IndisputableMonolith.NumberTheory.ZetaFromTheta
ZetaFromTheta supplies the Recognition Theta function as the kernel for the Mellin transform that produces the completed zeta function and its functional equation. Researchers tracking the RS-native zeta program cite the module to obtain the reflection identity from reciprocal symmetry without presupposing the classical zeta definition. The module imports the Mellin reflection theorem, the theta construction with eight-tick and phi-ladder weights, and the complex compatibility layer, then assembles the bridge theorems.
claimThe Recognition Theta function $T_{RS}(t)$ serves as the kernel for the Mellin transform, producing the completed zeta function that satisfies the functional equation $T_{RS}(t) = t^{-1} T_{RS}(1/t)$ and yields the completed zeta identity under the change of variable $s = 1/2 + it$.
background
Recognition Science derives the zeta function from the Mellin transform of a theta-style kernel built on the J-cost function. The Recognition Theta augments the basic sum with the eight-tick character and phi-ladder weights so that the kernel inherits modular invariance under $t$ to $1/t$. The module sits inside the NumberTheory domain and forms phase 4 of the RS zeta program.
proof idea
The module defines the theta kernel and proves Mellin admissibility. It then applies the reflection theorem imported from MellinTransform to obtain the completed zeta functional equation. Logic bridges certify equivalence to the classical statement while the complex compatibility layer anchors all objects to Mathlib's analytic substrate.
why it matters in Recognition Science
The module supplies the theta-to-zeta bridge required by HadamardFactorization for the unconditional RH closure track and by RecognitionTheta.MellinFactor for sub-conjecture A.3. It also feeds StripZeroFreeRegion for phase 5. The construction advances the RS zeta program by deriving the functional equation from the Recognition Theta kernel rather than from an external zeta definition.
scope and limits
- Does not compute explicit zeta values or series expansions.
- Does not establish zero-free regions inside the critical strip.
- Does not derive the Hadamard product representation.
- Does not address the distribution or multiplicity of zeros.
- Does not redefine holomorphy or contour integration.
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