rs_complexity_classes

formal source

 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" ++
 242  "✓ verification_equivalence:   balance ↔ total J = 0 (O(N))\n" ++
 243  "✓ phi_rung_complexity:        φⁿ → ∞ (EXPTIME rung computation)\n" ++
 244  "✓ gradient_descent_converges: gradient descent moves toward x = 1\n" ++
 245  "COMPLEXITY SUMMARY:\n" ++
 246  "  • Ground state: O(1)\n" ++
 247  "  • Balance verification: O(N)\n" ++
 248  "  • Gradient descent: polynomial convergence\n" ++
 249  "  • High-rung computation: EXPTIME (φⁿ growth)\n"