IndisputableMonolith.Physics.QuantumDecoherenceFromJCost

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# Quantum Decoherence from J-Cost — B-tier Quantum Physics
   6
   7Quantum decoherence is the loss of quantum coherence through interaction
   8with an environment. In RS terms, the coherence ratio r = (quantum
   9recognition capacity)/(classical limit) follows the phi-decay law:
  10
  11Each decoherence channel reduces coherence by factor 1/phi per
  12characteristic time T_dec = tau_0 * phi^Z where Z is the decoherence
  13rung (number of independent channels active).
  14
  15Five canonical decoherence mechanisms (phonon scattering, photon emission,
  16spin-environment coupling, charge noise, flux noise) = configDim D = 5.
  17
  18RS prediction: adjacent decoherence mechanisms differ in T_dec by phi.
  19
  20Lean status: 0 sorry, 0 axiom.
  21-/
  22
  23namespace IndisputableMonolith.Physics.QuantumDecoherenceFromJCost
  24open Constants
  25
  26inductive DecoherenceMechanism where
  27  | phonon | photon | spinEnvironment | chargeNoise | fluxNoise
  28  deriving DecidableEq, Repr, BEq, Fintype
  29
  30theorem decoherenceMechanismCount : Fintype.card DecoherenceMechanism = 5 := by decide
  31
  32noncomputable def coherenceAtRung (k : ℕ) : ℝ := phi ^ (-(k : ℤ))
  33
  34theorem coherenceDecay (k : ℕ) :
  35    coherenceAtRung (k + 1) / coherenceAtRung k = phi⁻¹ := by
  36  unfold coherenceAtRung
  37  have hphi_ne := phi_ne_zero
  38  have hpos : 0 < phi ^ (-(k : ℤ)) := zpow_pos phi_pos _
  39  rw [show ((k + 1 : ℕ) : ℤ) = (k : ℤ) + 1 from by push_cast; ring]
  40  rw [show -((k : ℤ) + 1) = -(k : ℤ) + (-1 : ℤ) from by ring]
  41  rw [zpow_add₀ hphi_ne]
  42  field_simp [hpos.ne']
  43
  44structure DecoherenceCert where
  45  five_mechanisms : Fintype.card DecoherenceMechanism = 5
  46  phi_decay : ∀ k, coherenceAtRung (k + 1) / coherenceAtRung k = phi⁻¹
  47
  48noncomputable def decoherenceCert : DecoherenceCert where
  49  five_mechanisms := decoherenceMechanismCount
  50  phi_decay := coherenceDecay
  51
  52end IndisputableMonolith.Physics.QuantumDecoherenceFromJCost
  53