def
definition
ic005_certificate
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IndisputableMonolith.Information.PhysicsComplexityStructure on GitHub at line 230.
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depends on
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all -
all -
rung -
jcost_squared_form -
all -
rung -
jcost_pos_away_from_one -
jcost_squared_form -
eval -
for -
balanced_config_zero_cost -
jcost_deriv_neg_of_lt_one -
jcost_deriv_pos_of_gt_one -
jcost_deriv_zero_at_one -
jcost_pos_away_from_one -
jcost_squared_form -
jcost_unique_minimum -
total_jcost_nonneg -
verification_equivalence -
rung -
rung -
all -
total -
total -
point -
eval -
gradient -
rung
formal source
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"
250
251#eval ic005_certificate
252
253end PhysicsComplexityStructure
254end Information
255end IndisputableMonolith