RecognitionThetaModularIdentity
plain-language theorem explainer
RecognitionThetaModularIdentity packages the existence of a continuous prefactor ρ such that the Recognition Theta satisfies Θ̃_RS(1/t) = ρ(t) Θ̃_RS(t) for all t > 0. Number theorists and Recognition Science researchers working on modular forms of cost functions would cite this as the formal statement of Sub-conjecture A.2. The declaration is a direct structure definition with no proof body or lemma applications.
Claim. There exists a continuous function ρ : ℝ → ℝ such that ∀ t > 0, Θ̃_RS(1/t) = ρ(t) ⋅ Θ̃_RS(t).
background
The Recognition Theta function Θ̃_RS(t) is defined as the tsum of recognitionThetaTerm t n, completing the cost theta Θ_J(t) = Σ e^{-t c(n)} by incorporating the 8-tick character chi8 and the phi-ladder weights via phiRung. Here phiRung n is the completely additive index with phiRung p = ⌊log_φ p⌋ for primes, while chi8 is the non-trivial real character mod 8. The module states structural properties such as phiRung_mul (complete additivity) and recognitionTheta_at_one (constant term evaluation) with zero sorrys.
proof idea
This is a definition that directly encodes the modular identity as a Prop structure. No lemmas are applied and no tactics are used; the body simply asserts existence of a continuous ρ satisfying the inversion equation for recognitionTheta.
why it matters
The declaration states Sub-conjecture A.2 of Paper I, supplying the modular inversion hypothesis required for the Mellin factorization in RecognitionThetaMellinFactor (Sub-conjecture A.3). It feeds RecognitionThetaModularAttackSurface and the equivalence theorems recognitionThetaModularIdentity_iff_prefactor and recognitionThetaModularIdentity_of_prefactor, which bridge to explicit prefactor data. The structure ties directly to the eight-tick octave (T7) and phi-ladder fixed point (T6) by positing the symmetry needed for Poisson summation over the ladder.
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