spb/prime-mystery-engine
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1{2 "title": "Certified prime desert via hybrid covering system",3 "author": "Simon-Pierre Boucher \u2014 contact@spboucher.ai",4 "date": "2026-08-06",5 "claim": "N and N+260 are consecutive primes (gap exactly 260).",6 "N": "1116336781708038449369693",7 "gap": 260,8 "merit": 4.69551,9 "csg_ratio": 0.084799,10 "construction": {11 "prime_set_max": 59,12 "primorial_P": "1922760350154212639070",13 "crt_residue_x": "1135778618595118709093",14 "shift_t": 580,15 "classes_a_p": {16 "2": 1,17 "3": 1,18 "5": 2,19 "7": 6,20 "11": 8,21 "13": 1,22 "17": 16,23 "19": 6,24 "23": 3,25 "29": 2,26 "31": 18,27 "37": 24,28 "41": 4,29 "43": 9,30 "47": 9,31 "53": 28,32 "59": 3833 },34 "holes": [35 36,36 54,37 68,38 78,39 108,40 110,41 114,42 116,43 126,44 128,45 180,46 194,47 198,48 200,49 218,50 23651 ],52 "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"53 },54 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"type": "covering_factor",1021 "q": 51022 },1023 "243": {1024 "type": "covering_factor",1025 "q": 21026 },1027 "244": {1028 "type": "covering_factor",1029 "q": 31030 },1031 "245": {1032 "type": "covering_factor",1033 "q": 21034 },1035 "246": {1036 "type": "covering_factor",1037 "q": 371038 },1039 "247": {1040 "type": "covering_factor",1041 "q": 21042 },1043 "248": {1044 "type": "covering_factor",1045 "q": 131046 },1047 "249": {1048 "type": "covering_factor",1049 "q": 21050 },1051 "250": {1052 "type": "covering_factor",1053 "q": 31054 },1055 "251": {1056 "type": "covering_factor",1057 "q": 21058 },1059 "252": {1060 "type": "covering_factor",1061 "q": 51062 },1063 "253": {1064 "type": "covering_factor",1065 "q": 21066 },1067 "254": {1068 "type": "covering_factor",1069 "q": 171070 },1071 "255": {1072 "type": "covering_factor",1073 "q": 21074 },1075 "256": {1076 "type": "covering_factor",1077 "q": 31078 },1079 "257": {1080 "type": "covering_factor",1081 "q": 21082 },1083 "258": {1084 "type": "covering_factor",1085 "q": 71086 },1087 "259": {1088 "type": "covering_factor",1089 "q": 21090 }1091 },1092 "endpoint_primality": {1093 "method": "deterministic Miller-Rabin, bases {2..37} (12 bases), valid for n < 3.317e24 (Sorenson-Webster 2015)",1094 "validity_bound": "3317044064679887385961981",1095 "both_endpoints_below_bound": true1096 },1097 "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."1098}