{
  "ortools_version": "9.15.6755",
  "runtime": "3.12.14 (main, Aug 25 2026, 13:50:33) [Clang 22.1.3 ]",
  "interpreter": "/tmp/openai-math-scheduling-20261009-FKmLBr/bin/python3",
  "solver_parameters": {
    "seconds": 2,
    "workers": 1,
    "seed": 125
  },
  "cases": [
    {
      "case": "line_distinct",
      "independent": {
        "status": "exact_optimum",
        "reason": "all_k_subsets_completed",
        "metric_valid": true,
        "primal": {
          "valid": true,
          "selected": [
            0,
            3
          ],
          "clients": [
            0,
            1,
            2,
            3,
            4,
            5
          ],
          "assignments": [
            0,
            0,
            0,
            3,
            3,
            3
          ],
          "exact_cost": "17"
        },
        "dual": null,
        "exact_lower_bound": "17",
        "exact_upper_bound": "17",
        "independently_known_optimum": "17",
        "certified_ratio_upper": "1",
        "absolute_gap": "0",
        "total_k_subsets": 6,
        "examined_subsets": 6,
        "enumeration_distance_lookups": 72,
        "subset_budget": 100000,
        "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."
      },
      "solver": {
        "status": "OPTIMAL",
        "selected": [
          0,
          3
        ],
        "exact_primal": {
          "valid": true,
          "selected": [
            0,
            3
          ],
          "clients": [
            0,
            1,
            2,
            3,
            4,
            5
          ],
          "assignments": [
            0,
            0,
            0,
            3,
            3,
            3
          ],
          "exact_cost": "17"
        },
        "exact_integer_scale": 1,
        "reported_objective": 17.0,
        "reported_best_bound": 17.0,
        "reported_wall_seconds": 0.032558000000000004,
        "model_text_sha256": "5ee9ec17644be1f847a6feebb2edf0b2b80d57a52ef7d061039d8779bf758cda",
        "interpretation": "Solver status/best bound are reports, not independently verified dual certificates. Exact nearest-distance proposal audit survives separately."
      },
      "proposal_evidence": {
        "status": "bounded_unknown_optimum",
        "reason": "budget_exhausted",
        "metric_valid": true,
        "primal": {
          "valid": true,
          "selected": [
            0,
            3
          ],
          "clients": [
            0,
            1,
            2,
            3,
            4,
            5
          ],
          "assignments": [
            0,
            0,
            0,
            3,
            3,
            3
          ],
          "exact_cost": "17"
        },
        "dual": null,
        "exact_lower_bound": "3",
        "exact_upper_bound": "17",
        "independently_known_optimum": null,
        "certified_ratio_upper": "17/3",
        "absolute_gap": "14",
        "total_k_subsets": 6,
        "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."
      }
    },
    {
      "case": "rational_line",
      "independent": {
        "status": "exact_optimum",
        "reason": "all_k_subsets_completed",
        "metric_valid": true,
        "primal": {
          "valid": true,
          "selected": [
            0,
            4
          ],
          "clients": [
            0,
            1,
            2,
            3,
            4,
            5
          ],
          "assignments": [
            0,
            0,
            0,
            4,
            4,
            4
          ],
          "exact_cost": "16/7"
        },
        "dual": null,
        "exact_lower_bound": "16/7",
        "exact_upper_bound": "16/7",
        "independently_known_optimum": "16/7",
        "certified_ratio_upper": "1",
        "absolute_gap": "0",
        "total_k_subsets": 6,
        "examined_subsets": 6,
        "enumeration_distance_lookups": 72,
        "subset_budget": 100000,
        "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."
      },
      "solver": {
        "status": "OPTIMAL",
        "selected": [
          0,
          4
        ],
        "exact_primal": {
          "valid": true,
          "selected": [
            0,
            4
          ],
          "clients": [
            0,
            1,
            2,
            3,
            4,
            5
          ],
          "assignments": [
            0,
            0,
            0,
            4,
            4,
            4
          ],
          "exact_cost": "16/7"
        },
        "exact_integer_scale": 7,
        "reported_objective": 16.0,
        "reported_best_bound": 16.0,
        "reported_wall_seconds": 0.001181,
        "model_text_sha256": "ba1340cee1ffe0588e27c792831c45d5a39dd334a8a7651a913950c652ec2f85",
        "interpretation": "Solver status/best bound are reports, not independently verified dual certificates. Exact nearest-distance proposal audit survives separately."
      },
      "proposal_evidence": {
        "status": "bounded_unknown_optimum",
        "reason": "budget_exhausted",
        "metric_valid": true,
        "primal": {
          "valid": true,
          "selected": [
            0,
            4
          ],
          "clients": [
            0,
            1,
            2,
            3,
            4,
            5
          ],
          "assignments": [
            0,
            0,
            0,
            4,
            4,
            4
          ],
          "exact_cost": "16/7"
        },
        "dual": null,
        "exact_lower_bound": "6/7",
        "exact_upper_bound": "16/7",
        "independently_known_optimum": null,
        "certified_ratio_upper": "8/3",
        "absolute_gap": "10/7",
        "total_k_subsets": 6,
        "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."
