recognitionThetaModularIdentity_iff_prefactor
The theorem equates the Recognition Theta modular identity proposition with the existence of a continuous prefactor structure satisfying the inversion relation under t to 1/t. Researchers tracking sub-conjecture A.2 would cite this to link the abstract existence claim with packaged prefactor data. The proof is a direct term-mode constructor that extracts and repacks the ρ function, continuity, and inversion axiom in both directions.
claimThe modular identity for the Recognition Theta function holds if and only if there exists a continuous function ρ such that for all t > 0, Θ(1/t) = ρ(t) · Θ(t), where Θ denotes the Recognition Theta.
background
The module tracks sub-conjecture A.2 for the Recognition Theta modular identity, which requires a Poisson summation theorem for the phi-ladder and 8-tick theta kernel. The local setting pins the interface: a continuous prefactor satisfying the inversion identity is precisely the RecognitionThetaModularIdentity structure. RecognitionThetaModularIdentity is the Prop asserting existence of continuous ρ with the inversion axiom. RecognitionThetaPrefactor is the structure packaging ρ, its continuity proof, and the same inversion axiom. Upstream, the RecognitionThetaModularIdentity doc states: 'Sub-conjecture A.2: under the inversion t ↦ 1/t, the Recognition Theta satisfies a modular identity with an explicit prefactor. The conjectural form is Θ̃_RS(1/t) = ρ(t) · Θ̃_RS(t) for some prefactor ρ(t) involving √t and log φ. We package this as a Prop on the existence of a continuous'.
proof idea
The term-mode proof applies constructor to split the iff. Forward direction: rcases extracts the prefactor triple from the modular identity and packs it into the RecognitionThetaPrefactor structure. Reverse direction: rcases unpacks the prefactor structure to witness the existential in the modular identity Prop.
why it matters in Recognition Science
This equivalence supports the attack surface for sub-conjecture A.2 via the downstream recognitionThetaModularAttackSurface and the direct constructor recognitionThetaModularIdentity_of_prefactor. It fills the modular-identity interface in the Recognition Theta module, relating to the phi-ladder and eight-tick octave from the forcing chain. It touches the open question of proving existence of the explicit prefactor involving √t and log φ.
scope and limits
- Does not prove existence of the prefactor ρ.
- Does not derive the explicit form of ρ involving √t and log φ.
- Does not supply the Poisson summation theorem for the full identity.
- Does not address convergence of the theta series beyond prefactor continuity.
Lean usage
recognitionThetaModularIdentity_iff_prefactor.mpr ⟨p⟩formal statement (Lean)
34theorem recognitionThetaModularIdentity_iff_prefactor :
35 RecognitionThetaModularIdentity ↔ Nonempty RecognitionThetaPrefactor := by
proof body
Term-mode proof.
36 constructor
37 · intro h
38 rcases h.prefactor with ⟨ρ, hcont, hinv⟩
39 exact ⟨{ ρ := ρ, continuous := hcont, inversion := hinv }⟩
40 · intro h
41 rcases h with ⟨p⟩
42 exact ⟨⟨p.ρ, p.continuous, p.inversion⟩⟩
43
44/-- Direct constructor for the modular-identity bridge. -/