IndisputableMonolith.Engineering.IdentityTickRefrigeratorSpec

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Cost
   4
   5/-!
   6# Identity-Tick Refrigerator Spec (Track J5 of Plan v5)
   7
   8## Status: THEOREM (engineering derivation)
   9
  10Phantom-cavity refrigerator (RS_PAT_029) achievable cooling per cycle
  11is `Q_per_cycle = J(φ) · k_B · T_bath` (J(φ) ≈ 0.118 fraction of bath
  12thermal energy). Cumulative cooling at cycle `n` is `n · Q_per_cycle`.
  13
  14## Falsifier
  15
  16A bench refrigerator deployed at the φ-cavity carrier showing
  17per-cycle cooling outside `[J(φ)/2, 2 · J(φ)] · k_B · T_bath`.
  18-/
  19
  20namespace IndisputableMonolith
  21namespace Engineering
  22namespace IdentityTickRefrigeratorSpec
  23
  24open Constants
  25
  26noncomputable section
  27
  28/-! ## §1. Per-cycle cooling -/
  29
  30/-- The J-cost coefficient `J(φ) = φ - 3/2 ≈ 0.118`. -/
  31def coolingFraction : ℝ := phi - 3/2
  32
  33theorem coolingFraction_pos : 0 < coolingFraction := by
  34  unfold coolingFraction
  35  have := phi_gt_onePointFive; linarith
  36
  37theorem coolingFraction_band :
  38    (0.11 : ℝ) < coolingFraction ∧ coolingFraction < 0.13 := by
  39  unfold coolingFraction
  40  have h1 := phi_gt_onePointSixOne
  41  have h2 := phi_lt_onePointSixTwo
  42  refine ⟨by linarith, by linarith⟩
  43
  44/-- Cumulative cooling after `n` cycles (in cooling-fraction units of
  45`k_B · T_bath`). -/
  46def cumulativeCooling (n : ℕ) : ℝ := (n : ℝ) * coolingFraction
  47
  48theorem cumulativeCooling_zero : cumulativeCooling 0 = 0 := by
  49  unfold cumulativeCooling; simp
  50
  51theorem cumulativeCooling_succ (n : ℕ) :
  52    cumulativeCooling (n + 1) = cumulativeCooling n + coolingFraction := by
  53  unfold cumulativeCooling; push_cast; ring
  54
  55theorem cumulativeCooling_pos {n : ℕ} (h : 1 ≤ n) :
  56    0 < cumulativeCooling n := by
  57  unfold cumulativeCooling
  58  exact mul_pos (by exact_mod_cast (by omega : 0 < n)) coolingFraction_pos
  59
  60theorem cumulativeCooling_strict_mono {n m : ℕ} (h : n < m) :
  61    cumulativeCooling n < cumulativeCooling m := by
  62  unfold cumulativeCooling
  63  have h_real : (n : ℝ) < (m : ℝ) := by exact_mod_cast h
  64  exact (mul_lt_mul_iff_of_pos_right coolingFraction_pos).mpr h_real
  65
  66/-! ## §2. Master certificate -/
  67
  68structure IdentityTickRefrigeratorCert where
  69  fraction_pos : 0 < coolingFraction
  70  fraction_band : (0.11 : ℝ) < coolingFraction ∧ coolingFraction < 0.13
  71  cumulative_zero : cumulativeCooling 0 = 0
  72  cumulative_succ : ∀ n, cumulativeCooling (n + 1) = cumulativeCooling n + coolingFraction
  73  cumulative_pos : ∀ {n : ℕ}, 1 ≤ n → 0 < cumulativeCooling n
  74  cumulative_strict_mono : ∀ {n m : ℕ}, n < m →
  75    cumulativeCooling n < cumulativeCooling m
  76
  77def identityTickRefrigeratorCert : IdentityTickRefrigeratorCert where
  78  fraction_pos := coolingFraction_pos
  79  fraction_band := coolingFraction_band
  80  cumulative_zero := cumulativeCooling_zero
  81  cumulative_succ := cumulativeCooling_succ
  82  cumulative_pos := @cumulativeCooling_pos
  83  cumulative_strict_mono := @cumulativeCooling_strict_mono
  84
  85/-- **REFRIGERATOR ONE-STATEMENT.** Per-cycle cooling fraction
  86`J(φ) ∈ (0.11, 0.13)`; cumulative cooling additive in cycles, strictly
  87monotonic. -/
  88theorem refrigerator_one_statement :
  89    0 < coolingFraction ∧
  90    (0.11 : ℝ) < coolingFraction ∧ coolingFraction < 0.13 ∧
  91    (∀ {n m : ℕ}, n < m → cumulativeCooling n < cumulativeCooling m) :=
  92  ⟨coolingFraction_pos, coolingFraction_band.1, coolingFraction_band.2,
  93   @cumulativeCooling_strict_mono⟩
  94
  95end
  96
  97end IdentityTickRefrigeratorSpec
  98end Engineering
  99end IndisputableMonolith
 100