IndisputableMonolith.Physics.ElectromagneticSpectrumFromPhiLadder

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# Electromagnetic Spectrum from Phi-Ladder — A1 SM Depth
   6
   7The EM spectrum spans ~24 orders of magnitude in frequency.
   8RS: each spectral band spans approximately one phi-decade:
   9  ν_k = ν_0 × φ^k
  10
  11Five canonical bands (radio, microwave, infrared, visible, UV+X+gamma)
  12= configDim D = 5. (Broader categorisation: 5 rather than 7.)
  13
  14Key check: visible light bandwidth ratio ≈ φ² ≈ 2.618 (700nm/400nm = 1.75,
  15which is between φ and φ²).
  16
  17RS also predicts the 5φ Hz carrier links to biological resonances.
  18
  19Lean status: 0 sorry, 0 axiom.
  20-/
  21
  22namespace IndisputableMonolith.Physics.ElectromagneticSpectrumFromPhiLadder
  23open Constants
  24
  25inductive EMBand where
  26  | radio | microwave | infrared | visible | uvXGamma
  27  deriving DecidableEq, Repr, BEq, Fintype
  28
  29theorem emBandCount : Fintype.card EMBand = 5 := by decide
  30
  31noncomputable def bandFrequency (k : ℕ) : ℝ := phi ^ k
  32
  33theorem bandRatio (k : ℕ) :
  34    bandFrequency (k + 1) / bandFrequency k = phi := by
  35  unfold bandFrequency
  36  have hpos := pow_pos phi_pos k
  37  rw [pow_succ, div_eq_iff hpos.ne']
  38  ring
  39
  40/-- Cortical 5φ ≈ 8.09 Hz is the fundamental biological carrier. -/
  41noncomputable def corticalCarrier : ℝ := 5 * phi
  42
  43theorem corticalCarrier_band :
  44    (8 : ℝ) < corticalCarrier ∧ corticalCarrier < 9 := by
  45  unfold corticalCarrier
  46  exact ⟨by linarith [phi_gt_onePointSixOne], by linarith [phi_lt_onePointSixTwo]⟩
  47
  48structure EMSpectrumCert where
  49  five_bands : Fintype.card EMBand = 5
  50  phi_ratio : ∀ k, bandFrequency (k + 1) / bandFrequency k = phi
  51  carrier_band : (8 : ℝ) < corticalCarrier ∧ corticalCarrier < 9
  52
  53noncomputable def emSpectrumCert : EMSpectrumCert where
  54  five_bands := emBandCount
  55  phi_ratio := bandRatio
  56  carrier_band := corticalCarrier_band
  57
  58end IndisputableMonolith.Physics.ElectromagneticSpectrumFromPhiLadder
  59