IndisputableMonolith.Applied.PhotobiomodulationDevice

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# Photobiomodulation Device: RS-Coherent Light Therapy
   6
   7This module formalizes the Recognition Science foundations for a photobiomodulation
   8(PBM) device operating in coherence with the φ-ladder.
   9
  10## Key Results
  11
  121. **φ-Energy Ladder**: E(n) = E_base · φⁿ defines discrete energy rungs
  132. **Therapeutic Wavelength**: Rung 6 gives E = φ eV → λ ≈ 766 nm (red/near-IR)
  143. **8-Beat Pattern Neutrality**: The modulation pattern satisfies 8-window neutrality
  154. **Brainwave Entrainment**: φⁿ Hz frequencies fall in known EEG bands
  16
  17## Physical Motivation
  18
  19The φ-energy ladder E(n) = E_coh · φⁿ predicts specific photon energies.
  20Since E_coh = φ⁻⁵ eV (BIOPHASE energy), rung 6 yields:
  21
  22    E(6) = φ⁻⁵ · φ⁶ = φ¹ eV ≈ 1.618 eV → λ ≈ 766 nm
  23
  24This wavelength sits in the red/near-IR photobiomodulation window (600–850 nm),
  25where clinical evidence for therapeutic efficacy is strongest.
  26
  27The 8-beat modulation pattern s(k) = cos(kπ/4) + (1/φ)·cos(kπ/2) satisfies
  28the fundamental 8-window neutrality constraint Σ s(k) = 0, preventing
  29accumulation of recognition strain during treatment.
  30
  31## References
  32
  33- RS_Biophase_Light_Device_Spec.tex (device specification)
  34- BiophaseCore.Constants (fundamental biophase parameters)
  35- Biology.BioClocking (φ-ladder timescales)
  36- Healing.HealingRate (8-tick rate bounds)
  37-/
  38
  39namespace IndisputableMonolith
  40namespace Applied
  41namespace Photobiomodulation
  42
  43open Constants
  44
  45/-! ## CODATA Physical Constants (SI, for wavelength calculation) -/
  46
  47/-- Planck constant (J·s), CODATA 2024 -/
  48def planck_h : ℝ := 6.62607015e-34
  49
  50/-- Speed of light (m/s), exact -/
  51def speed_of_light : ℝ := 299792458
  52
  53/-- Electron volt to joules conversion, exact -/
  54def eV_to_J : ℝ := 1.602176634e-19
  55
  56lemma planck_h_pos : 0 < planck_h := by norm_num [planck_h]
  57lemma speed_of_light_pos : 0 < speed_of_light := by norm_num [speed_of_light]
  58lemma eV_to_J_pos : 0 < eV_to_J := by norm_num [eV_to_J]
  59
  60/-! ## Section 1: The φ-Energy Ladder
  61
  62The φ-energy ladder maps integer rungs to photon energies.
  63E_base = φ⁻⁵ eV is the recognition coherence quantum.
  64Each rung scales by one power of φ. -/
  65
  66/-- Base energy of the φ-ladder: E_base = φ⁻⁵ eV (recognition coherence quantum).
  67    In joules: φ⁻⁵ × 1.602e-19. -/
  68noncomputable def E_base : ℝ := phi ^ (-(5 : ℝ)) * eV_to_J
  69
  70lemma E_base_pos : 0 < E_base :=
  71  mul_pos (Real.rpow_pos_of_pos phi_pos _) eV_to_J_pos
  72
  73/-- The φ-energy ladder: E(n) = E_base · φⁿ.
