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      "label": "Stabilization threshold n0 for Z(P(n,5)) = 12",
      "value": "23",
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      "location": "Theorem 5.1, abstract"
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    {
      "label": "Z(P(n,5)) for all n >= 23",
      "value": "12",
      "unit": "vertices",
      "location": "Theorem 5.1, abstract"
    },
    {
      "label": "Z(P(22,5))",
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      "unit": "vertices",
      "location": "Theorem 5.1 (sharpness), table of exact values"
    },
    {
      "label": "Boundary parameter p (first-force sources)",
      "value": "18",
      "unit": "vertices",
      "location": "Lemma 3.3, proof of Theorem 5.1"
    },
    {
      "label": "Finite window upper end N_max",
      "value": "154",
      "unit": "n",
      "location": "Lemma 3.2/3.3, proof of Theorem 5.1"
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    {
      "label": "DRAT-verified boundary certificates in the window 26..154",
      "value": "129",
      "unit": "certificates",
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    {
      "label": "Largest n with a fully certified value in the table",
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      "unit": "n",
      "location": "Table of exact values"
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    {
      "label": "Largest n certified by exhaustive enumeration",
      "value": "27",
      "unit": "n",
      "location": "Section 4.1"
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    {
      "label": "11-subsets enumerated at n = 25 (no forcing set)",
      "value": "37353738800",
      "unit": "subsets",
      "location": "Section 4.1 table, Theorem 5.1 step 4"
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    {
      "label": "11-subsets enumerated at n = 23 (no forcing set)",
      "value": "13340783196",
      "unit": "subsets",
      "location": "Section 4.1 table"
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    {
      "label": "beta_18(25): minimum boundary of an 18-set in P(25,5) (obstruction)",
      "value": "10",
      "unit": "vertices",
      "location": "Section 4.2, Section 7"
    },
    {
      "label": "Total drat-trim verification time for the p = 18 window",
      "value": "11700.7",
      "unit": "seconds",
      "location": "Section 4.3, verification scaling table"
    },
    {
      "label": "Order-independence test passes (T2)",
      "value": "1200",
      "unit": "seed sets",
      "location": "Computational framework section"
    },
    {
      "label": "Holdout predictions correct (all six holdout values equal 12)",
      "value": "6",
      "unit": "values",
      "location": "Section 4.5"
    },
    {
      "label": "Values Z(P(n,5)) for n = 11..22 (6,8,9,8,8,8,10,10,10,10,10,11)",
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      "location": "Theorem 5.1, table of exact values"
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    {
      "label": "Maximum number of occupied columns c_max for p = 18",
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      "location": "Lemma 3.2, proof of Theorem 5.1"
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    {
      "label": "Lower end n_1 of the certified boundary window",
      "value": "26",
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      "location": "Section 4.3, proof of Theorem 5.1"
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    {
      "label": "Total drat-trim verification time for the p = 18 window 26..154",
      "value": "11700.7",
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      "location": "Section 4.3, verification scaling table"
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    {
      "label": "Total CaDiCaL solve time for the p = 18 window 26..154",
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      "location": "Section 4.3, verification scaling table"
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    {
      "label": "Mean per-instance CaDiCaL solve time, p = 18 window",
      "value": "100.6",
      "unit": "seconds",
      "location": "Section 4.3"
    },
    {
      "label": "Mean per-instance drat-trim time, p = 18 window",
      "value": "90.7",
      "unit": "seconds",
      "location": "Section 4.3"
    },
    {
      "label": "Total DRAT proof lines, p = 18 window 26..154",
      "value": "346502796",
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      "location": "Section 4.3"
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    {
      "label": "Total size of DRAT proofs on disk, p = 18 window 26..154",
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    {
      "label": "CNF variables of F(154,18)",
      "value": "14323",
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    {
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      "value": "432226268255",
      "unit": "subsets",
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    {
      "label": "Second exhaustive enumeration (first-force normalization): total sets tested, 11 <= n <= 40",
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      "location": "Computational Result 6.1, abstract"
    },
    {
      "label": "Second exhaustive enumeration: total wall-clock time",
      "value": "2345.7",
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      "location": "Computational Result 6.1"
    },
    {
      "label": "Second exhaustive enumeration: 11-sets tested at n = 23 (none forcing)",
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      "location": "Table 2 (tab_brute_force)"
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    {
      "label": "Second exhaustive enumeration: 11-sets tested at n = 25 (none forcing)",
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      "label": "Second exhaustive enumeration: forcing 12-sets found at n = 23 (positive control)",
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      "label": "Largest n with Z >= 12 re-established by the second exhaustive enumeration",
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      "location": "Computational Result 6.1, Table 1, Limitations"
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    {
      "label": "Exact beta_18(n) for every 26 <= n <= 40 (zero slack)",
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      "location": "Observation 6.6"
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    {
      "label": "Boundary-encoding soundness check: (n,p) instances compared with exhaustive beta_p(n)",
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      "location": "Section 7 item (c), Table 6"
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    {
      "label": "Boundary-encoding soundness check: boundary-table cells compared (all consistent)",
      "value": "50",
      "unit": "cells",
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    {
      "label": "Enumeration-count audit: (n,s) rows checked against binomial and Burnside counts (no mismatch)",
