IndisputableMonolith.NumberTheory.RecognitionTheta.ModularIdentity

   1import IndisputableMonolith.NumberTheory.RecognitionTheta.Convergence
   2
   3/-!
   4  RecognitionTheta/ModularIdentity.lean
   5
   6  Track C, sub-conjecture A.2.
   7
   8  The RS theta modular identity needs a Poisson-summation theorem for the
   9  phi-ladder / 8-tick theta kernel. Mathlib has extensive Fourier analysis, but
  10  this project does not yet have the special lattice package required for
  11  `recognitionTheta`.
  12
  13  This module pins down the exact interface: a continuous prefactor satisfying
  14  the inversion identity is precisely the `RecognitionThetaModularIdentity`
  15  structure from `RecognitionTheta.lean`.
  16-/
  17
  18namespace IndisputableMonolith
  19namespace NumberTheory
  20namespace RecognitionTheta
  21namespace ModularIdentity
  22
  23noncomputable section
  24
  25/-- Candidate modular prefactor data for the Recognition Theta identity. -/
  26structure RecognitionThetaPrefactor where
  27  ρ : ℝ → ℝ
  28  continuous : Continuous ρ
  29  inversion :
  30    ∀ t : ℝ, 0 < t →
  31      recognitionTheta (1 / t) = ρ t * recognitionTheta t
  32
  33/-- Prefactor data is exactly the existing modular-identity structure. -/
  34theorem recognitionThetaModularIdentity_iff_prefactor :
  35    RecognitionThetaModularIdentity ↔ Nonempty RecognitionThetaPrefactor := by
  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. -/
  45def recognitionThetaModularIdentity_of_prefactor
  46    (p : RecognitionThetaPrefactor) :
  47    RecognitionThetaModularIdentity :=
  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. -/
  55structure RecognitionThetaModularAttackSurface where
  56  prefactor_equivalence :
  57    RecognitionThetaModularIdentity ↔ Nonempty RecognitionThetaPrefactor
  58  constructor :
  59    RecognitionThetaPrefactor → RecognitionThetaModularIdentity
  60
  61def recognitionThetaModularAttackSurface :
  62    RecognitionThetaModularAttackSurface where
  63  prefactor_equivalence := recognitionThetaModularIdentity_iff_prefactor
  64  constructor := recognitionThetaModularIdentity_of_prefactor
  65
  66end
  67
  68end ModularIdentity
  69end RecognitionTheta
  70end NumberTheory
  71end IndisputableMonolith
  72