RecognitionThetaModularAttackSurface
plain-language theorem explainer
RecognitionThetaModularAttackSurface packages the logical equivalence between the Recognition Theta modular identity conjecture and the existence of a nonempty continuous prefactor satisfying the inversion relation. Researchers tracking sub-conjecture A.2 in Recognition Science cite it to isolate the interface from the missing Poisson summation construction. It is realized as a record type whose two fields link the Prop to the prefactor data structure.
Claim. Let $Θ$ denote the Recognition Theta function. The structure asserts that the modular identity $Θ(1/t) = ρ(t) · Θ(t)$ for all $t > 0$ holds if and only if a nonempty continuous prefactor $ρ : ℝ → ℝ$ exists, together with a constructor that maps any such prefactor to the identity statement.
background
The RecognitionTheta/ModularIdentity module tracks sub-conjecture A.2 for the Recognition Theta function. RecognitionThetaPrefactor is the data structure consisting of a map $ρ : ℝ → ℝ$, a continuity proof, and the inversion property that recognitionTheta(1/t) equals ρ(t) times recognitionTheta(t) for every positive t. The upstream RecognitionThetaModularIdentity is the proposition that such a continuous prefactor exists, packaged as an existential statement over ρ.
proof idea
The declaration is a structure definition that directly records the biconditional between the modular identity proposition and the nonempty prefactor data type, together with the constructor map from prefactor to identity.
why it matters
This supplies the machine-readable status for sub-conjecture A.2, so downstream code depends only on a continuous prefactor satisfying inversion rather than the full Poisson summation construction. It feeds the definition of recognitionThetaModularAttackSurface. The open question remains the Poisson-summation proof for the phi-ladder / 8-tick theta kernel required to realize the prefactor.
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