RecognitionThetaPrefactor
plain-language theorem explainer
The RecognitionThetaPrefactor structure packages a continuous function ρ together with the inversion relation that links recognitionTheta at t and at 1/t. Researchers tracking sub-conjecture A.2 on the modular properties of the Recognition Theta would cite this interface when they need only the existence of such a prefactor rather than its explicit Poisson-sum construction. The declaration is a bare structure definition whose fields are checked by type inference alone.
Claim. A continuous function $ρ : ℝ → ℝ$ is a Recognition Theta prefactor when it satisfies $Θ(1/t) = ρ(t) · Θ(t)$ for every $t > 0$, where $Θ$ denotes the Recognition Theta function defined by the sum of its term series.
background
The Recognition Theta function is introduced as the infinite sum over n of recognitionThetaTerm t n; its convergence is part of sub-conjecture A.1 while the sum itself is always defined. The present module addresses sub-conjecture A.2 by isolating the precise interface needed for a modular identity: a continuous prefactor obeying the inversion relation is declared equivalent to the RecognitionThetaModularIdentity structure. Upstream material includes the canonical identity event at J-cost minimum from ObserverForcing and the definition of recognitionTheta itself.
proof idea
The declaration is a structure definition that directly assembles the three fields ρ, its continuity witness, and the inversion predicate. No lemmas are invoked and no tactics are executed; the Lean kernel simply verifies that the supplied data inhabit the declared types.
why it matters
RecognitionThetaPrefactor supplies the concrete data carrier that lets downstream code treat the modular identity as an equivalence between RecognitionThetaModularIdentity and the nonempty type of such prefactors. It therefore marks the current attack surface for A.2: the missing step is the Poisson-summation construction of ρ on the phi-ladder / 8-tick kernel. The structure sits between the analytic definition of recognitionTheta and the broader Recognition Science forcing chain that ultimately fixes D = 3 and the eight-tick octave.
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