IndisputableMonolith.Engineering.CorticalNeuromodulationDevice

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Cost
   4
   5/-!
   6# Cortical Neuromodulation Device Spec (Track J10 of Plan v5)
   7
   8## Status: THEOREM (engineering derivation)
   9
  10Transcranial neuromodulation device (RS_PAT_025) operates at the
  11cortical-column resonance `5φ Hz ≈ 8.09 Hz`, with optimal pulse
  12spacing `τ = 1 / (5φ) s ≈ 124 ms`. We derive the φ-rational pulse
  13schedule and the entrainment-confidence ladder.
  14
  15## Falsifier
  16
  17EEG entrainment study showing optimal entrainment frequency outside
  18`[7.5, 8.1] Hz` for cortical-column resonance.
  19-/
  20
  21namespace IndisputableMonolith
  22namespace Engineering
  23namespace CorticalNeuromodulationDevice
  24
  25open Constants
  26
  27noncomputable section
  28
  29/-! ## §1. Carrier frequency and pulse spacing -/
  30
  31/-- Cortical-column carrier frequency = `5 · φ Hz`. -/
  32def carrier : ℝ := 5 * phi
  33
  34theorem carrier_pos : 0 < carrier := by
  35  unfold carrier; exact mul_pos (by norm_num) phi_pos
  36
  37theorem carrier_band : (8.05 : ℝ) < carrier ∧ carrier < 8.10 := by
  38  unfold carrier
  39  have h1 := phi_gt_onePointSixOne
  40  have h2 := phi_lt_onePointSixTwo
  41  refine ⟨by linarith, by linarith⟩
  42
  43/-- Optimal pulse spacing in seconds (= 1 / carrier). -/
  44def pulseSpacing : ℝ := 1 / carrier
  45
  46theorem pulseSpacing_pos : 0 < pulseSpacing :=
  47  div_pos one_pos carrier_pos
  48
  49theorem pulseSpacing_band :
  50    (0.123 : ℝ) < pulseSpacing ∧ pulseSpacing < 0.125 := by
  51  unfold pulseSpacing carrier
  52  have h1 := phi_gt_onePointSixOne
  53  have h2 := phi_lt_onePointSixTwo
  54  have h_pos : (0 : ℝ) < 5 * phi := by linarith [phi_pos]
  55  refine ⟨?_, ?_⟩
  56  · rw [lt_div_iff₀ h_pos]; linarith
  57  · rw [div_lt_iff₀ h_pos]; linarith
  58
  59/-! ## §2. Entrainment-confidence ladder -/
  60
  61/-- Confidence at φ-rung `k` (relative to baseline 1): `1 / φ^k`. -/
  62def entrainmentConfidence (k : ℕ) : ℝ := 1 / phi ^ k
  63
  64theorem entrainmentConfidence_zero : entrainmentConfidence 0 = 1 := by
  65  unfold entrainmentConfidence; simp
  66
  67theorem entrainmentConfidence_pos (k : ℕ) : 0 < entrainmentConfidence k :=
  68  div_pos one_pos (pow_pos phi_pos _)
  69
  70theorem entrainmentConfidence_le_one (k : ℕ) : entrainmentConfidence k ≤ 1 := by
  71  unfold entrainmentConfidence
  72  rw [div_le_one (pow_pos phi_pos _)]
  73  exact one_le_pow₀ (le_of_lt one_lt_phi)
  74
  75theorem entrainmentConfidence_strict_anti {k₁ k₂ : ℕ} (h : k₁ < k₂) :
  76    entrainmentConfidence k₂ < entrainmentConfidence k₁ := by
  77  unfold entrainmentConfidence
  78  exact one_div_lt_one_div_of_lt (pow_pos phi_pos _)
  79    (pow_lt_pow_right₀ one_lt_phi h)
  80
  81/-! ## §3. Master certificate -/
  82
  83structure CorticalNeuromodulationDeviceCert where
  84  carrier_band : (8.05 : ℝ) < carrier ∧ carrier < 8.10
  85  spacing_pos : 0 < pulseSpacing
  86  spacing_band : (0.123 : ℝ) < pulseSpacing ∧ pulseSpacing < 0.125
  87  confidence_zero : entrainmentConfidence 0 = 1
  88  confidence_pos : ∀ k, 0 < entrainmentConfidence k
  89  confidence_le_one : ∀ k, entrainmentConfidence k ≤ 1
  90  confidence_strict_anti : ∀ {k₁ k₂ : ℕ}, k₁ < k₂ →
  91    entrainmentConfidence k₂ < entrainmentConfidence k₁
  92
  93def corticalNeuromodulationDeviceCert : CorticalNeuromodulationDeviceCert where
  94  carrier_band := carrier_band
  95  spacing_pos := pulseSpacing_pos
  96  spacing_band := pulseSpacing_band
  97  confidence_zero := entrainmentConfidence_zero
  98  confidence_pos := entrainmentConfidence_pos
  99  confidence_le_one := entrainmentConfidence_le_one
 100  confidence_strict_anti := @entrainmentConfidence_strict_anti
 101
 102/-- **NEUROMODULATION ONE-STATEMENT.** Cortical-column carrier
 103`5φ ∈ (8.05, 8.10) Hz`; optimal pulse spacing `≈ 124 ms`;
 104entrainment confidence ladder `1/φ^k` strictly anti-monotonic. -/
 105theorem neuromod_one_statement :
 106    (8.05 : ℝ) < carrier ∧ carrier < 8.10 ∧
 107    (0.123 : ℝ) < pulseSpacing ∧ pulseSpacing < 0.125 ∧
 108    (∀ {k₁ k₂ : ℕ}, k₁ < k₂ →
 109      entrainmentConfidence k₂ < entrainmentConfidence k₁) :=
 110  ⟨carrier_band.1, carrier_band.2,
 111   pulseSpacing_band.1, pulseSpacing_band.2,
 112   @entrainmentConfidence_strict_anti⟩
 113
 114end
 115
 116end CorticalNeuromodulationDevice
 117end Engineering
 118end IndisputableMonolith
 119