recognitionThetaModularIdentity_of_prefactor
This definition constructs the modular identity structure for the recognition theta function directly from a prefactor that is continuous and obeys the inversion relation. Number theorists or Recognition Science researchers addressing sub-conjecture A.2 would cite it to bridge candidate prefactors to the required identity interface. The proof is a one-line wrapper applying the mpr direction of the equivalence lemma to package the prefactor data.
claimLet $p$ be a continuous function $ρ:ℝ→ℝ$ such that for all $t>0$, recognition theta at $1/t$ equals $ρ(t)$ times recognition theta at $t$. Then $p$ determines a modular identity structure for the recognition theta function.
background
The module addresses sub-conjecture A.2: the modular identity for the recognition theta function on the phi-ladder with an 8-tick kernel. Mathlib supplies general Fourier tools, yet the project still lacks the lattice-specific Poisson summation package needed to construct the prefactor explicitly. The local setting therefore pins the interface exactly: any continuous prefactor obeying the inversion identity is declared to be the RecognitionThetaModularIdentity structure. RecognitionThetaPrefactor is the structure carrying a continuous real function ρ together with the inversion axiom ∀ t>0, recognitionTheta(1/t)=ρ(t)·recognitionTheta(t).
proof idea
One-line wrapper that applies the mpr constructor of the equivalence lemma recognitionThetaModularIdentity_iff_prefactor to the supplied prefactor p, directly yielding the modular identity structure.
why it matters in Recognition Science
The definition supplies the constructor field inside recognitionThetaModularAttackSurface, the machine-readable status record for sub-conjecture A.2. It therefore lets all downstream code treat any qualifying continuous prefactor as the modular identity without further proof obligations. The parent theorem is the attack-surface record; the framework landmark is the missing Poisson-summation step for the phi-ladder / 8-tick theta kernel. It touches the open question of constructing the prefactor via Poisson summation rather than assuming it.
scope and limits
- Does not prove existence of any non-constant continuous prefactor obeying the inversion.
- Does not supply the Poisson summation formula for the recognition theta kernel.
- Does not derive numerical values for constants or connect to spatial dimensions.
- Does not establish analytic continuation or convergence properties of the theta function.
Lean usage
def attackSurface : RecognitionThetaModularAttackSurface := { prefactor_equivalence := recognitionThetaModularIdentity_iff_prefactor, constructor := recognitionThetaModularIdentity_of_prefactor }formal statement (Lean)
45def recognitionThetaModularIdentity_of_prefactor
46 (p : RecognitionThetaPrefactor) :
47 RecognitionThetaModularIdentity :=
proof body
Definition body.
48 recognitionThetaModularIdentity_iff_prefactor.mpr ⟨p⟩
49
50/-! ## Current A.2 attack surface -/
51
52/-- Machine-readable A.2 status: all downstream code only needs a continuous
53prefactor satisfying inversion; the missing theorem is the Poisson-summation
54construction of that prefactor. -/