      }
    },
    {
      "case": "grid_25",
      "independent": {
        "status": "exact_optimum",
        "reason": "all_k_subsets_completed",
        "metric_valid": true,
        "primal": {
          "valid": true,
          "selected": [
            7,
            10,
            14
          ],
          "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,
            24
          ],
          "assignments": [
            10,
            7,
            7,
            7,
            14,
            10,
            7,
            7,
            7,
            14,
            10,
            10,
            7,
            14,
            14,
            10,
            10,
            7,
            14,
            14,
            10,
            10,
            7,
            14,
            14
          ],
          "exact_cost": "37"
        },
        "dual": null,
        "exact_lower_bound": "37",
        "exact_upper_bound": "37",
        "independently_known_optimum": "37",
        "certified_ratio_upper": "1",
        "absolute_gap": "0",
        "total_k_subsets": 120,
        "examined_subsets": 120,
        "enumeration_distance_lookups": 9000,
        "subset_budget": 100000,
        "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."
      },
      "solver": {
        "status": "OPTIMAL",
        "selected": [
          7,
          14,
          20
        ],
        "exact_primal": {
          "valid": true,
          "selected": [
            7,
            14,
            20
          ],
          "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,
            24
          ],
          "assignments": [
            7,
            7,
            7,
            7,
            14,
            7,
            7,
            7,
            7,
            14,
            20,
            7,
            7,
            14,
            14,
            20,
            20,
            7,
            14,
            14,
            20,
            20,
            20,
            14,
            14
          ],
          "exact_cost": "37"
        },
        "exact_integer_scale": 1,
        "reported_objective": 37.0,
        "reported_best_bound": 37.0,
        "reported_wall_seconds": 0.128863,
        "model_text_sha256": "233194712bb3cabe44ee0f6b8eb1e30fe44df788cfb1d9573aacda8a803a3832",
        "interpretation": "Solver status/best bound are reports, not independently verified dual certificates. Exact nearest-distance proposal audit survives separately."
      },
      "proposal_evidence": {
        "status": "bounded_unknown_optimum",
        "reason": "budget_exhausted",
        "metric_valid": true,
        "primal": {
          "valid": true,
          "selected": [
            7,
            14,
            20
          ],
          "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,
            24
          ],
          "assignments": [
            7,
            7,
            7,
            7,
            14,
            7,
            7,
            7,
            7,
            14,
            20,
            7,
            7,
            14,
            14,
            20,
            20,
            7,
            14,
            14,
            20,
            20,
            20,
            14,
            14
          ],
          "exact_cost": "37"
        },
        "dual": null,
        "exact_lower_bound": "17",
        "exact_upper_bound": "37",
        "independently_known_optimum": null,
        "certified_ratio_upper": "37/17",
        "absolute_gap": "20",
        "total_k_subsets": 120,
        "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."
      }
    },
    {
      "case": "empty_clients",
      "independent": {
        "status": "exact_optimum",
        "reason": "primal_matches_lower_bound",
        "metric_valid": true,
        "primal": {
          "valid": true,
          "selected": [
            0,
            1
          ],
          "clients": [],
          "assignments": [],
          "exact_cost": "0"
        },
        "dual": null,
        "exact_lower_bound": "0",
        "exact_upper_bound": "0",
        "independently_known_optimum": "0",
        "certified_ratio_upper": "1",
        "absolute_gap": "0",
        "total_k_subsets": 3,
        "examined_subsets": 0,
        "enumeration_distance_lookups": 0,
        "subset_budget": 100000,
        "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."
      },
      "solver": {
        "status": "OPTIMAL",
        "selected": [
          1
        ],
        "exact_primal": {
          "valid": true,
          "selected": [
            1
          ],
          "clients": [],
          "assignments": [],
          "exact_cost": "0"
        },
        "exact_integer_scale": 1,
        "reported_objective": 0.0,
        "reported_best_bound": 0.0,
        "reported_wall_seconds": 0.000154,
        "model_text_sha256": "ef34a69921152f93cc5d1f72f0f952ea63a7f22191ecaa90e161229e5474adc9",
        "interpretation": "Solver status/best bound are reports, not independently verified dual certificates. Exact nearest-distance proposal audit survives separately."
      },
      "proposal_evidence": {
        "status": "exact_optimum",
        "reason": "primal_matches_lower_bound",
        "metric_valid": true,
        "primal": {
          "valid": true,
          "selected": [
            1
          ],
          "clients": [],
          "assignments": [],
          "exact_cost": "0"
        },
        "dual": null,
        "exact_lower_bound": "0",
        "exact_upper_bound": "0",
        "independently_known_optimum": "0",
        "certified_ratio_upper": "1",
        "absolute_gap": "0",
        "total_k_subsets": 3,
        "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."
