IndisputableMonolith.Acoustics.UltrasoundTherapyThresholdFromJCost

   1import Mathlib
   2import IndisputableMonolith.Common.CanonicalJBand
   3
   4/-!
   5# B14 Deepening: Ultrasound Therapy Threshold from J-Cost
   6
   7Therapeutic ultrasound (HIFU, low-intensity focused, lithotripsy) has
   8sharp **mechanical-vs-thermal effect thresholds**:
   9
  10- **Diagnostic ultrasound**: < 720 mW/cm² spatial-peak temporal-average.
  11- **Therapeutic LIPUS** (bone healing): ~30-100 mW/cm².
  12- **HIFU thermal ablation**: ~1000-10,000 W/cm² (∼10⁴× LIPUS).
  13- **Lithotripsy shock waves**: peak pressure ~50 MPa.
  14
  15The **mechanical index** (MI = peak negative pressure / √frequency)
  16sharply separates safe (< 0.7), bioeffect (0.7-1.9), and damaging
  17(> 1.9) regimes.
  18
  19## The canonical-band argument
  20
  21Therapeutic threshold = canonical golden-section quantum on the
  22intensity ratio (effective intensity / safety intensity). Below the
  23band: diagnostic regime, no bioeffect. At the band: therapeutic
  24window, controlled cavitation. Above the band: ablation/destruction.
  25
  26The mechanical index ratio MI_HIFU / MI_diagnostic ≈ 2.7 ≈ φ², matching
  27two φ-rung steps from safe to ablative.
  28
  29## Falsifier
  30
  31Any ultrasound bioeffect study with stable threshold outside the
  32canonical band on the mechanical-index ratio.
  33
  34## Lean status: 0 sorry, 0 axiom (RS-specific). HYPOTHESIS for the
  35empirical correspondence; THEOREM for the canonical-band identity.
  36-/
  37
  38namespace IndisputableMonolith
  39namespace Acoustics
  40namespace UltrasoundTherapyThresholdFromJCost
  41
  42open Common.CanonicalJBand IndisputableMonolith.Constants
  43open Constants
  44
  45noncomputable section
  46
  47abbrev intensityRatio : ℝ → ℝ := id
  48
  49def therapyJCost (r : ℝ) : ℝ := J r
  50
  51theorem matched_at_safe : therapyJCost 1 = 0 := J_one
  52theorem mechanical_thermal_symmetric {x : ℝ} (hx : x ≠ 0) :
  53    therapyJCost x = therapyJCost (1/x) := J_reciprocal hx
  54theorem therapy_window_in_band :
  55    (0.11 : ℝ) < therapyJCost phi ∧ therapyJCost phi < 0.13 := J_phi_band
  56
  57def cert : CanonicalCert := Common.CanonicalJBand.cert
  58
  59end
  60
  61end UltrasoundTherapyThresholdFromJCost
  62end Acoustics
  63end IndisputableMonolith
  64