jcost_complexity_gap

formal source

 208/-- **THEOREM IC-005.17**: The J-cost squared form bounds from below.
 209    J(x) ≥ 0 with equality only at x = 1. This is the "complexity gap" between
 210    the ground state and any imbalanced state. -/
 211theorem jcost_complexity_gap (x : ℝ) (hx : x > 0) (hne : x ≠ 1) :
 212    Jcost x > Jcost 1 := by
 213  rw [Cost.Jcost_unit0]
 214  exact jcost_pos_away_from_one x hx hne
 215
 216/-! ## V. The RS Complexity Classification -/
 217
 218/-- Summary of RS complexity classes. -/
 219def rs_complexity_classes : List String := [
 220  "Ground state (x=1): unique, 0 cost, O(1) to verify",
 221  "Local dynamics: 8-tick update, O(1) per tick",
 222  "Balance verification: O(N) linear scan",
 223  "J-cost minimization: convex, polynomial gradient descent",
 224  "φ-rung computation: EXPTIME (φⁿ grows without bound)",
 225  "Global RS configuration: NP-hard analog (exponentially many states)"
 226]
 227
 228/-! ## Summary Certificate -/
 229
 230def ic005_certificate : String :=
 231  "═══════════════════════════════════════════════════════════\n" ++
 232  "  IC-005: PHYSICS COMPLEXITY — STATUS: DERIVED\n" ++
 233  "═══════════════════════════════════════════════════════════\n" ++
 234  "✓ jcost_unique_minimum:       J(1) ≤ J(x) for all x > 0\n" ++
 235  "✓ jcost_squared_form:         J(x) = (x-1)²/(2x)\n" ++
 236  "✓ jcost_pos_away_from_one:    J(x) > 0 for x ≠ 1\n" ++
 237  "✓ jcost_deriv_zero_at_one:    J'(1) = 0 (critical point)\n" ++
 238  "✓ jcost_deriv_pos_of_gt_one:  J'(x) > 0 for x > 1\n" ++
 239  "✓ jcost_deriv_neg_of_lt_one:  J'(x) < 0 for 0 < x < 1\n" ++
 240  "✓ total_jcost_nonneg:         Σ J(xᵢ) ≥ 0\n" ++
 241  "✓ balanced_config_zero_cost:  all xᵢ = 1 → Σ J = 0\n" ++