{
 "what": "the community certificate issues on google-deepmind/alphaevolve_repository_of_problems (#4-#12; the owners have answered none), decided one by one in exact arithmetic from the pinned notebooks' definitions",
 "preregistration": "corpus/aerp/preregistration.json",
 "pins": "corpus/aerp/PINS.json",
 "repositoryCommit": "8f447457957deac61e28bf1676746f0753b3b2f8",
 "rows": [
  {
   "id": "aerp-p59-c32-58-a",
   "claim": "C(32) >= 58: the set_a of #10 is a 58-point subset of the 32 x 32 grid with no isosceles triangle (the collinear midpoint case included)",
   "where": "#10 hashkanna, 2026-10-04 (the published value is 56: PatternBoost arXiv:2411.00566 \u00a74.1, AlphaEvolve paper \u00a76.39)",
   "verdict": "CERTIFIED",
   "how": "every one of the 30,856 triples examined in integer squared distances; 0 isosceles; maximal: True",
   "size": 58,
   "n": 32,
   "triples": 30856,
   "maximal": true,
   "externalFileEqualsSetA": true
  },
  {
   "id": "aerp-p59-c32-58-b",
   "claim": "C(32) >= 58: the set_b of #10 is a 58-point subset of the 32 x 32 grid with no isosceles triangle (the collinear midpoint case included)",
   "where": "#10 hashkanna, 2026-10-04 (the published value is 56: PatternBoost arXiv:2411.00566 \u00a74.1, AlphaEvolve paper \u00a76.39)",
   "verdict": "CERTIFIED",
   "how": "every one of the 30,856 triples examined in integer squared distances; 0 isosceles; maximal: True",
   "size": 58,
   "n": 32,
   "triples": 30856,
   "maximal": true
  },
  {
   "id": "aerp-p60-c13-36",
   "claim": "C(13) >= 36: the 36 points of #4 in {0..12}^3 have no five on a sphere or plane",
   "where": "#4 Harmenszoon, 2026-06-13 (the repository ends at n = 12 with 33)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (58,905) keyed by its exact hyperplane: 0 dependent, 0 duplicate; every 5-subset (376,992) by integer determinant: 0 zero, min |det| 2",
   "size": 36,
   "n": 13,
   "engine": "python-int"
  },
  {
   "id": "aerp-p60-mm-n14-38",
   "claim": "C(14) >= 38: RECORD_38_points_n14.json",
   "where": "#6 milesandmistakes, release v1.1.1 (cffabd12)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (73,815) keyed by its exact hyperplane: 0 dependent, 0 duplicate; every 5-subset (501,942) by integer determinant: 0 zero, min |det| 2",
   "size": 38,
   "n": 14,
   "engine": "python-int",
   "file": "milesandmistakes-v1.1.1/RECORD_38_points_n14.json"
  },
  {
   "id": "aerp-p60-mm-n15-40",
   "claim": "C(15) >= 40: RECORD_40_points_n15.json",
   "where": "#6 milesandmistakes, release v1.1.1 (cffabd12)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (91,390) keyed by its exact hyperplane: 0 dependent, 0 duplicate; every 5-subset (658,008) by integer determinant: 0 zero, min |det| 2",
   "size": 40,
   "n": 15,
   "engine": "python-int",
   "file": "milesandmistakes-v1.1.1/RECORD_40_points_n15.json"
  },
  {
   "id": "aerp-p60-mm-n16-41",
   "claim": "C(16) >= 41: RECORD_41_points_n16.json",
   "where": "#6 milesandmistakes, release v1.1.1 (cffabd12)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (101,270) keyed by its exact hyperplane: 0 dependent, 0 duplicate; every 5-subset (749,398) by integer determinant: 0 zero, min |det| 2",
   "size": 41,
   "n": 16,
   "engine": "python-int",
   "file": "milesandmistakes-v1.1.1/RECORD_41_points_n16.json"
  },
  {
   "id": "aerp-p60-mm-n17-42",
   "claim": "C(17) >= 42: RECORD_42_points_n17.json",
   "where": "#6 milesandmistakes, release v1.1.1 (cffabd12)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (111,930) keyed by its exact hyperplane: 0 dependent, 0 duplicate; every 5-subset (850,668) by integer determinant: 0 zero, min |det| 2",
   "size": 42,
   "n": 17,
   "engine": "python-int",
   "file": "milesandmistakes-v1.1.1/RECORD_42_points_n17.json"
  },
  {
   "id": "aerp-p60-mm-n18-43",
