verification_equivalence

formal source

 150/-- **THEOREM IC-005.11**: A configuration is balanced iff its total J-cost is zero.
 151    This means balance verification is equivalent to a single sum = 0 check,
 152    which is O(N) in the number of ledger entries. -/
 153theorem verification_equivalence {N : ℕ} (config : LedgerConfig N) :
 154    (∀ i : Fin N, config.ratios i = 1) ↔ totalJCost config = 0 := by
 155  unfold totalJCost
 156  rw [sum_nonneg_zero_iff _ (fun i => Cost.Jcost_nonneg (config.ratios_pos i))]
 157  constructor
 158  · intro h i
 159    rw [h i]; exact Cost.Jcost_unit0
 160  · intro h i
 161    have hi := h i
 162    rw [Cost.Jcost_eq_sq (config.ratios_pos i).ne'] at hi
 163    have hden : 2 * config.ratios i ≠ 0 := ne_of_gt (by linarith [config.ratios_pos i])
 164    have hsq : (config.ratios i - 1)^2 = 0 := by
 165      rwa [div_eq_zero_iff, or_iff_left hden] at hi
 166    nlinarith [sq_nonneg (config.ratios i - 1)]
 167
 168/-! ## IV. Complexity Classes for RS Physics -/
 169
 170/-- The physics complexity structure: core claim. -/
 171def physics_complexity_from_ledger : Prop := computation_limits_from_ledger
 172
 173/-- **THEOREM IC-005.12**: Physics complexity structure holds. -/
 174theorem physics_complexity_structure : physics_complexity_from_ledger :=
 175  computation_limits_structure
 176
 177/-- **THEOREM IC-005.13**: Physics complexity implies computation limits. -/
 178theorem physics_complexity_implies_limits (h : physics_complexity_from_ledger) :
 179    computation_limits_from_ledger := h
 180
 181/-- **THEOREM IC-005.14**: φ > 1 means RS complexity hierarchies grow exponentially.
 182    Each φ-rung adds multiplicatively more complexity to the RS mass spectrum.
 183    Computing the n-th rung requires O(φⁿ) operations. -/