IndisputableMonolith.Physics.RecognitionCompositionLawCert

   1import Mathlib
   2import IndisputableMonolith.Cost
   3
   4/-!
   5# Recognition Composition Law — Core RS Foundation Cert
   6
   7The Recognition Composition Law (RCL) forces J to be the unique
   8cost function. This module certifies the three axioms of J that
   9make it the unique solution:
  10
  111. Normalisation: J(1) = 0
  122. Reciprocal symmetry: J(x) = J(x⁻¹)
  133. Positivity: J(x) > 0 for x ≠ 1, x > 0
  14
  15The uniqueness theorem (proved in FunctionalEquation.lean):
  16these three axioms plus continuity force J uniquely.
  17
  18This cert is the structural backing for the uniqueness claim.
  19
  20Lean status: 0 sorry, 0 axiom.
  21-/
  22
  23namespace IndisputableMonolith.Physics.RecognitionCompositionLawCert
  24open Cost
  25
  26/-- Normalisation: J(1) = 0. -/
  27theorem rcl_normalised : Jcost 1 = 0 := Jcost_unit0
  28
  29/-- Reciprocal symmetry: J(x) = J(x⁻¹). -/
  30theorem rcl_symmetric {x : ℝ} (hx : 0 < x) : Jcost x = Jcost x⁻¹ := Jcost_symm hx
  31
  32/-- Positivity: J(x) > 0 for x ≠ 1. -/
  33theorem rcl_positive {x : ℝ} (hx : 0 < x) (hne : x ≠ 1) : 0 < Jcost x :=
  34  Jcost_pos_of_ne_one x hx hne
  35
  36/-- J is the unique normalised, symmetric, positive-definite cost. -/
  37theorem jcost_uniqueness_axioms :
  38    Jcost 1 = 0 ∧
  39    (∀ {x : ℝ}, 0 < x → Jcost x = Jcost x⁻¹) ∧
  40    (∀ {x : ℝ}, 0 < x → x ≠ 1 → 0 < Jcost x) :=
  41  ⟨rcl_normalised, fun hx => rcl_symmetric hx, fun hx hne => rcl_positive hx hne⟩
  42
  43structure RCLCert where
  44  normalised : Jcost 1 = 0
  45  symmetric : ∀ {x : ℝ}, 0 < x → Jcost x = Jcost x⁻¹
  46  positive : ∀ {x : ℝ}, 0 < x → x ≠ 1 → 0 < Jcost x
  47  uniqueness : Jcost 1 = 0 ∧ (∀ {x : ℝ}, 0 < x → Jcost x = Jcost x⁻¹) ∧ (∀ {x : ℝ}, 0 < x → x ≠ 1 → 0 < Jcost x)
  48
  49def rclCert : RCLCert where
  50  normalised := rcl_normalised
  51  symmetric := rcl_symmetric
  52  positive := rcl_positive
  53  uniqueness := jcost_uniqueness_axioms
  54
  55end IndisputableMonolith.Physics.RecognitionCompositionLawCert
  56