTheta assembles the Recognition Theta as the kernel for the Mellin transform to produce the completed zeta function in the RS-native program. Researchers tracking the zeta construction phases would cite it when linking theta kernels to functional equations. The module is an import-and-bridge assembly that relies on upstream reflection and compatibility results without new internal proofs.
claimThe Recognition Theta function $\tilde{\Theta}_{RS}(t)$ acts as the kernel for the Mellin transform, yielding the completed zeta function whose functional equation follows from reciprocal symmetry under $t \mapsto 1/t$.
background
This module sits in Phase 3-4 of the RS-native zeta program. It imports LogicComplexCompat to adopt Mathlib's $\mathbb{C}$ as the analytic substrate without redefining holomorphy or contour integration, MellinTransform for the reflection theorem derived from reciprocal symmetry and kernel substitution, and RecognitionTheta for the candidate completion of the cost theta function.
RecognitionTheta states that $\tilde{\Theta}_{RS}(t)$ completes $\Theta_J(t) = \sum e^{-t c(n)}$ by incorporating the 8-tick character (T7) and phi-ladder weight (T6) to inherit a modular identity under $t \mapsto 1/t$. MellinTransform deliberately separates the algebraic/RS content from analytic details.
proof idea
This is a definition module, no proofs. It imports three upstream modules (LogicComplexCompat, MellinTransform, RecognitionTheta) and declares the theta-style kernel together with admissible structures and completed-zeta bridges that inherit reflection from the imported reciprocal-symmetry theorem.
why it matters in Recognition Science
The module feeds HadamardFactorization (Track D of the RH unconditional-closure plan), RecognitionTheta.MellinFactor (Track C, sub-conjecture A.3), and StripZeroFreeRegion (Phase 5). It supplies the explicit kernel step that converts the Recognition Theta modular identity into the completed zeta functional equation, advancing the T6-T7 landmarks of the forcing chain.
scope and limits
- Does not compute explicit zeta zeros or numerical values.
- Does not prove the Riemann hypothesis or zero-free regions.
- Does not redefine holomorphy or contour integration.
- Does not supply the Hadamard product factorization.
used by (3)
depends on (3)
declarations in this module (15)
-
def
thetaKernel -
structure
ThetaMellinAdmissible -
theorem
theta_mellin_reflection -
structure
ThetaCompletedZetaBridge -
theorem
completed_zeta_functional_equation_from_mellin -
theorem
completed_zeta_functional_equation_mathlib -
theorem
logic_completed_zeta_functional_equation -
structure
LogicThetaZetaBridge -
theorem
logic_completed_zeta_functional_equation_from_theta -
structure
ZetaFromThetaPhase4Cert -
def
zetaFromThetaPhase4Cert -
structure
CompletedZetaFunctionalEquationCert -
def
completedZetaFunctionalEquationCert -
def
completedZetaFunctionalEquationCert_of_thetaBridge -
theorem
completed_zeta_functional_equation_from_cert