IndisputableMonolith.Foundation.PhysicsLogicRealization

   1import IndisputableMonolith.Foundation.OrderedLogicRealization
   2
   3/-!
   4  PhysicsLogicRealization.lean
   5
   6  Lightweight physics realization hook for Universal Forcing.
   7
   8  The full physics forcing chain is large and currently imports modules with
   9  unrelated build fragility. This module gives the stable interface: identity
  10  ticks as the step action, recognition states as the carrier, and equality
  11  cost as the minimal realization of physical tick arithmetic.
  12-/
  13
  14namespace IndisputableMonolith
  15namespace Foundation
  16namespace PhysicsLogicRealization
  17
  18/-- Minimal recognition-state skeleton indexed by identity ticks. -/
  19structure PhysicsState where
  20  tick : ArithmeticFromLogic.LogicNat
  21  deriving DecidableEq
  22
  23/-- Equality cost on physics states. -/
  24def physicsCost (x y : PhysicsState) : Nat :=
  25  if x = y then 0 else 1
  26
  27@[simp] theorem physicsCost_self (x : PhysicsState) : physicsCost x x = 0 := by
  28  simp [physicsCost]
  29
  30theorem physicsCost_symm (x y : PhysicsState) : physicsCost x y = physicsCost y x := by
  31  by_cases h : x = y
  32  · subst h
  33    simp [physicsCost]
  34  · have h' : y ≠ x := by intro hyx; exact h hyx.symm
  35    simp [physicsCost, h, h']
  36
  37/-- Identity-tick successor. -/
  38def tickStep (x : PhysicsState) : PhysicsState :=
  39  ⟨ArithmeticFromLogic.LogicNat.succ x.tick⟩
  40
  41/-- Interpret the free arithmetic orbit as identity-tick states. -/
  42def physicsInterpret (n : ArithmeticFromLogic.LogicNat) : PhysicsState :=
  43  ⟨n⟩
  44
  45/-- Physics realization skeleton. -/
  46def physicsRealization : LogicRealization where
  47  Carrier := PhysicsState
  48  Cost := Nat
  49  zeroCost := inferInstance
  50  compare := physicsCost
  51  zero := ⟨ArithmeticFromLogic.LogicNat.zero⟩
  52  step := tickStep
  53  Orbit := ArithmeticFromLogic.LogicNat
  54  orbitZero := ArithmeticFromLogic.LogicNat.zero
  55  orbitStep := ArithmeticFromLogic.LogicNat.succ
  56  interpret := physicsInterpret
  57  interpret_zero := rfl
  58  interpret_step := by intro n; rfl
  59  orbit_no_confusion := by
  60    intro n h
  61    exact ArithmeticFromLogic.LogicNat.zero_ne_succ n h
  62  orbit_step_injective := ArithmeticFromLogic.LogicNat.succ_injective
  63  orbit_induction := by
  64    intro P h0 hs n
  65    exact ArithmeticFromLogic.LogicNat.induction (motive := P) h0 hs n
  66  orbitEquivLogicNat := Equiv.refl ArithmeticFromLogic.LogicNat
  67  orbitEquiv_zero := rfl
  68  orbitEquiv_step := by intro n; rfl
  69  identity := physicsCost_self
  70  nonContradiction := physicsCost_symm
  71  excludedMiddle := True
  72  composition := True
  73  actionInvariant := True
  74  nontrivial := by
  75    refine ⟨⟨ArithmeticFromLogic.LogicNat.succ ArithmeticFromLogic.LogicNat.zero⟩, ?_⟩
  76    simp [physicsCost]
  77
  78/-- Physics tick interpretation is faithful. -/
  79theorem physics_faithful :
  80    LogicRealization.FaithfulArithmeticInterpretation physicsRealization where
  81  injective := by
  82    intro a b h
  83    cases h
  84    rfl
  85  zero_step_noncollapse := by
  86    intro n h
  87    have htick := congrArg PhysicsState.tick h
  88    exact ArithmeticFromLogic.LogicNat.zero_ne_succ n htick
  89
  90/-- Physics realization has invariant extracted arithmetic. -/
  91noncomputable def physics_arithmetic_invariant (R : LogicRealization.{0, 0}) :
  92    (UniversalForcing.arithmeticOf physicsRealization).peano.carrier
  93      ≃ (UniversalForcing.arithmeticOf R).peano.carrier :=
  94  ArithmeticOf.equivOfInitial
  95    (UniversalForcing.arithmeticOf physicsRealization)
  96    (UniversalForcing.arithmeticOf R)
  97
  98end PhysicsLogicRealization
  99end Foundation
 100end IndisputableMonolith
 101