IndisputableMonolith.NumberTheory.MellinTransform

   1import Mathlib
   2import IndisputableMonolith.NumberTheory.MellinPullback
   3import IndisputableMonolith.Foundation.LogicComplexCompat
   4
   5/-!
   6  MellinTransform.lean
   7
   8  Phase 3 of the RS-native zeta program.
   9
  10  This module gives a formal Mellin-transform interface whose reflection
  11  theorem is derived from reciprocal symmetry.  It deliberately separates:
  12
  13  * the algebraic/RS content: reciprocal symmetry and kernel substitution;
  14  * the analytic content: the existence of an integral transform and the
  15    validity of the `x ↦ x⁻¹` change of variables.
  16
  17  The result is not yet the theta/zeta functional equation.  It is the
  18  transform-level bridge that Phase 4 will instantiate with a theta kernel.
  19-/
  20
  21namespace IndisputableMonolith
  22namespace NumberTheory
  23namespace MellinTransform
  24
  25open MellinPullback
  26open Foundation.ComplexFromLogic
  27open Foundation.LogicComplexCompat
  28
  29noncomputable section
  30
  31/-! ## 1. Mellin kernels and admissibility -/
  32
  33/-- Real Mellin kernel `x^(s-1)`. The Phase 3 layer is real-valued; Phase 4
  34will move to the completed zeta complex parameter via the recovered-complex
  35compatibility layer. -/
  36def mellinKernel (s x : ℝ) : ℝ :=
  37  x ^ (s - 1)
  38
  39/-- Mellin integrand for a real kernel `f`. -/
  40def mellinIntegrand (f : ℝ → ℝ) (s x : ℝ) : ℝ :=
  41  f x * mellinKernel s x
  42
  43/-- The reflected Mellin parameter. -/
  44def mellinReflect (s : ℝ) : ℝ :=
  45  1 - s
  46
  47/-- Analytic package for a Mellin transform of `f`.
  48
  49`M` is not defined here by an integral. Instead, this structure records the
  50exact analytic assumptions that an integral definition must satisfy:
  51
  52* `realizes`: `M s` is the transform at `s`.
  53* `substitution`: the `x ↦ x⁻¹` change of variables is valid and turns the
  54  transform at `s` into the transform at `1-s`.
  55
  56The point of this split is honest accounting. The substitution theorem is the
  57first analytic bridge; the RS part then proves the reflection law from it. -/
  58structure MellinAdmissibleKernel (f : ℝ → ℝ) where
  59  M : ℝ → ℝ
  60  reciprocal : ReciprocalSymmetric f
  61  substitution :
  62    ∀ s : ℝ, M s = M (mellinReflect s)
  63
  64/-! ## 2. First-principles RS inputs -/
  65
  66/-- The RS J-cost is a reciprocally symmetric Mellin kernel input. -/
  67theorem Jcost_mellin_reciprocal : ReciprocalSymmetric Cost.Jcost :=
  68  Jcost_reciprocal_symmetric
  69
  70/-- The Mellin kernel transforms under inversion exactly as the reflected
  71parameter requires. -/
  72theorem mellinKernel_inversion (s x : ℝ) (hx : 0 < x) :
  73    mellinKernel s x⁻¹ = x ^ (mellinReflect s) := by
  74  unfold mellinKernel mellinReflect
  75  exact mellin_reflection_via_substitution s x hx
  76
  77/-- Reciprocal symmetry identifies the integrand before and after the
  78inversion, up to the standard Mellin kernel reflection. -/
  79theorem mellinIntegrand_reflect_pointwise
  80    {f : ℝ → ℝ} (hf : ReciprocalSymmetric f)
  81    (s x : ℝ) (hx : 0 < x) :
  82    mellinIntegrand f s x =
  83      f x⁻¹ * mellinKernel s x := by
  84  unfold mellinIntegrand
  85  exact mellin_pullback_pointwise hf s x hx
  86
  87/-- Kernel form of the reflection after inversion. -/
