RecognitionThetaMellinFactor
plain-language theorem explainer
RecognitionThetaMellinFactor packages Sub-conjecture A.3 as a Prop asserting that the Mellin transform of the Recognition Theta function factors as zeta(s) times a non-vanishing meromorphic G_RS(s) inheriting reflection symmetry from the modular identity. Number theorists closing the zeta bridge in Recognition Science cite this when moving from the theta sum to the completed L-function. The declaration is realized as a direct structure definition whose body reduces to an existential quantifier over G together with a trivial True placeholder.
Claim. There exists a function $G : ℂ → ℂ$ with $G ≠ 0$ such that the Mellin transform of the Recognition Theta function factors as $ζ(s) · G(s)$, where $G$ is meromorphic and inherits the reflection symmetry of the modular identity under the Recognition Theta construction.
background
The Recognition Theta function Θ̃_RS(t) completes the cost theta sum Θ_J(t) = Σ e^{-t c(n)} by adjoining the 8-tick character chi8 and the phi-rung weights phiRung n so that the sum satisfies a modular identity under t ↦ 1/t. The module defines phiRung as the completely additive index with r(p) = ⌊log_φ p⌋ for primes, chi8 as the non-trivial real character mod 8, recognitionThetaTerm as the weighted summand, and recognitionTheta t as the resulting tsum. Sub-conjectures A.1 (convergence), A.2 (modular identity), and A.3 (Mellin factorization) are introduced as hypothesis structures; the present declaration is exactly A.3. Upstream results supply the gravitational constant G in RS-native units and the functional-equation reparametrization used to relate the theta sum to its Mellin transform.
proof idea
The declaration is a structure definition whose body consists of an existential quantifier over a non-zero G : ℂ → ℂ together with the placeholder True. No lemmas or tactics are invoked; the construction directly encodes the statement of Sub-conjecture A.3 without analytic content.
why it matters
It supplies the hypothesis structure for Sub-conjecture A.3 and is referenced by the downstream attack-surface structure and the placeholder theorem that equate the abstract factor to the completed-zeta bridge. The declaration therefore sits at the interface between the phi-ladder arithmetic (T6) and the eight-tick modular identity (T7), preparing the ground for the Mellin factorization that would link Recognition Theta to the Riemann zeta function in the Recognition Science program.
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