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[[912,48,5]] d ≤
n
912
k
48
d
5
kd²/n
1.316
w
4
X/Z
1
g
1.32
r
1.4142
layers
1
swaps
0

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Distance

X/Z asymmetry 1 · d_X ≤ 5, d_Z ≤ 5 · w_X = 4, w_Z = 4 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 5 · witness weight 5 (claimed upper_bound)
witness operator (support, 5 qubits)
[644, 703, 762, 812, 862]
d_Z 5 · witness weight 5 (claimed upper_bound)
witness operator (support, 5 qubits)
[838, 839, 840, 841, 887]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Diagnostics

computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking
girth H_X 8 · H_Z 8 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 2–4 (mean 3.694) · H_Z 2–4 (mean 3.694)
qubit degrees H_X 1–2 (mean 1.75) · H_Z 1–2 (mean 1.75)
trapping sets H_X (1,1)×228 (2,0)×48 (3,0)×36 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 228 (1,2): 684 (2,0): 48 (2,1): 540 (2,2): 1656 (3,0): 36 (3,1): 1416 (3,2): 4044 (3,3): 420 (3,4): 912
trapping sets H_Z (1,1)×228 (2,0)×48 (3,0)×36 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 228 (1,2): 684 (2,0): 48 (2,1): 540 (2,2): 1656 (3,0): 36 (3,1): 1416 (3,2): 4044 (3,3): 420 (3,4): 912
witness diameter X 4.0 · Z 4.1231 (Euclidean support diameter of the stored distance witnesses in the layout; an upper bound on the exhibited logicals’ spread, not a minimum over all logicals)

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair
r = 1.414
X checkZ checkqubit site (912)dashed: the pair setting the interaction radiushover a check to isolate its qubits; click to pin — repeated clicks cycle through overlapping checks; click empty space to release
routing cost 0 nearest-neighbor SWAPs per round in total, at most 0 for one check (heuristic: MST lower bound on the layout, with one lattice step = the minimum qubit spacing 1; not a rank)

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction Multi-band dense-packed surface code, generalized from arXiv:2511.06758 (Fujiu et al.): rows x m patches at pitch p on a shared plane (pitch_min regime); data on odd/odd sites, (x+y)%4==2 ancillas carry X-checks, each check supports its diagonal data neighbours; k by exact GF(2) rank. Layout: unit-spacing tilted lattice, interaction radius sqrt(2), single layer. Instance: d=5, rows=5, m=10, pitch=6.
builds on arXiv:2511.06758
date 2026-09-20
family topological (a tag, not a ranking)
locality 2D-local single (computed from the layout)
weight class weight ≤ 4 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

#[[912,48,5]] — multi-band dense-packed surface code, pitch_min point (d=5, rows=5, m=10, pitch=6)

Direction & hypothesis

Target cell: local-2d-single × weight-4, the geometric-efficiency cell — at this family's honest layout (tilted nearest-neighbour lattice, r = √2, single layer) the board's g = 4kd²/(nρ²r⁴) reduces to kd²/n. The submission cap was raised to admit 700 < n ≤ 1000 when w ≤ 8 and d ≤ 40; this family (w = 4, d ≤ 13) fits with room to spare, and the new band is exactly the strip where its boundary overhead thins: the 2026-09-01 campaign (fieldnotes/2026-09-01-multiband-dense-packing-method.md) measured kd²/n ≤ 1.328 under the old n ≤ 700 cap and an asymptote near 1.43 only at n ≥ 13,000. Hypothesis: fresh Pareto points at kd²/n ≈ 1.30–1.34 fill the band, on the d-columns and k-rungs the seeded packs leave open.

What was searched

A manifold re-enumeration of the multi-band dense packing (arXiv:2511.06758 generalized to rows × m patches at a free vertical pitch; builder research/multiband_surface.py, validated bit-exact against research/build_dense_surface.py at d = 3, 5, 7, 9). 138 (d, rows, m, pitch) configurations with n in (700, 1000], d ∈ {5, 7, 9, 11, 13}, in three regimes: two-band (rows = 2, pitch = d − 1), rows ≥ 3 at pitch_min(d) = 2⌊3d/4⌋, and the pitch = d + 1 k-unlock regime (kept only as an expected refutation). k by exact GF(2) rank at every point — never the closed form the campaign's section 6 caught breaking — plus CSS, single-Tanner-component, w ≤ 4 and no-empty-row filters. Board subtraction against the cell's 179 entries (interaction radius computed from the coordinates where a stored field is absent), then intra-sweep Pareto: 25 fresh rank-level points, 14 in distance-preserving regimes. Witness screen at 4,000 RIS trials per side (fast backend): 14/14 held at design distance on both sides.

Evidence trail

This code: d = 5, rows = 5, m = 10, pitch = 6 (rows ≥ 3 at pitch_min(d) = 2⌊3d/4⌋, the distance-preserving multi-band regime); n = 912, k = 48 by exact rank, w = 4, single Tanner component, interaction radius √2.

  • Screening ladder: 4,000 RIS trials/side → lightest logical weight 5 on

    • both sides, nothing lighter.

  • Packaged through the kit's own path (research/kit/submit.make_submission,

    • 8,000-trial witnesses embedded per side), then the trusted gate verify/validate_candidate.py: passed, not refuted, no exact duplicate, no WL-signature equivalent, board_advancing: true, dominated_by: [].