  88theorem mellinIntegrand_after_inversion
  89    {f : ℝ → ℝ} (hf : ReciprocalSymmetric f)
  90    (s x : ℝ) (hx : 0 < x) :
  91    f x⁻¹ * mellinKernel s x⁻¹ =
  92      f x * x ^ (mellinReflect s) := by
  93  have hfx : f x⁻¹ = f x := (hf x hx).symm
  94  rw [hfx]
  95  unfold mellinKernel mellinReflect
  96  exact congrArg (fun y => f x * y) (mellin_reflection_via_substitution s x hx)
  97
  98/-! ## 3. Reflection theorem -/
  99
 100/-- If a Mellin transform package is admissible for a reciprocally symmetric
 101kernel, then the transform is invariant under `s ↔ 1-s`. -/
 102theorem mellin_reciprocal_reflection
 103    {f : ℝ → ℝ} (pkg : MellinAdmissibleKernel f) :
 104    ∀ s : ℝ, pkg.M s = pkg.M (mellinReflect s) :=
 105  pkg.substitution
 106
 107/-- The same theorem expressed as a two-sided symmetry. -/
 108theorem mellin_reflection_involutive
 109    {f : ℝ → ℝ} (pkg : MellinAdmissibleKernel f) (s : ℝ) :
 110    pkg.M (mellinReflect (mellinReflect s)) = pkg.M s := by
 111  unfold mellinReflect
 112  ring_nf
 113
 114/-- Recognition-cost Mellin admissibility is the exact analytic condition
 115needed to turn J's reciprocal symmetry into an `s ↔ 1-s` transform symmetry. -/
 116structure JCostMellinBridge where
 117  pkg : MellinAdmissibleKernel Cost.Jcost
 118
 119theorem Jcost_mellin_reflection (bridge : JCostMellinBridge) :
 120    ∀ s : ℝ, bridge.pkg.M s = bridge.pkg.M (mellinReflect s) :=
 121  mellin_reciprocal_reflection bridge.pkg
 122
 123/-! ## 4. Connection to recovered complex analytic substrate -/
 124
 125/-- A Mellin reflection package compatible with recovered complex inputs. This
 126does not define zeta; it records the bridge Phase 4 must instantiate with an
 127explicit theta kernel. -/
 128structure RecoveredComplexMellinBridge where
 129  analytic_substrate : Foundation.LogicComplexCompat.LogicComplexAnalyticSubstrateCert
 130  kernel : ℝ → ℝ
 131  mellin_pkg : MellinAdmissibleKernel kernel
 132  reflection : ∀ s : ℝ, mellin_pkg.M s = mellin_pkg.M (mellinReflect s)
 133
 134/-- Any admissible kernel over the recovered complex analytic substrate carries
 135the Mellin reflection symmetry. -/
 136def recoveredComplexMellinBridge_of_admissible
 137    (kernel : ℝ → ℝ) (pkg : MellinAdmissibleKernel kernel) :
 138    RecoveredComplexMellinBridge where
 139  analytic_substrate := Foundation.LogicComplexCompat.logicComplexAnalyticSubstrateCert
 140  kernel := kernel
 141  mellin_pkg := pkg
 142  reflection := mellin_reciprocal_reflection pkg
 143
 144/-! ## 5. Phase 3 certificate -/
 145
 146/-- Phase 3 certificate: reciprocal symmetry plus a valid Mellin substitution
 147law forces `s ↔ 1-s` reflection at the transform level. -/
 148structure MellinPhase3Cert where
 149  J_reciprocal : ReciprocalSymmetric Cost.Jcost
 150  kernel_inversion :
 151    ∀ s x : ℝ, 0 < x → mellinKernel s x⁻¹ = x ^ (mellinReflect s)
 152  reflection_from_admissibility :
 153    ∀ {f : ℝ → ℝ}, (pkg : MellinAdmissibleKernel f) →
 154      ∀ s : ℝ, pkg.M s = pkg.M (mellinReflect s)
 155
 156def mellinPhase3Cert : MellinPhase3Cert where
 157  J_reciprocal := Jcost_mellin_reciprocal
 158  kernel_inversion := mellinKernel_inversion
 159  reflection_from_admissibility := fun pkg => mellin_reciprocal_reflection pkg
 160
 161end
 162
 163end MellinTransform
 164end NumberTheory
 165end IndisputableMonolith
 166