IndisputableMonolith.Physics.EM.PhotonEnergyFrequency

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# E = ℏω for a recognition mode (statement-locked skeleton)
   6
   7Curated light target 4.1. There is no photon frequency object in the library yet,
   8so this authors one (the DFT-8 octave angular frequency) and derives the
   9energy-frequency relation `E = ℏω` with `ℏ = φ⁻⁵`.
  10
  11Non-vacuity is the point: energy is DEFINED as the time-derivative of the
  12accumulated recognition action (`ℏ` times the accumulated phase `ω·t`), NOT as `ℏω`
  13by fiat. The theorem then proves that this independently-defined energy equals `ℏω`.
  14`ℏ` is the action per radian of phase; energy is the rate of action accumulation.
  15
  16Anchors: `Constants.hbar`, `Constants.hbar_eq_phi_inv_fifth`, `Constants.tau0`.
  17-/
  18
  19noncomputable section
  20
  21namespace IndisputableMonolith
  22namespace Physics
  23namespace EM
  24namespace PhotonEnergyFrequency
  25
  26open IndisputableMonolith.Constants
  27
  28/-- The recognition action accumulated by a mode of angular frequency `ω` over time
  29`t`: `ℏ` times the accumulated phase `ω·t`. `ℏ` is the action per radian. -/
  30def photonAction (omega t : ℝ) : ℝ := hbar * (omega * t)
  31
  32/-- The energy of a recognition mode: the rate of action accumulation, the time
  33derivative of `photonAction`. (Independently defined; NOT `ℏω` by definition.) -/
  34def photonEnergy (omega : ℝ) : ℝ := deriv (fun t => photonAction omega t) 0
  35
  36/-- The DFT-8 octave angular frequency of mode `k`: a mode completing `k` cycles per
  37eight-tick octave has angular frequency `2π k / (8 τ₀)`. Kinematic; no energy input. -/
  38def octaveAngularFreq (k : ℕ) : ℝ := 2 * Real.pi * (k : ℝ) / (8 * tau0)
  39
  40/-- **E = ℏω.** The energy of a recognition mode equals `ℏ` times its angular
  41frequency. Non-vacuous: `photonEnergy` is the time-derivative of the accumulated
  42action, and this theorem proves it equals `ℏω`. -/
  43theorem photon_energy_eq_hbar_omega (omega : ℝ) :
  44    photonEnergy omega = hbar * omega := by
  45  unfold photonEnergy photonAction
  46  have h0 : HasDerivAt (fun t : ℝ => omega * t) omega 0 := by
  47    simpa using (hasDerivAt_id (0 : ℝ)).const_mul omega
  48  exact (h0.const_mul hbar).deriv
  49
  50/-- The octave mode energy in RS φ-form: `E_k = φ⁻⁵ · ω_k` (since `ℏ = φ⁻⁵`). -/
  51theorem photon_octave_energy_phi (k : ℕ) :
  52    photonEnergy (octaveAngularFreq k) = phi ^ (-(5 : ℝ)) * octaveAngularFreq k := by
  53  rw [photon_energy_eq_hbar_omega, hbar_eq_phi_inv_fifth]
  54
  55/-- Certificate: the energy-frequency law and its RS φ-form. -/
  56theorem photonEnergyCert :
  57    (∀ omega : ℝ, photonEnergy omega = hbar * omega)
  58      ∧ (∀ k : ℕ, photonEnergy (octaveAngularFreq k)
  59            = phi ^ (-(5 : ℝ)) * octaveAngularFreq k) :=
  60  ⟨photon_energy_eq_hbar_omega, photon_octave_energy_phi⟩
  61
  62end PhotonEnergyFrequency
  63end EM
  64end Physics
  65end IndisputableMonolith
  66