  74    Maps real-valued rung n to photon energy in joules. -/
  75noncomputable def phi_energy_rung (n : ℝ) : ℝ := E_base * phi ^ n
  76
  77lemma phi_energy_rung_pos (n : ℝ) : 0 < phi_energy_rung n :=
  78  mul_pos E_base_pos (Real.rpow_pos_of_pos phi_pos n)
  79
  80/-- Adjacent rungs scale by φ. -/
  81theorem phi_energy_rung_step (n : ℝ) :
  82    phi_energy_rung (n + 1) = phi_energy_rung n * phi := by
  83  unfold phi_energy_rung
  84  rw [Real.rpow_add phi_pos, Real.rpow_one]
  85  ring
  86
  87/-- Rung 0 recovers E_base. -/
  88theorem phi_energy_rung_zero : phi_energy_rung 0 = E_base := by
  89  simp [phi_energy_rung]
  90
  91/-! ## Section 2: Therapeutic Wavelength (Rung 6 → 766 nm)
  92
  93Rung 6 on the φ-energy ladder gives E = φ eV, because
  94E(6) = E_base · φ⁶ = φ⁻⁵ · φ⁶ eV = φ eV.
  95The corresponding wavelength λ = hc/(φ eV) ≈ 766 nm falls
  96squarely in the red/near-IR photobiomodulation window. -/
  97
  98/-- Photobiomodulation energy: E_PBM = φ eV. -/
  99noncomputable def E_PBM : ℝ := phi * eV_to_J
 100
 101lemma E_PBM_pos : 0 < E_PBM :=
 102  mul_pos phi_pos eV_to_J_pos
 103
 104/-- E_PBM bounded by φ bounds: E_PBM ∈ (1.61, 1.62) eV. -/
 105lemma E_PBM_bounds :
 106    1.61 * eV_to_J < E_PBM ∧ E_PBM < 1.62 * eV_to_J :=
 107  ⟨mul_lt_mul_of_pos_right phi_gt_onePointSixOne eV_to_J_pos,
 108   mul_lt_mul_of_pos_right phi_lt_onePointSixTwo eV_to_J_pos⟩
 109
 110/-- E_PBM equals rung 6 on the φ-energy ladder:
 111    E_base · φ⁶ = φ⁻⁵ eV · φ⁶ = φ eV = E_PBM. -/
 112theorem E_PBM_is_rung_6 : E_PBM = phi_energy_rung 6 := by
 113  unfold E_PBM phi_energy_rung E_base
 114  have key : phi ^ (-(5 : ℝ)) * phi ^ (6 : ℝ) = phi := by
 115    rw [← Real.rpow_add phi_pos,
 116        (by norm_num : (-(5 : ℝ) + (6 : ℝ) : ℝ) = (1 : ℝ)),
 117        Real.rpow_one]
 118  have h : phi ^ (-(5 : ℝ)) * eV_to_J * phi ^ (6 : ℝ) = phi * eV_to_J := by
 119    calc phi ^ (-(5 : ℝ)) * eV_to_J * phi ^ (6 : ℝ)
 120        = phi ^ (-(5 : ℝ)) * phi ^ (6 : ℝ) * eV_to_J := by ring
 121      _ = phi * eV_to_J := by rw [key]
 122  exact h.symm
 123
 124/-- Division bounds helper: transfers E_PBM bounds to any positive numerator. -/
 125private lemma div_bounds_of_E_PBM {K : ℝ} (hK : 0 < K) :
 126    K / (1.62 * eV_to_J) < K / E_PBM ∧
 127      K / E_PBM < K / (1.61 * eV_to_J) := by
 128  have ⟨h_lower, h_upper⟩ := E_PBM_bounds
 129  have h_upper_pos : 0 < 1.62 * eV_to_J :=
 130    mul_pos (by norm_num) eV_to_J_pos
 131  have h_lower_pos : 0 < 1.61 * eV_to_J :=
 132    mul_pos (by norm_num) eV_to_J_pos
 133  have h1 := one_div_lt_one_div_of_lt E_PBM_pos h_upper
 134  have h2 := one_div_lt_one_div_of_lt h_lower_pos h_lower
 135  constructor
 136  · have := mul_lt_mul_of_pos_left h1 hK
 137    simpa [div_eq_mul_inv, mul_comm, mul_left_comm, mul_assoc]
 138  · have := mul_lt_mul_of_pos_left h2 hK
 139    simpa [div_eq_mul_inv, mul_comm, mul_left_comm, mul_assoc]
 140
 141/-- Therapeutic wavelength: λ_PBM = hc / E_PBM (meters). -/