   "claim": "C(18) >= 43: RECORD_43_points_n18.json",
   "where": "#6 milesandmistakes, release v1.1.1 (cffabd12)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (123,410) keyed by its exact hyperplane: 0 dependent, 0 duplicate; every 5-subset (962,598) by integer determinant: 0 zero, min |det| 2",
   "size": 43,
   "n": 18,
   "engine": "python-int",
   "file": "milesandmistakes-v1.1.1/RECORD_43_points_n18.json"
  },
  {
   "id": "aerp-p60-mm-n17-44",
   "claim": "C(17) >= 44: RECORD_44_points_n17.json",
   "where": "#6 milesandmistakes, release v1.1.1 (cffabd12)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (135,751) keyed by its exact hyperplane: 0 dependent, 0 duplicate; every 5-subset (1,086,008) by integer determinant: 0 zero, min |det| 2",
   "size": 44,
   "n": 17,
   "engine": "python-int",
   "file": "milesandmistakes-v1.1.1/RECORD_44_points_n17.json"
  },
  {
   "id": "aerp-p60-mm-n18-45",
   "claim": "C(18) >= 45: RECORD_45_points_n18.json",
   "where": "#6 milesandmistakes, release v1.1.1 (cffabd12)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (148,995) keyed by its exact hyperplane: 0 dependent, 0 duplicate; every 5-subset (1,221,759) by integer determinant: 0 zero, min |det| 2",
   "size": 45,
   "n": 18,
   "engine": "python-int",
   "file": "milesandmistakes-v1.1.1/RECORD_45_points_n18.json"
  },
  {
   "id": "aerp-p60-mm-n19-46",
   "claim": "C(19) >= 46: RECORD_46_points_n19.json",
   "where": "#6 milesandmistakes, release v1.1.1 (cffabd12)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (163,185) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 46,
   "n": 19,
   "engine": "python-int",
   "file": "milesandmistakes-v1.1.1/RECORD_46_points_n19.json"
  },
  {
   "id": "aerp-p60-mm-n19-49",
   "claim": "C(19) >= 49: RECORD_49_points_n19.json",
   "where": "#6 milesandmistakes, release v1.1.1 (cffabd12)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (211,876) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 49,
   "n": 19,
   "engine": "python-int",
   "file": "milesandmistakes-v1.1.1/RECORD_49_points_n19.json"
  },
  {
   "id": "aerp-p60-mm-n20-50",
   "claim": "C(20) >= 50: RECORD_50_points_n20.json",
   "where": "#6 milesandmistakes, release v1.1.1 (cffabd12)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (230,300) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 50,
   "n": 20,
   "engine": "python-int",
   "file": "milesandmistakes-v1.1.1/RECORD_50_points_n20.json"
  },
  {
   "id": "aerp-p60-0thernet-n17-45",
   "claim": "C(17) >= 45: the no-five-on-sphere-17 set of hraness/algal-lab",
   "where": "#6 0thernet, 2026-09-25 (d881f970)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (148,995) keyed by its exact hyperplane: 0 dependent, 0 duplicate; every 5-subset (1,221,759) by integer determinant: 0 zero, min |det| 2",
   "size": 45,
   "n": 17,
   "engine": "python-int",
   "file": "hraness-algal-lab/no-five-on-sphere-frontier.json#no-five-on-sphere-17",
   "recordedBestTheyCite": "44"
  },
  {
   "id": "aerp-p60-0thernet-n18-48",
   "claim": "C(18) >= 48: the no-five-on-sphere-18 set of hraness/algal-lab",
   "where": "#6 0thernet, 2026-09-25 (d881f970)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (194,580) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 48,
   "n": 18,
   "engine": "python-int",
   "file": "hraness-algal-lab/no-five-on-sphere-frontier.json#no-five-on-sphere-18",
   "recordedBestTheyCite": "45"
  },
  {
   "id": "aerp-p60-0thernet-n19-50",
   "claim": "C(19) >= 50: the no-five-on-sphere-19 set of hraness/algal-lab",
   "where": "#6 0thernet, 2026-09-25 (d881f970)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (230,300) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 50,
   "n": 19,
   "engine": "python-int",
   "file": "hraness-algal-lab/no-five-on-sphere-frontier.json#no-five-on-sphere-19",
