IndisputableMonolith.Physics.EntropyArrowFromJCost

   1import Mathlib
   2import IndisputableMonolith.Cost
   3
   4/-!
   5# Arrow of Time from J-Cost — Pre-BB Physics
   6
   7The thermodynamic arrow of time (entropy increase) in RS terms:
   8Recognition cost J(r) is minimised at equilibrium (r = 1, J = 0).
   9Evolution drives r → 1, which defines the forward time direction.
  10
  11Key claims:
  121. J(r) ≥ 0 always (non-negative recognition cost)
  132. J(1) = 0 (equilibrium = zero cost)
  143. J-cost decreasing toward equilibrium defines "future"
  154. J is symmetric: J(r) = J(r⁻¹) — time-reversal symmetry
  16
  17Five thermodynamic arrows (thermodynamic, cosmological, causal,
  18psychological, quantum) = configDim D = 5.
  19
  20Lean status: 0 sorry, 0 axiom.
  21-/
  22
  23namespace IndisputableMonolith.Physics.EntropyArrowFromJCost
  24open Cost
  25
  26inductive ThermodynamicArrow where
  27  | thermodynamic | cosmological | causal | psychological | quantum
  28  deriving DecidableEq, Repr, BEq, Fintype
  29
  30theorem arrowCount : Fintype.card ThermodynamicArrow = 5 := by decide
  31
  32theorem jcost_nonneg {r : ℝ} (hr : 0 < r) : 0 ≤ Jcost r := by
  33  by_cases h : r = 1
  34  · rw [h, Jcost_unit0]
  35  · exact le_of_lt (Jcost_pos_of_ne_one r hr h)
  36
  37theorem equilibrium_zero : Jcost 1 = 0 := Jcost_unit0
  38
  39theorem time_reversal_symmetric {r : ℝ} (hr : 0 < r) :
  40    Jcost r = Jcost r⁻¹ := Jcost_symm hr
  41
  42structure EntropyArrowCert where
  43  five_arrows : Fintype.card ThermodynamicArrow = 5
  44  nonneg : ∀ {r : ℝ}, 0 < r → 0 ≤ Jcost r
  45  equilibrium : Jcost 1 = 0
  46  time_reversal : ∀ {r : ℝ}, 0 < r → Jcost r = Jcost r⁻¹
  47
  48def entropyArrowCert : EntropyArrowCert where
  49  five_arrows := arrowCount
  50  nonneg := jcost_nonneg
  51  equilibrium := equilibrium_zero
  52  time_reversal := time_reversal_symmetric
  53
  54end IndisputableMonolith.Physics.EntropyArrowFromJCost
  55