 142noncomputable def lambda_PBM : ℝ := planck_h * speed_of_light / E_PBM
 143
 144lemma lambda_PBM_pos : 0 < lambda_PBM := by
 145  unfold lambda_PBM
 146  exact div_pos (mul_pos planck_h_pos speed_of_light_pos) E_PBM_pos
 147
 148/-- λ_PBM falls in the therapeutic photobiomodulation window (750–780 nm). -/
 149theorem lambda_PBM_in_therapeutic_window :
 150    750e-9 < lambda_PBM ∧ lambda_PBM < 780e-9 := by
 151  unfold lambda_PBM
 152  have h_hc_pos : 0 < planck_h * speed_of_light :=
 153    mul_pos planck_h_pos speed_of_light_pos
 154  have ⟨h_lower, h_upper⟩ := div_bounds_of_E_PBM h_hc_pos
 155  constructor
 156  · calc (750e-9 : ℝ)
 157        < planck_h * speed_of_light / (1.62 * eV_to_J) := by
 158          norm_num [planck_h, speed_of_light, eV_to_J]
 159      _ < planck_h * speed_of_light / E_PBM := h_lower
 160  · calc planck_h * speed_of_light / E_PBM
 161        < planck_h * speed_of_light / (1.61 * eV_to_J) := h_upper
 162      _ < (780e-9 : ℝ) := by norm_num [planck_h, speed_of_light, eV_to_J]
 163
 164/-- Tighter approximation: λ_PBM ≈ 766 nm (within ±5 nm). -/
 165theorem lambda_PBM_approx : abs (lambda_PBM - 766e-9) < 5e-9 := by
 166  unfold lambda_PBM
 167  have h_hc_pos : 0 < planck_h * speed_of_light :=
 168    mul_pos planck_h_pos speed_of_light_pos
 169  have ⟨h_lower, h_upper⟩ := div_bounds_of_E_PBM h_hc_pos
 170  have h_lo_ref :
 171      (761e-9 : ℝ) < planck_h * speed_of_light / (1.62 * eV_to_J) := by
 172    norm_num [planck_h, speed_of_light, eV_to_J]
 173  have h_hi_ref :
 174      planck_h * speed_of_light / (1.61 * eV_to_J) < (771e-9 : ℝ) := by
 175    norm_num [planck_h, speed_of_light, eV_to_J]
 176  have h_gt := lt_trans h_lo_ref h_lower
 177  have h_lt := lt_trans h_upper h_hi_ref
 178  exact abs_lt.mpr ⟨by linarith, by linarith⟩
 179
 180/-! ## Section 3: 8-Beat Modulation Pattern
 181
 182The RS-coherent modulation pattern is derived from a superposition of
 183DFT modes: s(k) = cos(kπ/4) + (1/φ)·cos(kπ/2).
 184
 185Using the identities 1/φ = φ - 1 and standard cosine values:
 186- s(0) = 1 + 1/φ = φ
 187- s(1) = √2/2
 188- s(2) = 0 - 1/φ = 1 - φ
 189- s(3) = -√2/2
 190- s(4) = -1 + 1/φ = φ - 2
 191- s(5) = -√2/2
 192- s(6) = 0 - 1/φ = 1 - φ
 193- s(7) = √2/2
 194
 195The φ terms and √2/2 terms each cancel pairwise,
 196giving Σ s(k) = 0 — exact 8-window neutrality. -/
 197
 198/-- The RS-coherent 8-beat modulation pattern values.
 199    Derived from s(k) = cos(kπ/4) + (1/φ)·cos(kπ/2). -/
 200noncomputable def rs_pattern : Fin 8 → ℝ
 201  | ⟨0, _⟩ => phi
 202  | ⟨1, _⟩ => Real.sqrt 2 / 2
 203  | ⟨2, _⟩ => 1 - phi
 204  | ⟨3, _⟩ => -(Real.sqrt 2 / 2)
 205  | ⟨4, _⟩ => phi - 2
 206  | ⟨5, _⟩ => -(Real.sqrt 2 / 2)
 207  | ⟨6, _⟩ => 1 - phi
 208  | ⟨7, _⟩ => Real.sqrt 2 / 2
 209
 210/-- **8-WINDOW NEUTRALITY**: The RS modulation pattern sums to zero over
 211    one complete 8-beat cycle. This is the fundamental recognition ledger
 212    constraint — the pattern produces no net strain accumulation.
