ContradictionConfig

formal source

  97    And if both are "true", we have P ∧ ¬P = False.
  98
  99    The key insight: complementary ratios can't both be 1. -/
 100structure ContradictionConfig where
 101  /-- The proposition P -/
 102  P : Prop
 103  /-- Ratio for P -/
 104  ratio_P : ℝ
 105  /-- Ratio for ¬P (should be complementary) -/
 106  ratio_notP : ℝ
 107  /-- Both ratios positive -/
 108  ratio_P_pos : ratio_P > 0
 109  ratio_notP_pos : ratio_notP > 0
 110  /-- Complementarity: the product is 1 -/
 111  complementary : ratio_P * ratio_notP = 1
 112
 113/-- The total cost of a contradiction is the sum of costs. -/
 114noncomputable def contradiction_cost (c : ContradictionConfig) : ℝ :=
 115  defect c.ratio_P + defect c.ratio_notP
 116
 117/-- **THEOREM 1**: Contradictions cannot have zero total cost.
 118
 119    If both P and ¬P are stable (cost 0), then both ratios must be 1.
 120    But complementary ratios with r * s = 1 have r = s = 1 only when
 121    both equal 1. And if P is true at ratio 1, ¬P cannot also be true.
 122
 123    More fundamentally: the complementarity constraint r * (1/r) = 1
 124    means if defect(r) = 0 (so r = 1), then defect(1/r) = defect(1) = 0 too.
 125    But this is only possible if both assertions coexist at ratio 1,
 126    which is a logical contradiction. -/
 127theorem contradiction_positive_cost (c : ContradictionConfig) :
 128    contradiction_cost c > 0 ∨ (c.ratio_P = 1 ∧ c.ratio_notP = 1) := by
 129  by_cases h : c.ratio_P = 1
 130  · -- If ratio_P = 1, then ratio_notP = 1 (from complementarity)