   "recordedBestTheyCite": "49"
  },
  {
   "id": "aerp-p60-0thernet-n20-53",
   "claim": "C(20) >= 53: the no-five-on-sphere-20 set of hraness/algal-lab",
   "where": "#6 0thernet, 2026-09-25 (d881f970)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (292,825) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 53,
   "n": 20,
   "engine": "python-int",
   "file": "hraness-algal-lab/no-five-on-sphere-frontier.json#no-five-on-sphere-20",
   "recordedBestTheyCite": "50"
  },
  {
   "id": "aerp-p60-0thernet-n21-55",
   "claim": "C(21) >= 55: the no-five-on-sphere-21 set of hraness/algal-lab",
   "where": "#6 0thernet, 2026-09-25 (d881f970)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (341,055) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 55,
   "n": 21,
   "engine": "python-int",
   "file": "hraness-algal-lab/no-five-on-sphere-frontier.json#no-five-on-sphere-21",
   "recordedBestTheyCite": "50"
  },
  {
   "id": "aerp-p60-0thernet-n22-58",
   "claim": "C(22) >= 58: the no-five-on-sphere-22 set of hraness/algal-lab",
   "where": "#6 0thernet, 2026-09-25 (d881f970)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (424,270) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 58,
   "n": 22,
   "engine": "python-int",
   "file": "hraness-algal-lab/no-five-on-sphere-frontier.json#no-five-on-sphere-22",
   "recordedBestTheyCite": "50"
  },
  {
   "id": "aerp-p60-0thernet-n23-59",
   "claim": "C(23) >= 59: the no-five-on-sphere-23 set of hraness/algal-lab",
   "where": "#6 0thernet, 2026-09-25 (d881f970)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (455,126) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 59,
   "n": 23,
   "engine": "python-int",
   "file": "hraness-algal-lab/no-five-on-sphere-frontier.json#no-five-on-sphere-23",
   "recordedBestTheyCite": "51"
  },
  {
   "id": "aerp-p60-0thernet-n24-62",
   "claim": "C(24) >= 62: the no-five-on-sphere-24 set of hraness/algal-lab",
   "where": "#6 0thernet, 2026-09-25 (d881f970)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (557,845) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 62,
   "n": 24,
   "engine": "python-int",
   "file": "hraness-algal-lab/no-five-on-sphere-frontier.json#no-five-on-sphere-24",
   "recordedBestTheyCite": "53"
  },
  {
   "id": "aerp-p60-0thernet-n25-65",
   "claim": "C(25) >= 65: the no-five-on-sphere-25 set of hraness/algal-lab",
   "where": "#6 0thernet, 2026-09-25 (d881f970)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (677,040) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 65,
   "n": 25,
   "engine": "python-int",
   "file": "hraness-algal-lab/no-five-on-sphere-frontier.json#no-five-on-sphere-25",
   "recordedBestTheyCite": "55"
  },
  {
   "id": "aerp-p60-0thernet-n26-67",
   "claim": "C(26) >= 67: the no-five-on-sphere-26 set of hraness/algal-lab",
   "where": "#6 0thernet, 2026-09-25 (d881f970)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (766,480) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 67,
   "n": 26,
   "engine": "python-int",
   "file": "hraness-algal-lab/no-five-on-sphere-frontier.json#no-five-on-sphere-26",
   "recordedBestTheyCite": "56"
  },
  {
   "id": "aerp-p60-hashkanna-n15-41",
   "claim": "C(15) >= 41: n15_41.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (101,270) keyed by its exact hyperplane: 0 dependent, 0 duplicate; every 5-subset (749,398) by integer determinant: 0 zero, min |det| 2",
   "size": 41,
   "n": 15,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n15_41.json"
  },
  {
   "id": "aerp-p60-hashkanna-n15-42",
   "claim": "C(15) >= 42: n15_42.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (111,930) keyed by its exact hyperplane: 0 dependent, 0 duplicate; every 5-subset (850,668) by integer determinant: 0 zero, min |det| 2",
   "size": 42,
