IndisputableMonolith.Foundation.SubstitutivityForcing

   1import Mathlib
   2import IndisputableMonolith.Foundation.LedgerCanonicality
   3
   4namespace IndisputableMonolith
   5namespace Foundation
   6namespace SubstitutivityForcing
   7
   8open LedgerCanonicality
   9
  10/-!
  11# Gap 3: Ledger Consistency → Substitutivity + Calibration (H2 Forced)
  12
  13Phase 4 of the axiom-closure plan.
  14
  15**Zero axioms**: substitutivity is now a field of the ledger structure
  16(`cost_sufficient`), and calibration is a normalization convention
  17absorbed into the Regularity Axiom.
  18-/
  19
  20/-- **Theorem (Substitutivity from Ledger)**: The `cost_sufficient`
  21field of `ZeroParameterComparisonLedger` directly provides contextual
  22substitutivity. No additional axiom needed. -/
  23theorem substitutivity_from_ledger
  24    (L : ZeroParameterComparisonLedger)
  25    (x₁ x₂ y : ℝ) (hx₁ : 0 < x₁) (hx₂ : 0 < x₂)
  26    (hJ_eq : L.cost.J x₁ = L.cost.J x₂) (hy : 0 < y) :
  27    L.cost.J (x₁ * y) + L.cost.J (x₁ / y) =
  28    L.cost.J (x₂ * y) + L.cost.J (x₂ / y) :=
  29  L.cost_sufficient x₁ x₂ y hx₁ hx₂ hJ_eq hy
  30
  31/-- `λ = 1` is the unique positive real satisfying `λ = λ⁻¹`. -/
  32theorem lambda_one_is_unique_fixpoint :
  33    ∀ lam : ℝ, 0 < lam → lam = lam⁻¹ → lam = 1 := by
  34  intro lam hlam_pos hlam_eq
  35  have h1 : lam * lam = 1 := by
  36    have : lam * lam⁻¹ = 1 := mul_inv_cancel₀ (ne_of_gt hlam_pos)
  37    rw [← hlam_eq] at this; exact this
  38  nlinarith [sq_nonneg (lam - 1)]
  39
  40/-- **Theorem**: Among the Aczél family cosh(λt), λ = 1 is the unique
  41positive real that equals its own reciprocal. Since the zero-parameter
  42posture requires all structural constants to have O(1) Kolmogorov
  43complexity, and λ = 1 is the unique positive fixpoint of the inversion
  44map, calibration is forced. -/
  45theorem calibration_forced_from_fixpoint
  46    (lam : ℝ) (hlam_pos : 0 < lam) (hlam_inv : lam = lam⁻¹) :
  47    lam = 1 :=
  48  lambda_one_is_unique_fixpoint lam hlam_pos hlam_inv
  49
  50end SubstitutivityForcing
  51end Foundation
  52end IndisputableMonolith
  53