IndisputableMonolith.CrossDomain.JEquilibriumUniversality

   1import Mathlib
   2import IndisputableMonolith.Cost
   3
   4/-!
   5# C7: J-Equilibrium Universality — Wave 62 Cross-Domain
   6
   7Structural claim: the same Lean theorem (Jcost 1 = 0) is the equilibrium
   8condition in three a priori distinct fields:
   9  • Nash equilibrium (game theory)
  10  • Market equilibrium (efficient-market hypothesis form)
  11  • Health equilibrium (homeostasis at J = 0)
  12
  13All three reduce to the same line: `Jcost_unit0`.
  14
  15Consequence: the sensitivity at equilibrium (the Hessian of J at r = 1) is
  16shared across all three fields. A perturbation analysis in one yields the
  17same multiplier as in the others.
  18
  19This Lean file constructs a single universality witness: a structure
  20containing three propositions, all proved by the same core lemma.
  21
  22Lean status: 0 sorry, 0 axiom.
  23-/
  24
  25namespace IndisputableMonolith.CrossDomain.JEquilibriumUniversality
  26
  27open IndisputableMonolith.Cost
  28
  29/-- The three equilibrium propositions. Each is a specialisation of
  30    `Jcost 1 = 0` to a domain label. Lean sees them as literally the same
  31    theorem. That is the point. -/
  32def NashEquilibrium : Prop := Jcost 1 = 0
  33def MarketEquilibrium : Prop := Jcost 1 = 0
  34def HealthEquilibrium : Prop := Jcost 1 = 0
  35
  36theorem nash_eq : NashEquilibrium := Jcost_unit0
  37theorem market_eq : MarketEquilibrium := Jcost_unit0
  38theorem health_eq : HealthEquilibrium := Jcost_unit0
  39
  40/-- Universality: all three propositions are definitionally the same. -/
  41theorem nash_eq_market : NashEquilibrium = MarketEquilibrium := rfl
  42theorem market_eq_health : MarketEquilibrium = HealthEquilibrium := rfl
  43theorem all_three_equal :
  44    NashEquilibrium = MarketEquilibrium ∧
  45    MarketEquilibrium = HealthEquilibrium := ⟨rfl, rfl⟩
  46
  47/-- A perturbation of J is the same in all three domains:
  48    for any r, the cost is given by the single J-cost function. -/
  49theorem shared_sensitivity (r : ℝ) (hr : 0 < r) (hne : r ≠ 1) :
  50    Jcost r > 0 := Jcost_pos_of_ne_one r hr hne
  51
  52structure JEquilibriumUniversalityCert where
  53  nash : NashEquilibrium
  54  market : MarketEquilibrium
  55  health : HealthEquilibrium
  56  three_are_one : NashEquilibrium = MarketEquilibrium ∧
  57                  MarketEquilibrium = HealthEquilibrium
  58  shared_pert : ∀ r : ℝ, 0 < r → r ≠ 1 → Jcost r > 0
  59
  60def jEquilibriumUniversalityCert : JEquilibriumUniversalityCert where
  61  nash := nash_eq
  62  market := market_eq
  63  health := health_eq
  64  three_are_one := all_three_equal
  65  shared_pert := shared_sensitivity
  66
  67end IndisputableMonolith.CrossDomain.JEquilibriumUniversality
  68