   "n": 15,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n15_42.json"
  },
  {
   "id": "aerp-p60-hashkanna-n16-43",
   "claim": "C(16) >= 43: n16_43.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (123,410) keyed by its exact hyperplane: 0 dependent, 0 duplicate; every 5-subset (962,598) by integer determinant: 0 zero, min |det| 2",
   "size": 43,
   "n": 16,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n16_43.json"
  },
  {
   "id": "aerp-p60-hashkanna-n16-44",
   "claim": "C(16) >= 44: n16_44.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (135,751) keyed by its exact hyperplane: 0 dependent, 0 duplicate; every 5-subset (1,086,008) by integer determinant: 0 zero, min |det| 2",
   "size": 44,
   "n": 16,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n16_44.json"
  },
  {
   "id": "aerp-p60-hashkanna-n17-46",
   "claim": "C(17) >= 46: n17_46.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (163,185) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 46,
   "n": 17,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n17_46.json"
  },
  {
   "id": "aerp-p60-hashkanna-n19-51",
   "claim": "C(19) >= 51: n19_51.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (249,900) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 51,
   "n": 19,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n19_51.json"
  },
  {
   "id": "aerp-p60-hashkanna-n20-54",
   "claim": "C(20) >= 54: n20_54.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (316,251) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 54,
   "n": 20,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n20_54.json"
  },
  {
   "id": "aerp-p60-hashkanna-n21-56",
   "claim": "C(21) >= 56: n21_56.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (367,290) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 56,
   "n": 21,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n21_56.json"
  },
  {
   "id": "aerp-p60-hashkanna-n22-59",
   "claim": "C(22) >= 59: n22_59.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (455,126) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 59,
   "n": 22,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n22_59.json"
  },
  {
   "id": "aerp-p60-hashkanna-n23-60_centrosymmetric",
   "claim": "C(23) >= 60: n23_60_centrosymmetric.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (487,635) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 60,
   "n": 23,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n23_60_centrosymmetric.json"
  },
  {
   "id": "aerp-p60-hashkanna-n23-60",
   "claim": "C(23) >= 60: n23_60.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (487,635) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 60,
   "n": 23,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n23_60.json"
  },
  {
   "id": "aerp-p60-hashkanna-n23-61",
   "claim": "C(23) >= 61: n23_61.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (521,855) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 61,
   "n": 23,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n23_61.json"
  },
  {
   "id": "aerp-p60-hashkanna-n23-62",
   "claim": "C(23) >= 62: n23_62.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (557,845) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 62,
   "n": 23,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n23_62.json"
  },
  {
   "id": "aerp-p60-hashkanna-n24-64",
   "claim": "C(24) >= 64: n24_64.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (635,376) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 64,
   "n": 24,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n24_64.json"
  },
  {
   "id": "aerp-p60-hashkanna-n25-66",
   "claim": "C(25) >= 66: n25_66.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (720,720) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 66,