 213
 214    Algebraically: the φ terms cancel (φ + (1-φ) + (φ-2) + (1-φ) = 0)
 215    and the √2/2 terms cancel (alternating signs). -/
 216theorem rs_pattern_window_neutral :
 217    Finset.univ.sum rs_pattern = 0 := by
 218  rw [Finset.sum_fin_eq_sum_range]
 219  simp only [Finset.sum_range_succ, Finset.sum_range_zero, rs_pattern]
 220  rw [dif_pos (show (0 : ℕ) < 8 by omega),
 221      dif_pos (show (1 : ℕ) < 8 by omega),
 222      dif_pos (show (2 : ℕ) < 8 by omega),
 223      dif_pos (show (3 : ℕ) < 8 by omega),
 224      dif_pos (show (4 : ℕ) < 8 by omega),
 225      dif_pos (show (5 : ℕ) < 8 by omega),
 226      dif_pos (show (6 : ℕ) < 8 by omega),
 227      dif_pos (show (7 : ℕ) < 8 by omega)]
 228  ring
 229
 230/-- A window-neutral modulation pattern for an 8-tick cycle. -/
 231structure WindowNeutralPattern where
 232  values : Fin 8 → ℝ
 233  neutral : Finset.univ.sum values = 0
 234
 235/-- The RS pattern constitutes a valid window-neutral pattern. -/
 236noncomputable def rs_neutral_pattern : WindowNeutralPattern :=
 237  ⟨rs_pattern, rs_pattern_window_neutral⟩
 238
 239/-- The peak value of the pattern occurs at k=0 and equals φ. -/
 240theorem rs_pattern_peak : rs_pattern ⟨0, by omega⟩ = phi := rfl
 241
 242/-- The φ-indexed entries (k = 0, 2, 4, 6) independently sum to zero. -/
 243theorem rs_pattern_phi_components_neutral :
 244    rs_pattern ⟨0, by omega⟩ + rs_pattern ⟨2, by omega⟩ +
 245    rs_pattern ⟨4, by omega⟩ + rs_pattern ⟨6, by omega⟩ = 0 := by
 246  simp only [rs_pattern]
 247  ring
 248
 249/-- The √2/2-indexed entries (k = 1, 3, 5, 7) independently sum to zero. -/
 250theorem rs_pattern_sqrt_components_neutral :
 251    rs_pattern ⟨1, by omega⟩ + rs_pattern ⟨3, by omega⟩ +
 252    rs_pattern ⟨5, by omega⟩ + rs_pattern ⟨7, by omega⟩ = 0 := by
 253  simp only [rs_pattern]
 254  ring
 255
 256/-! ## Section 4: Brainwave Entrainment Frequencies
 257
 258The device modulation frequencies are powers of φ in Hz, which
 259fall naturally into known EEG bands:
 260- φ³ ≈ 4.24 Hz → theta (4–8 Hz)
 261- φ⁵ ≈ 11.09 Hz → alpha (8–13 Hz)
 262- φ⁸ ≈ 46.98 Hz → gamma (30–100 Hz)
 263
 264This is not tuned — it falls out of the φ-ladder structure. -/
 265
 266/-- φ³ ≈ 4.24 Hz falls in the theta EEG band (4–8 Hz).
 267    Theta waves: deep meditation, memory consolidation, healing. -/
 268theorem phi_cubed_in_theta_band : 4 < phi ^ 3 ∧ phi ^ 3 < 8 := by
 269  constructor
 270  · linarith [phi_cubed_bounds.1]
 271  · linarith [phi_cubed_bounds.2]
 272
 273/-- φ⁵ ≈ 11.09 Hz falls in the alpha EEG band (8–13 Hz).
