IndisputableMonolith.NumberTheory.ZeroCompositionInterface

   1import Mathlib
   2import IndisputableMonolith.Cost.FunctionalEquation
   3import IndisputableMonolith.Foundation.DiscretenessForcing
   4import IndisputableMonolith.NumberTheory.ZeroLocationCost
   5
   6/-!
   7# Zero Composition Interface
   8
   9**Classification: ALTERNATE** — not on the primary path to unconditional RH.
  10
  11This module isolates the exact theorem interface Vector C would need in order to
  12turn a zero-location observable into a critical-line forcing theorem.
  13
  14It is intentionally abstract: the existing RS machinery already proves that a
  15d'Alembert law with the right normalization and calibration forces `cosh`, so
  16the only real question is whether completed-ξ data can instantiate this
  17interface for actual zeta zeros.
  18
  19**Stage gate (April 2026):** `VectorCSymmetryOnlyNoGo` proves that pure FE
  20symmetry + RCL doubling data cannot produce a `ZeroCompositionWitness`.
  21Any successful Vector C path must import genuinely extra Euler/Hadamard-side
  22analytic input — the same data used by the primary routes. This module does
  23not supply new lemmas that reduce `EulerBoundaryBridgeAssumption` or
  24`HonestPhaseCostBridge`.
  25-/
  26
  27namespace IndisputableMonolith
  28namespace NumberTheory
  29
  30open IndisputableMonolith.Cost.FunctionalEquation
  31
  32noncomputable section
  33
  34private theorem cosh_eq_one_iff (t : ℝ) : Real.cosh t = 1 ↔ t = 0 := by
  35  constructor
  36  · intro h
  37    by_contra hne
  38    have hgt : 1 < Real.cosh t := Real.one_lt_cosh.mpr hne
  39    linarith
  40  · intro h
  41    simp [h]
  42
  43/-- The abstract zero-location composition law needed by Vector C. -/
  44structure ZeroCompositionLaw where
  45  H : ℝ → ℝ
  46  H_zero : H 0 = 1
  47  continuous : Continuous H
  48  dAlembert : ∀ t u : ℝ, H (t + u) + H (t - u) = 2 * H t * H u
  49  curvature : deriv (deriv H) 0 = 1
  50  smooth_hyp : dAlembert_continuous_implies_smooth_hypothesis H
  51  ode_hyp : dAlembert_to_ODE_hypothesis H
  52  cont_hyp : ode_regularity_continuous_hypothesis H
  53  diff_hyp : ode_regularity_differentiable_hypothesis H
  54  bootstrap_hyp : ode_linear_regularity_bootstrap_hypothesis H
  55
  56/-- Any instantiated zero-composition law is forced to be `cosh`. -/
  57theorem zeroCompositionLaw_forces_cosh (zc : ZeroCompositionLaw) :
  58    ∀ t : ℝ, zc.H t = Real.cosh t :=
  59  dAlembert_cosh_solution zc.H zc.H_zero zc.continuous zc.dAlembert
  60    zc.curvature zc.smooth_hyp zc.ode_hyp zc.cont_hyp zc.diff_hyp
  61    zc.bootstrap_hyp
  62
  63/-- Consequently, the minimum value `1` occurs exactly at `t = 0`. -/
  64theorem zeroCompositionLaw_forces_unique_minimum
  65    (zc : ZeroCompositionLaw) (t : ℝ) :
  66    zc.H t = 1 ↔ t = 0 := by
  67  rw [zeroCompositionLaw_forces_cosh zc t]
  68  exact cosh_eq_one_iff t
  69
  70/-- A zero-composition law forces the corresponding point onto the critical
  71line once the observable attains its minimum at that point's deviation. -/
  72theorem zeroCompositionLaw_forces_eta_zero
  73    (zc : ZeroCompositionLaw) (ρ : ℂ) :
  74    zc.H (zeroDeviation ρ) = 1 ↔ OnCriticalLine ρ := by
  75  constructor
  76  · intro h
  77    have hz : zeroDeviation ρ = 0 :=
  78      (zeroCompositionLaw_forces_unique_minimum zc (zeroDeviation ρ)).mp h
  79    exact (zeroDeviation_eq_zero_iff_on_critical_line ρ).mp hz
  80  · intro h
  81    have hz : zeroDeviation ρ = 0 :=
  82      (zeroDeviation_eq_zero_iff_on_critical_line ρ).mpr h
  83    exact (zeroCompositionLaw_forces_unique_minimum zc (zeroDeviation ρ)).mpr hz
  84
  85/-- A concrete Vector C witness at a specific complex point. -/
  86structure ZeroCompositionWitness (ρ : ℂ) where
  87  law : ZeroCompositionLaw
  88  value_at_deviation : law.H (zeroDeviation ρ) = 1
  89
  90/-- Any such witness forces the corresponding point onto the critical line. -/
  91theorem zeroCompositionWitness_forces_on_critical_line
  92    {ρ : ℂ} (w : ZeroCompositionWitness ρ) :
  93    OnCriticalLine ρ :=
  94  (zeroCompositionLaw_forces_eta_zero w.law ρ).mp w.value_at_deviation
  95
  96/-- Therefore the zero-location defect must vanish there as well. -/
  97theorem zeroCompositionWitness_forces_zero_defect
  98    {ρ : ℂ} (w : ZeroCompositionWitness ρ) :
  99    zeroDefect ρ = 0 := by
 100  exact (zeroDefect_zero_iff_on_critical_line ρ).mpr
 101    (zeroCompositionWitness_forces_on_critical_line w)
 102
 103end
 104
 105end NumberTheory
 106end IndisputableMonolith
 107