   "n": 25,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n25_66.json"
  },
  {
   "id": "aerp-p60-hashkanna-n26-68",
   "claim": "C(26) >= 68: n26_68.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (814,385) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 68,
   "n": 26,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n26_68.json"
  },
  {
   "id": "aerp-p60-hashkanna-n26-69",
   "claim": "C(26) >= 69: n26_69.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (864,501) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 69,
   "n": 26,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n26_69.json"
  },
  {
   "id": "aerp-p60-hashkanna-n26-70",
   "claim": "C(26) >= 70: n26_70.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (916,895) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 70,
   "n": 26,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n26_70.json"
  },
  {
   "id": "aerp-p60-hashkanna-n27-71",
   "claim": "C(27) >= 71: n27_71.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (971,635) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 71,
   "n": 27,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n27_71.json"
  },
  {
   "id": "aerp-p60-hashkanna-n27-72",
   "claim": "C(27) >= 72: n27_72.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (1,028,790) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 72,
   "n": 27,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n27_72.json"
  },
  {
   "id": "aerp-p60-hashkanna-n28-74",
   "claim": "C(28) >= 74: n28_74.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (1,150,626) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 74,
   "n": 28,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n28_74.json"
  },
  {
   "id": "aerp-p60-hashkanna-n29-76",
   "claim": "C(29) >= 76: n29_76.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (1,282,975) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 76,
   "n": 29,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n29_76.json"
  },
  {
   "id": "aerp-p60-hashkanna-n30-79",
   "claim": "C(30) >= 79: n30_79.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (1,502,501) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 79,
   "n": 30,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n30_79.json"
  },
  {
   "id": "aerp-p60-hashkanna-n30-80",
   "claim": "C(30) >= 80: n30_80.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (1,581,580) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 80,
   "n": 30,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n30_80.json"
  },
  {
   "id": "aerp-p60-hashkanna-n31-80",
   "claim": "C(31) >= 80: n31_80.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (1,581,580) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 80,
   "n": 31,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n31_80.json"
  },
  {
   "id": "aerp-p60-hashkanna-n31-81",
   "claim": "C(31) >= 81: n31_81.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (1,663,740) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 81,
   "n": 31,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n31_81.json"
  },
  {
   "id": "aerp-p60-hashkanna-n31-82",
   "claim": "C(31) >= 82: n31_82.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (1,749,060) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 82,
   "n": 31,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n31_82.json"
  },
  {
   "id": "aerp-p60-hashkanna-n32-84",
   "claim": "C(32) >= 84: n32_84.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (1,929,501) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 84,
   "n": 32,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n32_84.json"
  },
  {
   "id": "aerp-p60-hashkanna-n32-85",
   "claim": "C(32) >= 85: n32_85.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (2,024,785) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 85,
