IndisputableMonolith.NumberTheory.ZetaFromTheta

   1import IndisputableMonolith.NumberTheory.MellinTransform
   2import IndisputableMonolith.NumberTheory.RecognitionTheta
   3import IndisputableMonolith.Foundation.LogicComplexCompat
   4
   5/-!
   6  ZetaFromTheta.lean
   7
   8  Phase 4 of the RS-native zeta program.
   9
  10  This module connects the Recognition Theta program to the completed zeta
  11  functional equation.  It does not pretend that the full theta/Mellin
  12  analytic proof has been formalized.  Instead it isolates the exact bridge:
  13  a theta-style Mellin transform, compatible with the completed zeta function,
  14  gives the functional equation.
  15
  16  The current unconditional functional equation is still Mathlib's
  17  `completedRiemannZeta_one_sub`; this module records that theorem under the
  18  recovered-complex substrate and names the theta/Mellin bridge that would make
  19  it RS-native.
  20-/
  21
  22namespace IndisputableMonolith
  23namespace NumberTheory
  24namespace ZetaFromTheta
  25
  26open MellinTransform
  27open RecognitionTheta
  28open Foundation.ComplexFromLogic
  29open Foundation.ComplexFromLogic.LogicComplex
  30open Foundation.LogicComplexCompat
  31
  32noncomputable section
  33
  34/-! ## 1. Recognition theta as a Mellin kernel -/
  35
  36/-- The theta-style kernel for the Mellin transform is the Recognition Theta
  37function. -/
  38def thetaKernel : ℝ → ℝ :=
  39  recognitionTheta
  40
  41/-- A theta Mellin package is a Mellin-admissible transform for the Recognition
  42Theta kernel. -/
  43structure ThetaMellinAdmissible where
  44  pkg : MellinAdmissibleKernel thetaKernel
  45
  46theorem theta_mellin_reflection (theta : ThetaMellinAdmissible) :
  47    ∀ s : ℝ, theta.pkg.M s = theta.pkg.M (mellinReflect s) :=
  48  mellin_reciprocal_reflection theta.pkg
  49
  50/-! ## 2. Bridge from theta Mellin data to completed zeta -/
  51
  52/-- The analytic bridge required to identify the Recognition Theta Mellin
  53transform with the completed zeta function.
  54
  55`completedMellin` is the complex-valued continuation of the theta Mellin
  56transform. The bridge records:
  57
  58* it agrees with `completedRiemannZeta`;
  59* the Mellin reflection symmetry transports to complex reflection `s ↦ 1-s`.
  60
  61This is the Phase 4 mathematical bite. -/
  62structure ThetaCompletedZetaBridge where
  63  theta : ThetaMellinAdmissible
  64  completedMellin : ℂ → ℂ
  65  completed_matches_zeta :
  66    ∀ s : ℂ, completedMellin s = completedRiemannZeta s
  67  completed_reflection_from_mellin :
  68    ∀ s : ℂ, completedMellin s = completedMellin (1 - s)
  69
  70/-- The theorem requested by Phase 4: theta Mellin data identified with
  71completed zeta gives the functional equation. -/
  72theorem completed_zeta_functional_equation_from_mellin
  73    (bridge : ThetaCompletedZetaBridge) (s : ℂ) :
  74    completedRiemannZeta s = completedRiemannZeta (1 - s) := by
  75  rw [← bridge.completed_matches_zeta s,
  76      bridge.completed_reflection_from_mellin s,
  77      bridge.completed_matches_zeta (1 - s)]
  78
  79/-- Mathlib's completed-zeta functional equation, restated as the current
  80unconditional analytic source. -/
  81theorem completed_zeta_functional_equation_mathlib (s : ℂ) :
  82    completedRiemannZeta s = completedRiemannZeta (1 - s) :=
  83  (completedRiemannZeta_one_sub s).symm
  84
  85/-! ## 3. Recovered-complex version -/
  86
  87/-- Completed-zeta functional equation on recovered-complex inputs. -/
  88theorem logic_completed_zeta_functional_equation (s : LogicComplex) :
  89    logicCompletedRiemannZeta s =
  90      logicCompletedRiemannZeta (fromComplex (1 - toComplex s)) := by
  91  simpa [logicCompletedRiemannZeta] using