 274    Alpha waves: relaxed wakefulness, optimal healing coherence.
 275    This is the recommended default device modulation. -/
 276theorem phi_fifth_in_alpha_band : 8 < phi ^ 5 ∧ phi ^ 5 < 13 := by
 277  constructor
 278  · linarith [phi_fifth_bounds.1]
 279  · linarith [phi_fifth_bounds.2]
 280
 281/-- φ⁸ ≈ 46.98 Hz falls in the gamma EEG band (30–100 Hz).
 282    Gamma waves: heightened awareness, cross-modal binding. -/
 283theorem phi_eighth_in_gamma_band : 30 < phi ^ 8 ∧ phi ^ 8 < 100 := by
 284  rw [phi_eighth_eq]
 285  constructor <;> nlinarith [phi_gt_onePointSixOne, phi_lt_onePointSixTwo]
 286
 287/-- The three device modes span distinct, non-overlapping frequency ranges. -/
 288theorem modes_span_distinct_bands :
 289    phi ^ 3 < phi ^ 5 ∧ phi ^ 5 < phi ^ 8 := by
 290  constructor
 291  · linarith [phi_cubed_bounds.2, phi_fifth_bounds.1]
 292  · linarith [phi_fifth_bounds.2, phi_eighth_in_gamma_band.1]
 293
 294/-! ## Section 5: Device Specification
 295
 296The complete RS-coherent PBM device specification.
 297All parameters derive from φ with zero free parameters. -/
 298
 299/-- RS-coherent photobiomodulation device specification.
 300    Every field is derived from the golden ratio φ. -/
 301structure PBMDeviceSpec where
 302  /-- Operating wavelength (meters) -/
 303  wavelength_m : ℝ
 304  /-- Window-neutral modulation pattern -/
 305  modulation_pattern : WindowNeutralPattern
 306  /-- Theta mode frequency (Hz) -/
 307  theta_freq_Hz : ℝ
 308  /-- Alpha mode frequency (Hz) -/
 309  alpha_freq_Hz : ℝ
 310  /-- Gamma mode frequency (Hz) -/
 311  gamma_freq_Hz : ℝ
 312  /-- Ticks per modulation cycle -/
 313  ticks_per_cycle : ℕ
 314
 315  wavelength_pos : 0 < wavelength_m
 316  wavelength_therapeutic : 750e-9 < wavelength_m ∧ wavelength_m < 780e-9
 317  ticks_is_eight : ticks_per_cycle = 8
 318  theta_in_band : 4 < theta_freq_Hz ∧ theta_freq_Hz < 8
 319  alpha_in_band : 8 < alpha_freq_Hz ∧ alpha_freq_Hz < 13
 320  gamma_in_band : 30 < gamma_freq_Hz ∧ gamma_freq_Hz < 100
 321  modes_ordered : theta_freq_Hz < alpha_freq_Hz ∧ alpha_freq_Hz < gamma_freq_Hz
 322
 323/-- The canonical RS-coherent PBM device.
 324    All parameters are derived from φ with zero free parameters. -/
 325noncomputable def rs_device : PBMDeviceSpec := {
 326  wavelength_m := lambda_PBM
 327  modulation_pattern := rs_neutral_pattern
 328  theta_freq_Hz := phi ^ 3
 329  alpha_freq_Hz := phi ^ 5
 330  gamma_freq_Hz := phi ^ 8
 331  ticks_per_cycle := 8
 332
 333  wavelength_pos := lambda_PBM_pos
 334  wavelength_therapeutic := lambda_PBM_in_therapeutic_window
 335  ticks_is_eight := rfl
 336  theta_in_band := phi_cubed_in_theta_band
 337  alpha_in_band := phi_fifth_in_alpha_band
 338  gamma_in_band := phi_eighth_in_gamma_band
 339  modes_ordered := modes_span_distinct_bands
 340}
 341
 342/-- The device uses exactly 8 ticks per modulation cycle,
 343    matching the fundamental recognition octave (T7: D=3 → 8 ticks). -/
 344theorem device_matches_octave : rs_device.ticks_per_cycle = 8 := rfl
 345
 346end Photobiomodulation
 347end Applied
 348end IndisputableMonolith
 349