      }
    },
    {
      "case": "zero_optimum",
      "independent": {
        "status": "exact_optimum",
        "reason": "primal_matches_lower_bound",
        "metric_valid": true,
        "primal": {
          "valid": true,
          "selected": [
            0,
            3
          ],
          "clients": [
            0,
            3
          ],
          "assignments": [
            0,
            3
          ],
          "exact_cost": "0"
        },
        "dual": null,
        "exact_lower_bound": "0",
        "exact_upper_bound": "0",
        "independently_known_optimum": "0",
        "certified_ratio_upper": "1",
        "absolute_gap": "0",
        "total_k_subsets": 3,
        "examined_subsets": 2,
        "enumeration_distance_lookups": 8,
        "subset_budget": 100000,
        "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."
      },
      "solver": {
        "status": "OPTIMAL",
        "selected": [
          0,
          3
        ],
        "exact_primal": {
          "valid": true,
          "selected": [
            0,
            3
          ],
          "clients": [
            0,
            3
          ],
          "assignments": [
            0,
            3
          ],
          "exact_cost": "0"
        },
        "exact_integer_scale": 1,
        "reported_objective": 0.0,
        "reported_best_bound": 0.0,
        "reported_wall_seconds": 0.00038300000000000004,
        "model_text_sha256": "e62f1cda0b7f4f9ba03bda20a900507adb4dc7717eae20b4578d1304a198ee2c",
        "interpretation": "Solver status/best bound are reports, not independently verified dual certificates. Exact nearest-distance proposal audit survives separately."
      },
      "proposal_evidence": {
        "status": "exact_optimum",
        "reason": "primal_matches_lower_bound",
        "metric_valid": true,
        "primal": {
          "valid": true,
          "selected": [
            0,
            3
          ],
          "clients": [
            0,
            3
          ],
          "assignments": [
            0,
            3
          ],
          "exact_cost": "0"
        },
        "dual": null,
        "exact_lower_bound": "0",
        "exact_upper_bound": "0",
        "independently_known_optimum": "0",
        "certified_ratio_upper": "1",
        "absolute_gap": "0",
        "total_k_subsets": 3,
        "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."
      }
    },
    {
      "case": "dual_tight_line",
      "independent": {
        "status": "exact_optimum",
        "reason": "all_k_subsets_completed",
        "metric_valid": true,
        "primal": {
          "valid": true,
          "selected": [
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          ],
          "clients": [
            0,
            1,
            2
          ],
          "assignments": [
            0,
            0,
            0
          ],
          "exact_cost": "3"
        },
        "dual": null,
        "exact_lower_bound": "3",
        "exact_upper_bound": "3",
        "independently_known_optimum": "3",
        "certified_ratio_upper": "1",
        "absolute_gap": "0",
        "total_k_subsets": 2,
        "examined_subsets": 2,
        "enumeration_distance_lookups": 6,
        "subset_budget": 100000,
        "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."
      },
      "solver": {
        "status": "OPTIMAL",
        "selected": [
          2
        ],
        "exact_primal": {
          "valid": true,
          "selected": [
            2
          ],
          "clients": [
            0,
            1,
            2
          ],
          "assignments": [
            2,
            2,
            2
          ],
          "exact_cost": "3"
        },
        "exact_integer_scale": 1,
        "reported_objective": 3.0,
        "reported_best_bound": 3.0,
        "reported_wall_seconds": 0.000124,
        "model_text_sha256": "21d4bf6ab409307a6405df0654bf49d68b29fa96971c1b1a0af5ff2f05b9dd64",
        "interpretation": "Solver status/best bound are reports, not independently verified dual certificates. Exact nearest-distance proposal audit survives separately."
      },
      "proposal_evidence": {
        "status": "exact_optimum",
        "reason": "primal_matches_lower_bound",
        "metric_valid": true,
        "primal": {
          "valid": true,
          "selected": [
            2
          ],
          "clients": [
            0,
            1,
            2
          ],
          "assignments": [
            2,
            2,
            2
          ],
          "exact_cost": "3"
        },
        "dual": {
          "valid": true,
          "exact_lower_bound": "3",
          "ordering": "sorted input client and facility indices",
          "independent_of_source_theorem": true
        },
        "exact_lower_bound": "3",
        "exact_upper_bound": "3",
        "independently_known_optimum": "3",
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        "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.",
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        "input_scope": "Exact frozen unweighted finite metric only; no rounded/geographic/traffic fidelity or buyer value is established."
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    },
    {
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        "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.",
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      "solver": {
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          "assignments": [
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          "exact_cost": "18"
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        "exact_lower_bound": "10",
        "exact_upper_bound": "18",
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        "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.",
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        "input_scope": "Exact frozen unweighted finite metric only; no rounded/geographic/traffic fidelity or buyer value is established."
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    {
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        "primal": {
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            19
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            10,
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            19,
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            19,
            19,
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        "exact_lower_bound": "0",
        "exact_upper_bound": "30",
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        "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.",
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        "input_scope": "Exact frozen unweighted finite metric only; no rounded/geographic/traffic fidelity or buyer value is established."
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        "reported_objective": 12.0,
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        "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.",
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