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651 ] }, { "X": [ 45, 76, 238, 289, 402, 455, 618 ], "Z": [ 37, 40, 71, 124, 161, 177, 181, 197, 234, 287, 318, 321, 376, 388, 407, 495, 497, 569, 579, 582, 620, 649, 651, 652 ] }, { "X": [ 77, 167, 239, 290, 403, 456, 619 ], "Z": [ 38, 41, 72, 125, 162, 178, 182, 198, 235, 288, 319, 322, 377, 389, 408, 496, 498, 570, 580, 583, 621, 650, 652, 653 ] }, { "X": [ 78, 240, 291, 404, 620 ], "Z": [ 39, 42, 73, 126, 163, 179, 183, 199, 236, 289, 320, 323, 378, 390, 409, 497, 499, 571, 581, 584, 622, 651, 653, 654 ] }, { "X": [ 167, 169, 241, 405, 568, 621 ], "Z": [ 40, 43, 74, 127, 164, 180, 184, 200, 237, 290, 321, 324, 379, 391, 410, 498, 500, 572, 582, 585, 623, 652, 654, 655 ] }, { "X": [ 49, 80, 170, 242, 406, 569, 622 ], "Z": [ 41, 44, 75, 128, 165, 181, 185, 201, 238, 291, 322, 325, 380, 392, 411, 499, 501, 573, 583, 586, 624, 653, 655, 656 ] }, { "X": [ 50, 169, 171, 243, 376, 407, 570 ], "Z": [ 42, 45, 76, 129, 166, 182, 186, 202, 239, 292, 323, 326, 381, 393, 412, 500, 502, 574, 584, 587, 625, 654, 656, 657 ] }, { "X": [ 51, 170, 172, 244, 408, 498, 571 ], "Z": [ 43, 46, 77, 130, 167, 183, 187, 203, 240, 293, 324, 327, 382, 394, 413, 501, 503, 575, 585, 588, 626, 655, 657, 658 ] }, { "X": [ 52, 171, 173, 258, 409, 572 ], "Z": [ 44, 47, 78, 131, 168, 184, 188, 204, 241, 294, 325, 328, 383, 395, 414, 502, 504, 576, 586, 589, 627, 656, 658, 659 ] }, { "X": [ 53, 172, 376, 500, 520, 573 ], "Z": [ 45, 48, 79, 132, 169, 185, 189, 205, 242, 295, 326, 329, 384, 396, 415, 503, 505, 577, 587, 590, 628, 657, 659, 660 ] }, { "X": [ 54, 173, 247, 380, 411, 501, 521, 574 ], "Z": [ 46, 49, 80, 133, 170, 186, 190, 206, 243, 296, 327, 330, 385, 397, 416, 504, 506, 578, 588, 591, 629, 658, 660, 661 ] } ] }, "distance": { "d": 95, "P": { "value": 95, "confidence": "upper_bound", "witness": { "X": [ 0, 18, 36, 43, 44, 48, 58, 59, 60, 66, 69, 70, 86, 89, 93, 102, 106, 107, 108, 111, 112, 114, 116, 118, 126, 128, 130, 133, 138, 140, 157, 159, 160, 163, 164, 165, 166, 179, 185, 197, 205, 207, 211, 229, 232, 254, 255, 257, 259, 268, 269, 273, 275, 276, 278, 294, 295, 298, 301, 302, 304, 305, 307, 319, 322, 326, 330 ], "Z": [ 0, 18, 36, 43, 44, 48, 58, 59, 60, 66, 69, 70, 86, 89, 93, 102, 106, 107, 108, 111, 112, 114, 116, 118, 126, 128, 130, 133, 138, 140, 157, 159, 160, 163, 164, 165, 166, 179, 185, 197, 205, 207, 211, 229, 232, 254, 255, 257, 259, 268, 269, 273, 275, 276, 278, 294, 295, 298, 301, 302, 304, 305, 307, 319, 322, 326, 330, 333, 335, 356, 359, 361, 380, 383, 385, 403, 404, 411, 446, 451, 453, 456, 500, 502, 522, 545, 547, 578, 589, 595, 618, 622, 642, 643, 648 ] } } }, "provenance": { "authors": [ "@vprusso" ], "construction": "Cyclic GB on Z_331, a=[0, 1, 3, 32, 70, 73, 83, 155, 157, 245, 264, 276], b=[0, 3, 34, 87, 124, 140, 144, 160, 197, 250, 281, 284]; 108 disjoint pairs with images X1=Z1Y2,X2=Y1Z2,Z1=Z1X2,Z2=X1Z2.", "origin": "submission", "date": "2026-10-11", "model": "GPT-6", "notes": "Finite pair-mask repair after a weight94 refutation of the previous mask. Preliminary research candidate; no full gate or exact distance." } }