IndisputableMonolith.CrossDomain.JPositivityUniversality

   1import Mathlib
   2import IndisputableMonolith.Cost
   3
   4/-!
   5# C16: J-Cost Positivity Universality — Wave 63 Cross-Domain
   6
   7Structural claim extending C7: the same single lemma `Jcost_pos_of_ne_one`
   8is the source of non-equilibrium cost in every RS domain where J-cost
   9applies. This is the off-equilibrium analogue of C7 (which covered
  10equilibrium Jcost(1) = 0).
  11
  12Universal lemma: for any r ∈ (0, ∞) with r ≠ 1, J(r) > 0.
  13
  14Specialisations (all the same theorem with domain labels):
  15  • Turbulent flow cost in hydrodynamics
  16  • Disease cost in medicine (deviation from homeostasis)
  17  • Off-target cost in CRISPR (imperfect guide match)
  18  • Off-equilibrium game theory
  19  • Market arbitrage gap
  20  • Biased reasoning (cognitive biases)
  21  • Recognition deficit (neurodevelopmental)
  22
  23All are definitionally the same proposition: `∀ r, 0 < r → r ≠ 1 → 0 < J r`.
  24
  25Lean status: 0 sorry, 0 axiom.
  26-/
  27
  28namespace IndisputableMonolith.CrossDomain.JPositivityUniversality
  29
  30open IndisputableMonolith.Cost
  31
  32/-- The universal non-equilibrium cost claim (definitional specialisation
  33    of `Jcost_pos_of_ne_one`). -/
  34def TurbulentCost : Prop := ∀ r : ℝ, 0 < r → r ≠ 1 → 0 < Jcost r
  35def DiseaseCost : Prop := ∀ r : ℝ, 0 < r → r ≠ 1 → 0 < Jcost r
  36def OffTargetCost : Prop := ∀ r : ℝ, 0 < r → r ≠ 1 → 0 < Jcost r
  37def OffEquilibriumGameCost : Prop := ∀ r : ℝ, 0 < r → r ≠ 1 → 0 < Jcost r
  38def MarketArbitrageGap : Prop := ∀ r : ℝ, 0 < r → r ≠ 1 → 0 < Jcost r
  39def BiasedReasoningCost : Prop := ∀ r : ℝ, 0 < r → r ≠ 1 → 0 < Jcost r
  40def RecognitionDeficit : Prop := ∀ r : ℝ, 0 < r → r ≠ 1 → 0 < Jcost r
  41
  42theorem turbulent_cost : TurbulentCost := fun r hr hne => Jcost_pos_of_ne_one r hr hne
  43theorem disease_cost : DiseaseCost := fun r hr hne => Jcost_pos_of_ne_one r hr hne
  44theorem off_target_cost : OffTargetCost := fun r hr hne => Jcost_pos_of_ne_one r hr hne
  45theorem off_equilibrium_game_cost : OffEquilibriumGameCost :=
  46  fun r hr hne => Jcost_pos_of_ne_one r hr hne
  47theorem market_arbitrage_gap : MarketArbitrageGap :=
  48  fun r hr hne => Jcost_pos_of_ne_one r hr hne
  49theorem biased_reasoning_cost : BiasedReasoningCost :=
  50  fun r hr hne => Jcost_pos_of_ne_one r hr hne
  51theorem recognition_deficit : RecognitionDeficit :=
  52  fun r hr hne => Jcost_pos_of_ne_one r hr hne
  53
  54/-! ## Universality: all seven are definitionally the same proposition. -/
  55
  56theorem all_seven_are_one :
  57    TurbulentCost = DiseaseCost ∧
  58    DiseaseCost = OffTargetCost ∧
  59    OffTargetCost = OffEquilibriumGameCost ∧
  60    OffEquilibriumGameCost = MarketArbitrageGap ∧
  61    MarketArbitrageGap = BiasedReasoningCost ∧
  62    BiasedReasoningCost = RecognitionDeficit :=
  63  ⟨rfl, rfl, rfl, rfl, rfl, rfl⟩
  64
  65/-- J-cost is symmetric around r = 1: J(r) = J(r⁻¹). -/
  66theorem symmetry_at_equilibrium (r : ℝ) (hr : 0 < r) : Jcost r = Jcost r⁻¹ :=
  67  Jcost_symm hr
  68
  69/-- The minimum of J on (0, ∞) is at r = 1, value 0. -/
  70theorem minimum_at_one : ∀ r : ℝ, 0 < r → Jcost 1 ≤ Jcost r := by
  71  intro r hr
  72  rw [Jcost_unit0]
  73  rcases eq_or_ne r 1 with heq | hne
  74  · rw [heq, Jcost_unit0]
  75  · exact le_of_lt (Jcost_pos_of_ne_one r hr hne)
  76
  77structure JPositivityUniversalityCert where
  78  turbulent : TurbulentCost
  79  disease : DiseaseCost
  80  off_target : OffTargetCost
  81  off_game : OffEquilibriumGameCost
  82  arbitrage : MarketArbitrageGap
  83  biased : BiasedReasoningCost
  84  deficit : RecognitionDeficit
  85  all_same : TurbulentCost = DiseaseCost ∧
  86             DiseaseCost = OffTargetCost ∧
  87             OffTargetCost = OffEquilibriumGameCost ∧
  88             OffEquilibriumGameCost = MarketArbitrageGap ∧
  89             MarketArbitrageGap = BiasedReasoningCost ∧
  90             BiasedReasoningCost = RecognitionDeficit
  91  minimum_at_1 : ∀ r : ℝ, 0 < r → Jcost 1 ≤ Jcost r
  92  symmetry : ∀ r : ℝ, 0 < r → Jcost r = Jcost r⁻¹
  93
  94def jPositivityUniversalityCert : JPositivityUniversalityCert where
  95  turbulent := turbulent_cost
  96  disease := disease_cost
  97  off_target := off_target_cost
  98  off_game := off_equilibrium_game_cost
  99  arbitrage := market_arbitrage_gap
 100  biased := biased_reasoning_cost
 101  deficit := recognition_deficit
 102  all_same := all_seven_are_one
 103  minimum_at_1 := minimum_at_one
 104  symmetry := symmetry_at_equilibrium
 105
 106end IndisputableMonolith.CrossDomain.JPositivityUniversality
 107