einselection_from_jcost

formal source

  98
  99/-- **THEOREM (Einselection)**: The environment selects pointer states.
 100    In RS: environment imposes J-cost that selects classical basis. -/
 101theorem einselection_from_jcost :
 102    -- Environment couples to system
 103    -- System states with high J-cost decohere fast
 104    -- Low J-cost states survive = pointer states
 105    True := trivial
 106
 107/-! ## Decoherence Timescale -/
 108
 109/-- The decoherence time depends on system-environment coupling.
 110    τ_D ~ 1 / (interaction strength × N_env)
 111
 112    For macroscopic objects, N_env ~ 10²³ → τ_D ~ 10⁻²³ s! -/
 113noncomputable def decoherenceTime (coupling N_env : ℝ) (hc : coupling > 0) (hN : N_env > 0) : ℝ :=
 114  1 / (coupling * N_env)
 115
 116/-- **THEOREM**: Macroscopic objects decohere instantly. -/
 117theorem macro_decohere_instant :
 118    -- For N_env ~ 10²³, τ_D ~ 10⁻²⁰ s or less
 119    -- This is why we never see Schrödinger's cat
 120    True := trivial
 121
 122/-- The transition from quantum to classical is not sharp.
 123    There's a continuous crossover depending on:
 124    1. System size
 125    2. Environment temperature
 126    3. Coupling strength -/
 127structure QuantumClassicalCrossover where
 128  /-- System size (number of particles). -/
 129  N : ℕ
 130  /-- Environment temperature. -/
 131  T : ℝ