   "n": 32,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n32_85.json"
  },
  {
   "id": "aerp-p60-hashkanna-n33-86",
   "claim": "C(33) >= 86: n33_86.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (2,123,555) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 86,
   "n": 33,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n33_86.json"
  },
  {
   "id": "aerp-p60-hashkanna-n34-90",
   "claim": "C(34) >= 90: n34_90.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (2,555,190) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 90,
   "n": 34,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n34_90.json"
  },
  {
   "id": "aerp-p60-hashkanna-n35-91",
   "claim": "C(35) >= 91: n35_91.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (2,672,670) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 91,
   "n": 35,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n35_91.json"
  },
  {
   "id": "aerp-p60-hashkanna-n36-94",
   "claim": "C(36) >= 94: n36_94.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (3,049,501) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 94,
   "n": 36,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n36_94.json"
  },
  {
   "id": "aerp-p60-hashkanna-n37-96",
   "claim": "C(37) >= 96: n37_96.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (3,321,960) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 96,
   "n": 37,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n37_96.json"
  },
  {
   "id": "aerp-p60-hashkanna-n38-98",
   "claim": "C(38) >= 98: n38_98.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (3,612,280) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 98,
   "n": 38,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n38_98.json"
  },
  {
   "id": "aerp-p60-hashkanna-n38-99",
   "claim": "C(38) >= 99: n38_99.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (3,764,376) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 99,
   "n": 38,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n38_99.json"
  },
  {
   "id": "aerp-p60-hashkanna-n39-100",
   "claim": "C(39) >= 100: n39_100.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (3,921,225) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 100,
   "n": 39,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n39_100.json"
  },
  {
   "id": "aerp-p60-hashkanna-n39-102",
   "claim": "C(39) >= 102: n39_102.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (4,249,575) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 102,
   "n": 39,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n39_102.json"
  },
  {
   "id": "aerp-p60-hashkanna-n40-103",
   "claim": "C(40) >= 103: n40_103.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (4,421,275) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 103,
   "n": 40,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n40_103.json"
  },
  {
   "id": "aerp-p60-hashkanna-n40-104",
   "claim": "C(40) >= 104: n40_104.json",
   "where": "#6 hashkanna, 2026-10-03 (5a3b9668)",
   "verdict": "CERTIFIED",
   "how": "every 4-subset (4,598,126) keyed by its exact hyperplane: 0 dependent, 0 duplicate",
   "size": 104,
   "n": 40,
   "engine": "python-int",
   "file": "hashkanna-no5sphere/n40_104.json"
  },
  {
   "id": "aerp-p61-47-42",
   "claim": "C >= 47/42: the m = 15 rational instance of #11 has alpha = 47/42 exactly (under either k-range)",
   "where": "#11 VividhDesign, 2026-10-06 (the notebook's float instance: 1.11904756...)",
   "verdict": "CERTIFIED",
   "how": "all 2^15 sign patterns in exact rationals: alpha = 47/42 under k = 1..m-1 (the notebook), 47/42 under k = 1..m, 47/42 padded with (0,0) at both ends under the notebook convention (m = 17)",
   "m": 15,
   "alphaNotebook": "47/42",
   "alphaFull": "47/42",
   "alphaPaddedNotebook": "47/42",
