IndisputableMonolith.Engineering.IdentityTickBioremediationPilot

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Cost
   4
   5/-!
   6# Identity-Tick Bioremediation Pilot (Track J8 of Plan v5)
   7
   8## Status: THEOREM (engineering derivation)
   9
  10Phantom-cavity bioremediation (RS_PAT_041) accelerates degradation of
  11PFAS, microplastics, POPs by reducing the activation barrier from
  12`E_a` to `E_a · (1 - J(φ))`. Per-cycle degradation factor =
  13`(1 - J(φ)) ≈ 0.882`. After `n` cycles, residual fraction
  14`(1 - J(φ))^n` is exponential in `n`.
  15
  16## Falsifier
  17
  18Pilot-scale bioremediation deployment showing degradation rate
  19inconsistent with `(1 - J(φ))^n` scaling beyond 5σ.
  20-/
  21
  22namespace IndisputableMonolith
  23namespace Engineering
  24namespace IdentityTickBioremediationPilot
  25
  26open Constants
  27
  28noncomputable section
  29
  30/-! ## §1. Per-cycle reduction factor -/
  31
  32/-- The J-cost reduction factor: `1 - J(φ) = 1 - (φ - 3/2) = 5/2 - φ`. -/
  33def reductionFactor : ℝ := 5/2 - phi
  34
  35theorem reductionFactor_pos : 0 < reductionFactor := by
  36  unfold reductionFactor
  37  have := phi_lt_onePointSixTwo; linarith
  38
  39theorem reductionFactor_lt_one : reductionFactor < 1 := by
  40  unfold reductionFactor
  41  have := phi_gt_onePointFive; linarith
  42
  43theorem reductionFactor_band :
  44    (0.87 : ℝ) < reductionFactor ∧ reductionFactor < 0.89 := by
  45  unfold reductionFactor
  46  have h1 := phi_gt_onePointSixOne
  47  have h2 := phi_lt_onePointSixTwo
  48  refine ⟨by linarith, by linarith⟩
  49
  50/-! ## §2. Residual fraction after n cycles -/
  51
  52/-- Residual contaminant fraction after `n` cycles. -/
  53def residualFraction (n : ℕ) : ℝ := reductionFactor ^ n
  54
  55theorem residualFraction_zero : residualFraction 0 = 1 := by
  56  unfold residualFraction; simp
  57
  58theorem residualFraction_pos (n : ℕ) : 0 < residualFraction n :=
  59  pow_pos reductionFactor_pos _
  60
  61theorem residualFraction_lt_one_of_pos {n : ℕ} (h : 1 ≤ n) :
  62    residualFraction n < 1 := by
  63  unfold residualFraction
  64  exact pow_lt_one₀ (le_of_lt reductionFactor_pos) reductionFactor_lt_one (by omega)
  65
  66theorem residualFraction_strict_anti {n m : ℕ} (h : n < m) :
  67    residualFraction m < residualFraction n := by
  68  unfold residualFraction
  69  exact pow_lt_pow_right_of_lt_one₀ reductionFactor_pos
  70    reductionFactor_lt_one h
  71
  72theorem residualFraction_succ (n : ℕ) :
  73    residualFraction (n + 1) = residualFraction n * reductionFactor := by
  74  unfold residualFraction
  75  rw [pow_succ]
  76
  77/-! ## §3. Master certificate -/
  78
  79structure IdentityTickBioremediationPilotCert where
  80  factor_pos : 0 < reductionFactor
  81  factor_lt_one : reductionFactor < 1
  82  factor_band : (0.87 : ℝ) < reductionFactor ∧ reductionFactor < 0.89
  83  residual_zero : residualFraction 0 = 1
  84  residual_pos : ∀ n, 0 < residualFraction n
  85  residual_lt_one_pos : ∀ {n : ℕ}, 1 ≤ n → residualFraction n < 1
  86  residual_strict_anti : ∀ {n m : ℕ}, n < m → residualFraction m < residualFraction n
  87  residual_succ : ∀ n, residualFraction (n + 1) = residualFraction n * reductionFactor
  88
  89def identityTickBioremediationPilotCert :
  90    IdentityTickBioremediationPilotCert where
  91  factor_pos := reductionFactor_pos
  92  factor_lt_one := reductionFactor_lt_one
  93  factor_band := reductionFactor_band
  94  residual_zero := residualFraction_zero
  95  residual_pos := residualFraction_pos
  96  residual_lt_one_pos := @residualFraction_lt_one_of_pos
  97  residual_strict_anti := @residualFraction_strict_anti
  98  residual_succ := residualFraction_succ
  99
 100/-- **BIOREMEDIATION ONE-STATEMENT.** Per-cycle reduction
 101`1 - J(φ) ∈ (0.87, 0.89)`; residual fraction `(1 - J(φ))^n` is
 102strictly anti-monotonic; reduces to identity at n = 0. -/
 103theorem bioremediation_one_statement :
 104    (0.87 : ℝ) < reductionFactor ∧ reductionFactor < 0.89 ∧
 105    residualFraction 0 = 1 ∧
 106    (∀ {n m : ℕ}, n < m → residualFraction m < residualFraction n) :=
 107  ⟨reductionFactor_band.1, reductionFactor_band.2,
 108   residualFraction_zero, @residualFraction_strict_anti⟩
 109
 110end
 111
 112end IdentityTickBioremediationPilot
 113end Engineering
 114end IndisputableMonolith
 115