{
  "status": "exact_optimum",
  "reason": "primal_matches_lower_bound",
  "metric_valid": true,
  "primal": {
    "valid": true,
    "selected": [
      1,
      2,
      3,
      4,
      6,
      7,
      10,
      13,
      15,
      16,
      19,
      22
    ],
    "clients": [
      0,
      1,
      2,
      3,
      4,
      5,
      6,
      7,
      8,
      9,
      10,
      11,
      12,
      13,
      14,
      15,
      16,
      17,
      18,
      19,
      20,
      21,
      22,
      23
    ],
    "assignments": [
      1,
      1,
      2,
      3,
      4,
      4,
      6,
      7,
      7,
      10,
      10,
      10,
      13,
      13,
      13,
      15,
      16,
      16,
      19,
      19,
      19,
      22,
      22,
      22
    ],
    "exact_cost": "12"
  },
  "dual": {
    "valid": true,
    "exact_lower_bound": "12",
    "ordering": "sorted input client and facility indices",
    "independent_of_source_theorem": true
  },
  "exact_lower_bound": "12",
  "exact_upper_bound": "12",
  "independently_known_optimum": "12",
  "certified_ratio_upper": "1",
  "absolute_gap": "0",
  "total_k_subsets": 2704156,
  "examined_subsets": 0,
  "enumeration_distance_lookups": 0,
  "subset_budget": 0,
  "lookup_budget": 5000000,
  "resource_scope": "Counts cover enumeration subsets and distance lookups only; metric parsing, O(n^3) triangle checks, primal/dual audits, Fraction bit arithmetic, allocations and runtime are not charged. Input dimensions/rationals have separate caps.",
  "enumeration_justification": "An at-most-k nonempty set can be augmented to k candidates without increasing uncapacitated nearest-distance cost. Exhausting exactly-k subsets therefore suffices.",
  "source_algorithm_implemented": false,
  "source_proof_verification": "not_run",
  "formal_python_verification": "not_run",
  "input_scope": "Exact frozen unweighted finite metric only; no rounded/geographic/traffic fidelity or buyer value is established."
}