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      "location": "Section 5 (exhaustive oracle), Table 6"
    },
    {
      "label": "Deletion lemma exhaustive small-set test D1: cases (no failure)",
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      "unit": "cases",
      "location": "Section 7 item (d), Table 6"
    },
    {
      "label": "Deletion lemma negative control D2: cases",
      "value": "810",
      "unit": "cases",
      "location": "Section 7 item (d), Table 6"
    },
    {
      "label": "Deletion lemma negative control D2: cases where the deleted column meets N[X]",
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    {
      "label": "Deletion lemma negative control D2: cases where the boundary size changed (expected)",
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    {
      "label": "Stored upper-bound witnesses replayed force by force (all ok)",
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    },
    {
      "label": "Largest n for which the P7 chronology was independently replayed",
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    },
    {
      "label": "Boundary certificates re-checked by the independent checker (all accepted)",
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    },
    {
      "label": "Fort-cover certificates re-checked by the independent checker (all accepted)",
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    {
      "label": "Simulated-annealing attack A (11-sets): instances",
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      "unit": "instances",
      "location": "Section 7 item (f)"
    },
    {
      "label": "Simulated-annealing attack B (18-sets): instances",
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      "unit": "instances",
      "location": "Section 7 item (f)"
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    {
      "label": "Random seed of the simulated-annealing attacks",
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      "unit": "seed",
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    },
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      "label": "Attack A best closure of an 11-set at n = 23 (of 46 vertices)",
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    {
      "label": "Attack A best closure of an 11-set at n = 25 (of 50 vertices)",
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    {
      "label": "Attack A best closure of an 11-set for every n >= 26",
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      "unit": "vertices",
      "location": "Section 7 item (f)"
    },
    {
      "label": "Exhaustive oracle: 11-subsets enumerated at n = 27 (largest instance)",
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      "location": "Computational Result 6.1, Table 2"
    },
    {
      "label": "Exhaustive oracle: orbit representatives tested at n = 27, s = 11",
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      "location": "Computational Result 6.1, Table 2"
    },
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      "label": "Exhaustive oracle: total wall-clock time, n = 11..27",
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      "unit": "seconds",
      "location": "Computational Result 6.1"
    },
    {
      "label": "Fort-cover certificate at n = 22: number of fort clauses",
      "value": "37521",
      "unit": "forts",
      "location": "Observation 6.7, Table 10"
    },
    {
      "label": "Fort-cover certificate at n = 22: DRAT proof lines",
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      "unit": "lines",
      "location": "Observation 6.7, Table 10"
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      "label": "Fort-cover solver UNSAT without proof log at n = 23: wall time",
      "value": "837",
      "unit": "seconds",
      "location": "Observation 6.7"
    },
    {
      "label": "Fort-cover solver UNSAT without proof log at n = 25: wall time",
      "value": "1796",
      "unit": "seconds",
      "location": "Observation 6.7"
    },
    {
      "label": "Largest number of forts generated in a timed-out holdout fort-cover search (n = 48)",
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      "unit": "forts",
      "location": "Observation 6.7"
    },
    {
      "label": "Round-based SAT encoding, n = 17: wall time of the completed rerun (UNSAT)",
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      "location": "Observation 6.7"
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    {
      "label": "Boundary window p = 18: minimum per-instance CaDiCaL solve time",
      "value": "17.4",
      "unit": "seconds",
      "location": "Computational Result 6.4"
    },
    {
      "label": "Boundary window p = 18: maximum per-instance CaDiCaL solve time",
      "value": "266.8",
      "unit": "seconds",
      "location": "Computational Result 6.4, Appendix B"
    },
    {
      "label": "Boundary window p = 18: minimum per-instance drat-trim time",
      "value": "13.3",
      "unit": "seconds",
      "location": "Computational Result 6.4"
    },
    {
      "label": "Boundary window p = 18: maximum per-instance drat-trim time",
      "value": "231.3",
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      "location": "Computational Result 6.4"
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    {
      "label": "Boundary window p = 18: shortest DRAT proof",
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    {
      "label": "Boundary window p = 18: longest DRAT proof",
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    {
      "label": "Boundary table cells computed (p = 12..24, n = 13..30)",
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      "label": "Number of fully certified entries in the exact-value table (n = 11..48)",
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      "label": "Forces in the P7 chronology on P(14,5) (Figure 2a)",
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      "location": "Figure 2 caption"
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    {
      "label": "Forces in the replayed chronology of the 8-vertex witness for P(15,5) (Figure 2b)",
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      "location": "Figure 2 caption"
    },
    {
      "label": "Verified literature sources (3 full text, 8 abstract, 9 metadata)",
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    },
    {
      "label": "Boundary certificates used for n >= 24 (129 in the window plus n = 24)",
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      "metricKey": "fortcover_n23_checker_seconds"
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    {
      "label": "Fort-cover DRAT proof at n = 23: size on disk",
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      "metricKey": "fortcover_n23_num_forts"
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      "P7_range": [
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      "P7_all_ok": true,
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