warWindowLow

formal source

  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