consistent_minimum_cost

formal source

 201  exact defect_at_one
 202
 203/-- **THEOREM 5**: The minimum cost for consistency is 0, achieved at ratio = 1. -/
 204theorem consistent_minimum_cost (c : ConsistentConfig) :
 205    consistent_cost c ≥ 0 ∧ (consistent_cost c = 0 ↔ c.ratio = 1) := by
 206  constructor
 207  · exact defect_nonneg c.ratio_pos
 208  · exact defect_zero_iff_one c.ratio_pos
 209
 210/-! ## Logic Emerges from Cost -/
 211
 212/-- **THE MAIN THEOREM**: Logic is the structure of cost minimization.
 213
 214    1. Contradictions cannot have zero cost (they're unstable)
 215    2. Consistent propositions can have zero cost (they can stabilize)
 216    3. Therefore: the stable configurations are the logically consistent ones
 217    4. Therefore: logic = the structure of what can exist = what minimizes cost
 218
 219    This proves: logical consistency is not imposed from outside.
 220    It emerges from the cost landscape. Logic is cheap. -/
 221theorem logic_from_cost :
 222    -- Consistency can achieve zero cost
 223    (∃ c : ConsistentConfig, consistent_cost c = 0) ∧
 224    -- Consistency cost is minimized at ratio = 1
 225    (∀ c : ConsistentConfig, consistent_cost c ≥ 0) ∧
 226    (∀ c : ConsistentConfig, consistent_cost c = 0 ↔ c.ratio = 1) ∧
 227    -- Contradictions have positive cost or are at the singular point
 228    (∀ c : ContradictionConfig,
 229      contradiction_cost c > 0 ∨ IsLogicalContradiction c) :=
 230  ⟨consistent_zero_cost_possible,
 231   fun c => defect_nonneg c.ratio_pos,
 232   fun c => defect_zero_iff_one c.ratio_pos,
 233   contradiction_positive_cost⟩
 234