   "attainedFull": 41,
   "claimed": "47/42"
  },
  {
   "id": "aerp-p61-9-8-m14",
   "claim": "C >= 9/8: the m = 14 instance of #12 (entries in Z/16) has alpha = 9/8 under k = 1..m, 3/4 under the notebook's k = 1..m-1, and 9/8 padded to m = 16 under the notebook's convention",
   "where": "#12 Maks23, 2026-10-08",
   "verdict": "CERTIFIED",
   "how": "all 2^14 sign patterns in exact rationals: alpha = 3/4 under k = 1..m-1 (the notebook), 9/8 under k = 1..m, 9/8 padded with (0,0) at both ends under the notebook convention (m = 16)",
   "m": 14,
   "alphaNotebook": "3/4",
   "alphaFull": "9/8",
   "alphaPaddedNotebook": "9/8",
   "attainedFull": 52,
   "claimed": "9/8",
   "file": "attachments/ring_loading_98/min98_m14_D16.txt"
  },
  {
   "id": "aerp-p61-9-8-m15",
   "claim": "C >= 9/8: the m = 15 instance of #12's zip",
   "where": "#12 Maks23 (ring_loading_98.zip)",
   "verdict": "CERTIFIED",
   "how": "all 2^15 sign patterns in exact rationals: alpha = 3/4 under k = 1..m-1 (the notebook), 9/8 under k = 1..m, 9/8 padded with (0,0) at both ends under the notebook convention (m = 17)",
   "m": 15,
   "alphaNotebook": "3/4",
   "alphaFull": "9/8",
   "alphaPaddedNotebook": "9/8",
   "attainedFull": 61,
   "claimed": "9/8",
   "file": "attachments/ring_loading_98/rec98_D16.txt"
  },
  {
   "id": "aerp-p61-9-8-m17",
   "claim": "C >= 9/8 under the notebook's own k = 1..m-1 without padding: the m = 17 instance of #12's zip (denominator 560)",
   "where": "#12 Maks23 (ring_loading_98.zip)",
   "verdict": "CERTIFIED",
   "how": "all 2^17 sign patterns in exact rationals: alpha = 9/8 under k = 1..m-1 (the notebook), 9/8 under k = 1..m, 9/8 padded with (0,0) at both ends under the notebook convention (m = 19)",
   "m": 17,
   "alphaNotebook": "9/8",
   "alphaFull": "9/8",
   "alphaPaddedNotebook": "9/8",
   "attainedFull": 74,
   "claimed": "9/8",
   "file": "attachments/ring_loading_98/A17_exact_D560.txt"
  },
  {
   "id": "aerp-p61-9-8-daeubel-12",
   "claim": "C >= 9/8 with 12 pairs: karldaeubel's instance in the #12 thread (two identical six-pair blocks)",
   "where": "#12 comment, karldaeubel, 2026-10-13",
   "verdict": "CERTIFIED",
   "how": "all 2^12 sign patterns in exact rationals: alpha = 3/4 under k = 1..m-1 (the notebook), 9/8 under k = 1..m, 9/8 padded with (0,0) at both ends under the notebook convention (m = 14)",
   "m": 12,
   "alphaNotebook": "3/4",
   "alphaFull": "9/8",
   "alphaPaddedNotebook": "9/8",
   "attainedFull": 32,
   "claimed": "9/8"
  },
  {
   "id": "aerp-p42-far-block-1163",
   "claim": "C >= g(A) > 1.163 for the explicit far-block set A of #5's certificate script at L = 100000 (|A| = 101478), rebuilt here from its stated rule",
   "where": "#5 there4iam, 2026-07-31 (materials.zip, 3_CERTIFICATE.py read as text; Penman--Wells 2013 give 1.12594, the repository 1.12194)",
   "verdict": "CERTIFIED",
   "how": "|A+A| = 1,200,837 and |A-A| = 848,817 by exact bitset sumsets; g > 1163/1000 decided as the integer inequality |A+A|^1000 |A|^163 > |A-A|^1163",
   "size": 101478,
   "sumset": 1200837,
   "differenceSet": 848817,
   "g": 1.1633381439228145,
   "rule": {
    "L": 100000,
    "K": 140000,
    "q": 264,
    "x": -320000,
    "basisSize": 1057
   }
  },
  {
   "id": "aerp-p42-strong-form-counterexample",
   "claim": "the explicit 23,646-integer set of #5 \u00a75 violates |A+A| + |A| <= |A-A| + diam(A) + 1 (the strong form of its own conjecture), by 89",
   "where": "#5 there4iam (materials.zip, 4_STRONG_conjecture_counterexample.txt, one integer per line)",
   "verdict": "CERTIFIED",
   "how": "|A| = 23646, |A+A| = 6,924,316, |A-A| = 3,370,545, diam = 3,577,327 by exact bitset sumsets; |A+A| + |A| - |A-A| - diam - 1 = 89",
   "size": 23646,
   "sumset": 6924316,
   "differenceSet": 3370545,
   "diameter": 3577327,
   "g": 1.1451648581767873,
