{
  "title": "Certified prime desert via hybrid covering system",
  "author": "Simon-Pierre Boucher \u2014 contact@spboucher.ai",
  "date": "2026-08-06",
  "claim": "N and N+260 are consecutive primes (gap exactly 260).",
  "N": "1116336781708038449369693",
  "gap": 260,
  "merit": 4.69551,
  "csg_ratio": 0.084799,
  "construction": {
    "prime_set_max": 59,
    "primorial_P": "1922760350154212639070",
    "crt_residue_x": "1135778618595118709093",
    "shift_t": 580,
    "classes_a_p": {
      "2": 1,
      "3": 1,
      "5": 2,
      "7": 6,
      "11": 8,
      "13": 1,
      "17": 16,
      "19": 6,
      "23": 3,
      "29": 2,
      "31": 18,
      "37": 24,
      "41": 4,
      "43": 9,
      "47": 9,
      "53": 28,
      "59": 38
    },
    "holes": [
      36,
      54,
      68,
      78,
      108,
      110,
      114,
      116,
      126,
      128,
      180,
      194,
      198,
      200,
      218,
      236
    ],
    "note": "positions with a covering factor are composite for EVERY shift t (proven); hole positions are certified composite for THIS t by explicit factor or strong MR witness"
  },
  "interior_certificates": {
    "1": {
      "type": "covering_factor",
      "q": 2
    },
    "2": {
      "type": "covering_factor",
      "q": 5
    },
    "3": {
      "type": "covering_factor",
      "q": 2
    },
    "4": {
      "type": "covering_factor",
      "q": 3
    },
    "5": {
      "type": "covering_factor",
      "q": 2
    },
    "6": {
      "type": "covering_factor",
      "q": 7
    },
    "7": {
      "type": "covering_factor",
      "q": 2
    },
    "8": {
      "type": "covering_factor",
      "q": 11
    },
    "9": {
      "type": "covering_factor",
      "q": 2
    },
    "10": {
      "type": "covering_factor",
      "q": 3
    },
    "11": {
      "type": "covering_factor",
      "q": 2
    },
    "12": {
      "type": "covering_factor",
      "q": 5
    },
    "13": {
      "type": "covering_factor",
      "q": 2
    },
    "14": {
      "type": "covering_factor",
      "q": 13
    },
    "15": {
      "type": "covering_factor",
      "q": 2
    },
    "16": {
      "type": "covering_factor",
      "q": 3
    },
    "17": {
      "type": "covering_factor",
      "q": 2
    },
    "18": {
      "type": "covering_factor",
      "q": 31
    },
    "19": {
      "type": "covering_factor",
      "q": 2
    },
    "20": {
      "type": "covering_factor",
      "q": 7
    },
    "21": {
      "type": "covering_factor",
      "q": 2
    },
    "22": {
      "type": "covering_factor",
      "q": 3
    },
    "23": {
      "type": "covering_factor",
      "q": 2
    },
    "24": {
      "type": "covering_factor",
      "q": 37
    },
    "25": {
      "type": "covering_factor",
      "q": 2
    },
    "26": {
      "type": "covering_factor",
      "q": 23
    },
    "27": {
      "type": "covering_factor",
      "q": 2
    },
    "28": {
      "type": "covering_factor",
      "q": 3
    },
    "29": {
      "type": "covering_factor",
      "q": 2
    },
    "30": {
      "type": "covering_factor",
      "q": 11
    },
    "31": {
      "type": "covering_factor",
      "q": 2
    },
    "32": {
      "type": "covering_factor",
      "q": 5
    },
    "33": {
      "type": "covering_factor",
      "q": 2
    },
    "34": {
      "type": "covering_factor",
      "q": 3
    },
    "35": {
      "type": "covering_factor",
      "q": 2
    },
    "36": {
      "type": "mr_witness",
      "a": 2
    },
    "37": {
      "type": "covering_factor",
      "q": 2
    },
    "38": {
      "type": "covering_factor",
      "q": 59
    },
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    "40": {
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      "q": 3
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      "q": 2
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      "q": 5
    },
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      "q": 19
    },
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      "q": 7
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    "53": {
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    },
    "54": {
      "type": "trial_factor",
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      "q": 13
    },
    "67": {
      "type": "covering_factor",
      "q": 2
    },
    "68": {
      "type": "trial_factor",
      "q": 18401
    },
    "69": {
      "type": "covering_factor",
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      "type": "covering_factor",
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    "257": {
      "type": "covering_factor",
      "q": 2
    },
    "258": {
      "type": "covering_factor",
      "q": 7
    },
    "259": {
      "type": "covering_factor",
      "q": 2
    }
  },
  "endpoint_primality": {
    "method": "deterministic Miller-Rabin, bases {2..37} (12 bases), valid for n < 3.317e24 (Sorenson-Webster 2015)",
    "validity_bound": "3317044064679887385961981",
    "both_endpoints_below_bound": true
  },
  "epistemic_status": "CERTIFIED construction; NOT a record (see records.md: merit record 41.94, largest maximal gap 1854). Baseline with same primes: classic primorial gives gap 61; pure covering (attempt 1) gave gap 90."
}