IndisputableMonolith.NumberTheory.RecognitionTheta.MellinFactor

   1import IndisputableMonolith.NumberTheory.RecognitionTheta.ModularIdentity
   2import IndisputableMonolith.NumberTheory.ZetaFromTheta
   3
   4/-!
   5  RecognitionTheta/MellinFactor.lean
   6
   7  Track C, sub-conjecture A.3.
   8
   9  The existing `RecognitionThetaMellinFactor` structure in
  10  `RecognitionTheta.lean` is only a placeholder: it asks for a nonzero
  11  function `G : ℂ → ℂ` and leaves the actual Mellin identity as `True`.
  12  This module makes that explicit by inhabiting the current structure and then
  13  defining the stronger interface needed for the RS-native theta/zeta bridge.
  14-/
  15
  16namespace IndisputableMonolith
  17namespace NumberTheory
  18namespace RecognitionTheta
  19namespace MellinFactor
  20
  21open ZetaFromTheta
  22
  23noncomputable section
  24
  25/-- The current abstract A.3 structure is inhabited by the constant `1`
  26function. This is a status theorem, not an analytic factorization theorem. -/
  27theorem recognitionThetaMellinFactor_placeholder :
  28    RecognitionThetaMellinFactor := by
  29  refine ⟨?g, ?hne, trivial⟩
  30  · exact fun _ : ℂ => 1
  31  · intro h
  32    have h0 := congrFun h 0
  33    norm_num at h0
  34
  35/-- Strong Mellin factorization data needed by the theta/zeta route.
  36
  37This strengthens the placeholder A.3 by naming the complex continuation of the
  38theta Mellin transform, its equality with `completedRiemannZeta`, and the
  39reflection inherited from A.2. -/
  40structure StrongRecognitionThetaMellinFactor where
  41  theta : ThetaMellinAdmissible
  42  completedMellin : ℂ → ℂ
  43  matches_completed_zeta :
  44    ∀ s : ℂ, completedMellin s = completedRiemannZeta s
  45  reflection :
  46    ∀ s : ℂ, completedMellin s = completedMellin (1 - s)
  47
  48/-- Strong Mellin factorization data is exactly enough to inhabit
  49`ThetaCompletedZetaBridge`. -/
  50def thetaCompletedZetaBridge_of_strongMellinFactor
  51    (factor : StrongRecognitionThetaMellinFactor) :
  52    ThetaCompletedZetaBridge where
  53  theta := factor.theta
  54  completedMellin := factor.completedMellin
  55  completed_matches_zeta := factor.matches_completed_zeta
  56  completed_reflection_from_mellin := factor.reflection
  57
  58/-- Conversely, the Phase 4 bridge is already the strong Mellin-factor package. -/
  59def strongMellinFactor_of_thetaCompletedZetaBridge
  60    (bridge : ThetaCompletedZetaBridge) :
  61    StrongRecognitionThetaMellinFactor where
  62  theta := bridge.theta
  63  completedMellin := bridge.completedMellin
  64  matches_completed_zeta := bridge.completed_matches_zeta
  65  reflection := bridge.completed_reflection_from_mellin
  66
  67theorem strongMellinFactor_iff_thetaCompletedZetaBridge :
  68    Nonempty StrongRecognitionThetaMellinFactor ↔
  69      Nonempty ThetaCompletedZetaBridge := by
  70  constructor
  71  · intro h
  72    rcases h with ⟨factor⟩
  73    exact ⟨thetaCompletedZetaBridge_of_strongMellinFactor factor⟩
  74  · intro h
  75    rcases h with ⟨bridge⟩
  76    exact ⟨strongMellinFactor_of_thetaCompletedZetaBridge bridge⟩
  77
  78/-! ## Current A.3 attack surface -/
  79
  80structure RecognitionThetaMellinFactorAttackSurface where
  81  placeholder_inhabited : RecognitionThetaMellinFactor
  82  strong_bridge_equivalence :
  83    Nonempty StrongRecognitionThetaMellinFactor ↔
  84      Nonempty ThetaCompletedZetaBridge
  85
  86def recognitionThetaMellinFactorAttackSurface :
  87    RecognitionThetaMellinFactorAttackSurface where
  88  placeholder_inhabited := recognitionThetaMellinFactor_placeholder
  89  strong_bridge_equivalence := strongMellinFactor_iff_thetaCompletedZetaBridge
  90
  91end
  92
  93end MellinFactor
  94end RecognitionTheta
  95end NumberTheory
  96end IndisputableMonolith
  97