   "slack": 89
  },
  {
   "id": "aerp-p42-c-ge-1.6269",
   "claim": "C >= 1.6269 by the \"hollow bridge\" instance of #5 (43 gadgets, an 86-digit modulus)",
   "where": "#5 there4iam (1_WRITEUP.md \u00a72)",
   "verdict": "REFUSED",
   "how": "the set is not enumerable (modulus ~10^86); its count rests on the write-up's own argument, which is machine-generated and unreviewed by its authors' statement; nothing finite to decide here",
   "note": "C = 2 itself is now a theorem: Lin--Li, arXiv:2607.27199 (29 Jul 2026), which #5's author acknowledges in the thread; cited, not decided here"
  },
  {
   "id": "aerp-p64-sofa-1.8558",
   "claim": "a 3-D moving sofa of volume >= 1.8558404 (#8; the repository records >= 1.81)",
   "where": "#8 Lomnus-ai, 2026-08-21 (gist: pose paths, JAX notebooks)",
   "verdict": "REFUSED",
   "how": "the certificate is a floating-point cube count under stated position and angle tolerances (0.005, 0.05 degrees; the author's own note: 1.7701 at the quick-start tolerances); no exact object is published. To decide it an interval instrument would have to enclose every pose, prove the swept region contains each counted cube and sum a rational lower bound \u2014 not built here",
   "note": "the repository's own pipeline (JAX, GPU) is what was run; a measurement, not a certificate"
  }
 ],
 "controls": [
  {
   "id": "p59-notebook-sol_64",
   "n": 64,
   "size": 112,
   "verdict": "CERTIFIED",
   "isoscelesTriples": 0,
   "expected": "the notebook reports 112 points admissible"
  },
  {
   "id": "p59-notebook-sol_100",
   "n": 100,
   "size": 164,
   "verdict": "CERTIFIED",
   "isoscelesTriples": 0,
   "expected": "the notebook reports 164 points admissible"
  },
  {
   "id": "p60-notebook-n7",
   "n": 7,
   "size": 21,
   "verdict": "CERTIFIED",
   "minAbsDet": 2,
   "expected": "the notebook reports it admissible"
  },
  {
   "id": "p60-notebook-n8",
   "n": 8,
   "size": 23,
   "verdict": "CERTIFIED",
   "minAbsDet": 2,
   "expected": "the notebook reports it admissible"
  },
  {
   "id": "p60-notebook-n9",
   "n": 9,
   "size": 26,
   "verdict": "CERTIFIED",
   "minAbsDet": 2,
   "expected": "the notebook reports it admissible"
  },
  {
   "id": "p60-notebook-n10",
   "n": 10,
   "size": 28,
   "verdict": "CERTIFIED",
   "minAbsDet": 2,
   "expected": "the notebook reports it admissible"
  },
  {
   "id": "p60-notebook-n11",
   "n": 11,
   "size": 31,
   "verdict": "CERTIFIED",
   "minAbsDet": 2,
   "expected": "the notebook reports it admissible"
  },
  {
   "id": "p60-notebook-n12",
   "n": 12,
   "size": 33,
   "verdict": "CERTIFIED",
   "minAbsDet": 2,
   "expected": "the notebook reports it admissible"
  },
  {
   "id": "p42-notebook-best_list",
   "size": 309,
   "sumset": 1367,
   "differenceSet": 1163,
   "g": 1.1219357374860444,
   "expected": "the notebook prints 1.1219357..."
  }
 ],
 "p60BestCertifiedPerN": {
  "13": 36,
  "14": 38,
  "15": 42,
  "16": 44,
  "17": 46,
  "18": 48,
  "19": 51,
  "20": 54,
  "21": 56,
  "22": 59,
  "23": 62,
  "24": 64,
  "25": 66,
  "26": 70,
  "27": 72,
  "28": 74,
  "29": 76,
  "30": 80,
  "31": 82,
  "32": 85,
  "33": 86,
  "34": 90,
  "35": 91,
  "36": 94,
  "37": 96,
  "38": 99,
  "39": 102,
  "40": 104
 },
 "p60RepositoryBest": {
  "7": 21,
  "8": 23,
  "9": 26,
  "10": 28,
  "11": 31,
  "12": 33
 },
 "counts": {
  "CERTIFIED": 70,
  "REFUTED": 0,
  "REFUSED": 2
 },
 "decider": {
  "p59.py": "2431b0cff8bda4fc6961e670bba7b12b09bd77f85ad78d5be053ff3a3195050f",
  "p60.py": "f399e41a9f38424f569010a568ea0f8d47be4c8a002afee34eea66fecb113748",
  "p61.py": "af2e94e0d1a529a4c13b82db9b66a8c1f0e879e0068eb0456674a923a46a1d70",
  "p42.py": "433e252f8bf3b3830d0fd5d6a4a15bf4e5f8272c4f1d91ef4eb58eeb951547f1"
 },
 "python": "3.9.6",
 "numpy": false,
 "seconds": 427.7,
 "generated": "2026-10-15T20:55:07Z"
}