IndisputableMonolith.InternationalRelations.PowerTransitionFromJCost

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Cost
   4
   5/-!
   6# Power Transition Theory from J-Cost (Plan v7 fifty-first pass)
   7
   8## Status: STRUCTURAL THEOREM (0 sorry, 0 axiom).
   9
  10## First International Relations module in the codebase.
  11
  12Power Transition Theory (Organski 1958; Lemke 2002): inter-state conflict
  13probability peaks when the rising power's relative capabilities cross
  14the `parity threshold` (ratio ≈ 1).
  15
  16RS prediction: the parity band is the J-cost zero set. A rising power
  17at `capabilities_ratio = 1` has zero J-cost (recognition equilibrium);
  18the system is maximally unstable at zero-cost because the cost surface
  19is at a local minimum and any perturbation raises cost equally in either
  20direction.
  21
  22Quantitatively, the "war window" is `r ∈ (1/φ, φ)`, the J-cost band
  23where `J(r) ≤ J(φ) ≈ 0.118`.
  24
  25## Falsifier
  26
  27Any empirical analysis of the Correlates of War (COW) dataset showing
  28that great-power conflict onset does NOT cluster near capability parity
  29ratios in `(0.618, 1.618)`.
  30-/
  31
  32namespace IndisputableMonolith
  33namespace InternationalRelations
  34namespace PowerTransitionFromJCost
  35
  36open Constants
  37open Cost
  38
  39noncomputable section
  40
  41/-- J-cost on the relative-capability ratio
  42    (rising_power_capabilities / incumbent_power_capabilities). -/
  43def capabilityCost (rising incumbent : ℝ) : ℝ :=
  44  Jcost (rising / incumbent)
  45
  46theorem capabilityCost_at_parity (c : ℝ) (h : c ≠ 0) :
  47    capabilityCost c c = 0 := by
  48  unfold capabilityCost; rw [div_self h]; exact Jcost_unit0
  49
  50theorem capabilityCost_nonneg (r i : ℝ) (hr : 0 < r) (hi : 0 < i) :
  51    0 ≤ capabilityCost r i := by
  52  unfold capabilityCost; exact Jcost_nonneg (div_pos hr hi)
  53
  54/-- War window: capability ratio in (1/φ, φ). -/
  55def warWindowLow : ℝ := phi⁻¹
  56def warWindowHigh : ℝ := phi
  57
  58theorem warWindowLow_pos : 0 < warWindowLow :=
  59  inv_pos.mpr phi_pos
  60
  61theorem warWindowHigh_gt_one : 1 < warWindowHigh := one_lt_phi
  62
  63theorem warWindow_ordered : warWindowLow < warWindowHigh := by
  64  unfold warWindowLow warWindowHigh
  65  -- phi⁻¹ < phi since phi > 1 implies phi * phi > 1 > phi * phi⁻¹ = 1... direct:
  66  have hphi_sq : phi ^ 2 = phi + 1 := phi_sq_eq
  67  rw [inv_eq_one_div]
  68  rw [div_lt_iff₀ phi_pos]
  69  nlinarith [phi_gt_onePointFive]
  70
  71/-- At the φ-boundary, J-cost equals the canonical recognition quantum. -/
  72theorem capabilityCost_at_phi_boundary :
  73    capabilityCost phi 1 = phi - 3 / 2 := by
  74  unfold capabilityCost; simp; exact Jcost_phi_val
  75
  76structure PowerTransitionCert where
  77  cost_at_parity : ∀ c : ℝ, c ≠ 0 → capabilityCost c c = 0
  78  cost_nonneg : ∀ r i : ℝ, 0 < r → 0 < i → 0 ≤ capabilityCost r i
  79  war_window_low_pos : 0 < warWindowLow
  80  war_window_high_gt_one : 1 < warWindowHigh
  81  war_window_ordered : warWindowLow < warWindowHigh
  82  boundary_cost : capabilityCost phi 1 = phi - 3 / 2
  83
  84noncomputable def cert : PowerTransitionCert where
  85  cost_at_parity := capabilityCost_at_parity
  86  cost_nonneg := capabilityCost_nonneg
  87  war_window_low_pos := warWindowLow_pos
  88  war_window_high_gt_one := warWindowHigh_gt_one
  89  war_window_ordered := warWindow_ordered
  90  boundary_cost := capabilityCost_at_phi_boundary
  91
  92theorem cert_inhabited : Nonempty PowerTransitionCert := ⟨cert⟩
  93
  94end
  95end PowerTransitionFromJCost
  96end InternationalRelations
  97end IndisputableMonolith
  98