files / problems / covering / compute / ok37_p_le17_results.json

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{
  "dependent_triples": 0,
  "ells": {
    "0": {
      "QM4_3_m4": false,
      "by_p": {
        "10": "UNSAT",
        "11": "UNSAT",
        "12": "UNSAT",
        "13": "UNSAT",
        "14": "UNSAT",
        "15": "UNSAT",
        "16": "UNSAT",
        "17": "UNSAT",
        "18": "SAT",
        "3": "UNSAT",
        "37": "SAT",
        "4": "UNSAT",
        "5": "UNSAT",
        "6": "UNSAT",
        "7": "UNSAT",
        "8": "UNSAT",
        "9": "UNSAT"
      },
      "certificate_p18": "/workspace/maths-ok37/problems/covering/compute/partition_H_OK37_p18_ell0.txt",
      "forced_singletons": 15,
      "impossible": false,
      "min_p": 18,
      "structural_lb": 15,
      "trivial_is_3ell": true,
      "unique_edges": 466
    },
    "1": {
      "QM4_3_m4": false,
      "by_p": {
        "10": "UNSAT",
        "11": "UNSAT",
        "12": "UNSAT",
        "13": "UNSAT",
        "14": "UNSAT",
        "15": "UNSAT",
        "16": "UNSAT",
        "17": "UNSAT",
        "18": "UNSAT",
        "19": "UNSAT",
        "20": "UNSAT",
        "21": "UNSAT",
        "22": "UNSAT",
        "23": "UNSAT",
        "24": "UNSAT",
        "25": "UNSAT",
        "26": "UNSAT",
        "27": "UNSAT",
        "28": "UNSAT",
        "29": "UNSAT",
        "3": "UNSAT",
        "30": "UNSAT",
        "31": "UNSAT",
        "32": "UNSAT",
        "33": "UNSAT",
        "34": "UNSAT",
        "35": "UNSAT",
        "36": "UNSAT",
        "37": "UNSAT",
        "4": "UNSAT",
        "5": "UNSAT",
        "6": "UNSAT",
        "7": "UNSAT",
        "8": "UNSAT",
        "9": "UNSAT"
      },
      "forced_singletons": 15,
      "impossible": true,
      "min_p": null,
      "reason": "syndrome 0 has no wt1/wt2/wt3 representation (no dependent triple)",
      "structural_lb": 38,
      "trivial_is_3ell": false,
      "unique_edges": 466
    },
    "2": {
      "QM4_3_m4": false,
      "by_p": {
        "10": "UNSAT",
        "11": "UNSAT",
        "12": "UNSAT",
        "13": "UNSAT",
        "14": "UNSAT",
        "15": "UNSAT",
        "16": "UNSAT",
        "17": "UNSAT",
        "18": "UNSAT",
        "19": "UNSAT",
        "20": "UNSAT",
        "21": "UNSAT",
        "22": "UNSAT",
        "23": "UNSAT",
        "24": "UNSAT",
        "25": "UNSAT",
        "26": "UNSAT",
        "27": "UNSAT",
        "28": "UNSAT",
        "29": "UNSAT",
        "3": "UNSAT",
        "30": "UNSAT",
        "31": "UNSAT",
        "32": "UNSAT",
        "33": "UNSAT",
        "34": "UNSAT",
        "35": "UNSAT",
        "36": "UNSAT",
        "37": "UNSAT",
        "4": "UNSAT",
        "5": "UNSAT",
        "6": "UNSAT",
        "7": "UNSAT",
        "8": "UNSAT",
        "9": "UNSAT"
      },
      "forced_singletons": 15,
      "impossible": true,
      "min_p": null,
      "reason": "syndrome 0 has no wt1/wt2/wt3 representation (no dependent triple)",
      "structural_lb": 38,
      "trivial_is_3ell": false,
      "unique_edges": 466
    },
    "3": {
      "QM4_3_m4": false,
      "by_p": {
        "10": "UNSAT",
        "11": "UNSAT",
        "12": "UNSAT",
        "13": "UNSAT",
        "14": "UNSAT",
        "15": "UNSAT",
        "16": "UNSAT",
        "17": "UNSAT",
        "18": "UNSAT",
        "19": "UNSAT",
        "20": "UNSAT",
        "21": "UNSAT",
        "22": "UNSAT",
        "23": "UNSAT",
        "24": "UNSAT",
        "25": "UNSAT",
        "26": "UNSAT",
        "27": "UNSAT",
        "28": "UNSAT",
        "29": "UNSAT",
        "3": "UNSAT",
        "30": "UNSAT",
        "31": "UNSAT",
        "32": "UNSAT",
        "33": "UNSAT",
        "34": "UNSAT",
        "35": "UNSAT",
        "36": "UNSAT",
        "37": "UNSAT",
        "4": "UNSAT",
        "5": "UNSAT",
        "6": "UNSAT",
        "7": "UNSAT",
        "8": "UNSAT",
        "9": "UNSAT"
      },
      "forced_singletons": 15,
      "impossible": true,
      "min_p": null,
      "reason": "syndrome 0 has no wt1/wt2/wt3 representation (no dependent triple)",
      "structural_lb": 38,
      "trivial_is_3ell": false,
      "unique_edges": 486
    }
  },
  "note": "No three columns sum to 0, so syndrome 0 has no wt2/wt3 representation; (3,ell) is impossible for ell>=1. For ell=0, unique conflict edges force 15 singleton blocks, Cadical proves p=17 UNSAT, and p=18 is SAT (minimum). Refinement: UNSAT at 17 => UNSAT for all p<=17. QM_4^3 at m=4 requires p(H,ell)<=2^4+1=17, so does not apply.",
  "trivial_p37": {
    "0": true,
    "1": false,
    "2": false,
    "3": false
  }
}