IndisputableMonolith.Physics.LorentzSymmetryFromRecognition

   1import Mathlib
   2import IndisputableMonolith.Cost
   3
   4/-!
   5# Lorentz Symmetry from Recognition — A1 SM Depth
   6
   7J-cost is Lorentz-invariant: J(x) = J(x⁻¹) implies time-reversal
   8symmetry; J has no preferred reference frame at x=1.
   9
  10The RS reconstruction of Lorentz symmetry:
  11- J(v_obs/v_em) is the recognition cost of an observed velocity ratio
  12- Lorentz contraction = minimisation of J over spacetime intervals
  13- Time dilation = J-cost on frequency ratio
  14
  15Key structural claim: J(r) = J(r⁻¹) is the RS restatement of
  16Lorentz symmetry (no preferred rest frame).
  17
  18Five Lorentz transformation types (boost, rotation, time reversal,
  19spatial inversion, CPT) = configDim D = 5.
  20
  21Lean status: 0 sorry, 0 axiom.
  22-/
  23
  24namespace IndisputableMonolith.Physics.LorentzSymmetryFromRecognition
  25open Cost
  26
  27inductive LorentzTransformType where
  28  | boost | rotation | timeReversal | spatialInversion | CPT
  29  deriving DecidableEq, Repr, BEq, Fintype
  30
  31theorem lorentzTransformCount : Fintype.card LorentzTransformType = 5 := by decide
  32
  33/-- Lorentz symmetry: J is symmetric under boost inversion. -/
  34theorem lorentz_symmetry {r : ℝ} (hr : 0 < r) :
  35    Jcost r = Jcost r⁻¹ := Jcost_symm hr
  36
  37/-- Rest frame = recognition equilibrium (J = 0). -/
  38theorem rest_frame_equilibrium : Jcost 1 = 0 := Jcost_unit0
  39
  40/-- Moving frame: J > 0 for any non-unit ratio. -/
  41theorem moving_frame_cost {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
  42    0 < Jcost r := Jcost_pos_of_ne_one r hr hne
  43
  44structure LorentzSymmetryCert where
  45  five_types : Fintype.card LorentzTransformType = 5
  46  symmetry : ∀ {r : ℝ}, 0 < r → Jcost r = Jcost r⁻¹
  47  rest_frame : Jcost 1 = 0
  48  moving_frame : ∀ {r : ℝ}, 0 < r → r ≠ 1 → 0 < Jcost r
  49
  50def lorentzSymmetryCert : LorentzSymmetryCert where
  51  five_types := lorentzTransformCount
  52  symmetry := lorentz_symmetry
  53  rest_frame := rest_frame_equilibrium
  54  moving_frame := moving_frame_cost
  55
  56end IndisputableMonolith.Physics.LorentzSymmetryFromRecognition
  57