  92    (completed_zeta_functional_equation_mathlib (toComplex s))
  93
  94/-- A recovered-complex theta/zeta bridge package. -/
  95structure LogicThetaZetaBridge where
  96  analytic_substrate : Foundation.LogicComplexCompat.LogicComplexAnalyticSubstrateCert
  97  theta_bridge : ThetaCompletedZetaBridge
  98
  99/-- If the theta/zeta Mellin bridge is supplied, recovered-complex completed
 100zeta satisfies the functional equation through that bridge. -/
 101theorem logic_completed_zeta_functional_equation_from_theta
 102    (bridge : LogicThetaZetaBridge) (s : LogicComplex) :
 103    logicCompletedRiemannZeta s =
 104      logicCompletedRiemannZeta (fromComplex (1 - toComplex s)) := by
 105  have h := completed_zeta_functional_equation_from_mellin
 106    bridge.theta_bridge (toComplex s)
 107  simpa [logicCompletedRiemannZeta] using h
 108
 109/-! ## 4. Phase 4 certificate -/
 110
 111structure ZetaFromThetaPhase4Cert where
 112  theta_kernel_defined : thetaKernel = recognitionTheta
 113  mathlib_functional_equation :
 114    ∀ s : ℂ, completedRiemannZeta s = completedRiemannZeta (1 - s)
 115  recovered_complex_functional_equation :
 116    ∀ s : LogicComplex,
 117      logicCompletedRiemannZeta s =
 118        logicCompletedRiemannZeta (fromComplex (1 - toComplex s))
 119  theta_bridge_implication :
 120    ∀ _bridge : ThetaCompletedZetaBridge,
 121      ∀ s : ℂ, completedRiemannZeta s = completedRiemannZeta (1 - s)
 122
 123def zetaFromThetaPhase4Cert : ZetaFromThetaPhase4Cert where
 124  theta_kernel_defined := rfl
 125  mathlib_functional_equation := completed_zeta_functional_equation_mathlib
 126  recovered_complex_functional_equation := logic_completed_zeta_functional_equation
 127  theta_bridge_implication := completed_zeta_functional_equation_from_mellin
 128
 129/-! ## 5. Phase 6: unconditional Mathlib FE bridge
 130
 131The strong `ThetaCompletedZetaBridge` requires the Recognition-Theta modular
 132identity (the reciprocal symmetry of `recognitionTheta`), which is not
 133proved.  The weaker certificate below isolates only the consequence the RH
 134recognition chain actually consumes — a `completedMellin : ℂ → ℂ` that agrees
 135with Mathlib's `completedRiemannZeta` and inherits the functional equation —
 136and is unconditionally inhabited from Mathlib. -/
 137
 138structure CompletedZetaFunctionalEquationCert where
 139  completedMellin : ℂ → ℂ
 140  completed_matches_zeta :
 141    ∀ s : ℂ, completedMellin s = completedRiemannZeta s
 142  completed_reflection :
 143    ∀ s : ℂ, completedMellin s = completedMellin (1 - s)
 144
 145/-- Unconditional inhabitation: take the completed zeta itself. -/
 146def completedZetaFunctionalEquationCert :
 147    CompletedZetaFunctionalEquationCert where
 148  completedMellin := completedRiemannZeta
 149  completed_matches_zeta := fun _ => rfl
 150  completed_reflection := completed_zeta_functional_equation_mathlib
 151
 152/-- The strong theta bridge implies the weaker certificate. -/
 153def completedZetaFunctionalEquationCert_of_thetaBridge
 154    (bridge : ThetaCompletedZetaBridge) :
 155    CompletedZetaFunctionalEquationCert where
 156  completedMellin := bridge.completedMellin
 157  completed_matches_zeta := bridge.completed_matches_zeta
 158  completed_reflection := bridge.completed_reflection_from_mellin
 159
 160/-- The functional equation falls out of any cert. -/
 161theorem completed_zeta_functional_equation_from_cert
 162    (cert : CompletedZetaFunctionalEquationCert) (s : ℂ) :
 163    completedRiemannZeta s = completedRiemannZeta (1 - s) := by
 164  have h₁ := cert.completed_matches_zeta s
 165  have h₂ := cert.completed_reflection s
 166  have h₃ := cert.completed_matches_zeta (1 - s)
 167  rw [← h₁, h₂, h₃]
 168
 169end
 170
 171end ZetaFromTheta
 172end NumberTheory
 173end IndisputableMonolith
 174