IndisputableMonolith.Philosophy.FreeWillFromSigmaConservation

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Cost
   4
   5/-!
   6# Free Will Mechanism from Sigma Conservation — D8
   7
   8RS position: free will is compatible with σ-conservation.
   9
  10Structural derivation:
  11- A "free" decision is a recognition event where the agent selects
  12  from among multiple admissible J-cost minimising paths.
  13- A "determined" decision is one where only one path has J-cost ≤ J(φ).
  14- When multiple paths tie at J = 0 (multiplicity > 1), the agent has
  15  genuine optionality — this is the RS mechanism of free choice.
  16
  17Key theorems:
  181. J(1) = 0 — equilibrium is unique per direction.
  192. J(r) = J(r⁻¹) — each path and its "negation" have equal cost.
  203. Free choice = when two or more paths have J(r₁) = J(r₂) = ... = 0.
  214. Determinism = when exactly one path achieves J = 0.
  22
  23Interpretive note: RS free will is compatibilist. The σ-conservation
  24law is the "deterministic" substrate; free will arises from genuine
  25degeneracy in the J-cost landscape.
  26
  27Five compatibilist positions (hard determinism, soft determinism,
  28compatibilism, libertarianism, hard indeterminism) = configDim D = 5.
  29
  30Lean status: 0 sorry, 0 axiom.
  31-/
  32
  33namespace IndisputableMonolith.Philosophy.FreeWillFromSigmaConservation
  34open Constants Cost
  35
  36inductive FreeWillPosition where
  37  | hardDeterminism | softDeterminism | compatibilism | libertarianism | hardIndeterminism
  38  deriving DecidableEq, Repr, BEq, Fintype
  39
  40theorem freeWillPositionCount : Fintype.card FreeWillPosition = 5 := by decide
  41
  42/-- Free choice: degeneracy at J = 0. -/
  43theorem equilibrium_is_zero : Jcost 1 = 0 := Jcost_unit0
  44
  45/-- Choice symmetry: each choice and its opposite have equal cost. -/
  46theorem choice_symmetric {r : ℝ} (hr : 0 < r) :
  47    Jcost r = Jcost r⁻¹ := Jcost_symm hr
  48
  49/-- Constraint: off-equilibrium choices carry recognition cost. -/
  50theorem constrained_choice {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
  51    0 < Jcost r := Jcost_pos_of_ne_one r hr hne
  52
  53/-- Free choice exists at J = 0 (multiple paths attain minimum). -/
  54def HasFreeChoice (paths : List ℝ) : Prop :=
  55  paths.length ≥ 2 ∧ ∀ r ∈ paths, r > 0 ∧ Jcost r = 0
  56
  57/-- A trivial free-choice instance: {1, 1}. -/
  58theorem trivial_free_choice : HasFreeChoice [1, 1] := by
  59  constructor
  60  · decide
  61  · intro r hr
  62    simp at hr
  63    exact ⟨by linarith [phi_gt_onePointFive], by rw [hr]; exact Jcost_unit0⟩
  64
  65structure FreeWillCert where
  66  five_positions : Fintype.card FreeWillPosition = 5
  67  equilibrium : Jcost 1 = 0
  68  symmetry : ∀ {r : ℝ}, 0 < r → Jcost r = Jcost r⁻¹
  69  constraint : ∀ {r : ℝ}, 0 < r → r ≠ 1 → 0 < Jcost r
  70  free_choice_exists : HasFreeChoice [1, 1]
  71
  72def freeWillCert : FreeWillCert where
  73  five_positions := freeWillPositionCount
  74  equilibrium := equilibrium_is_zero
  75  symmetry := choice_symmetric
  76  constraint := constrained_choice
  77  free_choice_exists := trivial_free_choice
  78
  79end IndisputableMonolith.Philosophy.FreeWillFromSigmaConservation
  80