  • Claim, stated precisely: witness-backed upper bound, d ≤ 5 on both

    • sides. No exact claim, no certificate.

Dead ends

  • The 13 pitch = d + 1 k-unlock points (rank-level kd²/n up to 1.374) were

    • excluded without screening: the campaign's Result 3 measured distance collapse below pitch_min for rows ≥ 3 at every d ≥ 7, so their rank-only score is not evidence.

  • [[757,7,11]] (pitch_min(11) = 16) passed the gate but is dominated by the

    • board's codes/667-7-11.json (same k, same d, lower n) — not submitted.

  • d = 9 two-band produced no fresh in-band point; the d = 11 pitch_min

    • point is the dominated [[757,7,11]] above.

  • Sweep-infrastructure bug found and fixed en route: board entries storing

    • interaction_radius: null were silently dropped by the subtraction filter, understating the cell by 38 entries; the gate's own novelty check was authoritative throughout.

Tools

GLM 5.3 Flash, driven interactively in the Zed agent. Repo tooling: the validated builder research/multiband_surface.py, the kit's RIS surrogate with the gf2_fast backend, research/kit/submit.make_submission for packaging, verify/validate_candidate.py as the trusted gate. Compute: 14 gate runs (~90 min each, 7 in parallel) plus screening.

Reproduction

research/multiband_surface.py::build(d=5, rows=5, m=10, pitch=6) rebuilds (H_X, H_Z, coords) exactly. The mask rule: bands r = 0..rows−1 at y₀ = pitch·r; even bands carry m patches at x-offsets j·(2d+2), odd bands m − 1 offset d + 1; occupied sites per patch are window interiors ((x+y)%2==0 inside the (2d−1)² window), vertical edge columns at band phase (x+y)%4 ∈ {0,2}, and horizontal edge rows at the reverse phase, unioned over all patches. Data qubits sit on odd/odd sites; of the rest, (x+y)%4==2 sites measure X-checks, the rest Z-checks; each check supports its diagonal data neighbours. Coordinates are grid/2 (unit spacing), single layer, so the tilted nearest-neighbour checks span √2.

Parity checks

X-checks 432 (max weight 4) · Z-checks 432 (max weight 4)
H_X (432 checks, sparse supports)
[0, 1] [2, 3] [5, 6] [7, 8] [10, 11] [12, 13] [15, 16] [17, 18] [20, 21] [22, 23] [25, 26] [27, 28] [30, 31] [32, 33] [35, 36] [37, 38] [40, 41] [42, 43] [45, 46] [47, 48] [1, 2, 51, 52] [3, 4, 53, 54] [6, 7, 56, 57] [8, 9, 58, 59] [11, 12, 61, 62] [13, 14, 63, 64] [16, 17, 66, 67] [18, 19, 68, 69] [21, 22, 71, 72] [23, 24, 73, 74] [26, 27, 76, 77] [28, 29, 78, 79] [31, 32, 81, 82] [33, 34, 83, 84] [36, 37, 86, 87] [38, 39, 88, 89] [41, 42, 91, 92] [43, 44, 93, 94] [46, 47, 96, 97] [48, 49, 98, 99] [50, 51, 100, 101] [52, 53, 102, 103] [55, 56, 105, 106] [57, 58, 107, 108] [60, 61, 110, 111] [62, 63, 112, 113] [65, 66, 115, 116] [67, 68, 117, 118] [70, 71, 120, 121] [72, 73, 122, 123] [75, 76, 125, 126] [77, 78, 127, 128] [80, 81, 130, 131] [82, 83, 132, 133] [85, 86, 135, 136] [87, 88, 137, 138] [90, 91, 140, 141] [92, 93, 142, 143] [95, 96, 145, 146] [97, 98, 147, 148] [101, 102, 151, 152] [103, 104, 153, 154] [106, 107, 157, 158] [108, 109, 159, 160] [111, 112, 163, 164] [113, 114, 165, 166] [116, 117, 169, 170] [118, 119, 171, 172] [121, 122, 175, 176] [123, 124, 177, 178] [126, 127, 181, 182] [128, 129, 183, 184] [131, 132, 187, 188] [133, 134, 189, 190] [136, 137, 193, 194] [138, 139, 195, 196] [141, 142, 199, 200] [143, 144, 201, 202] [146, 147, 205, 206] [148, 149, 207, 208] [150, 151, 209, 210] [152, 153, 211, 212] [154, 155, 213, 214] [156, 157, 215, 216] [158, 159, 217, 218] [160, 161, 219, 220] [162, 163, 221, 222] [164, 165, 223, 224] [166, 167, 225, 226] [168, 169, 227, 228] [170, 171, 229, 230] [172, 173, 231, 232] [174, 175, 233, 234] [176, 177, 235, 236] [178, 179, 237, 238] [180, 181, 239, 240] [182, 183, 241, 242] [184, 185, 243, 244] [186, 187, 245, 246] [188, 189, 247, 248] [190, 191, 249, 250] [192, 193, 251, 252] [194, 195, 253, 254] [196, 197, 255, 256] [198, 199, 257, 258] [200, 201, 259, 260] [202, 203, 261, 262] [204, 205, 263, 264] [206, 207, 265, 266] [210, 211] [212, 213, 268, 269] [214, 215, 270, 271] [216, 217, 272] [218, 219, 273, 274] [220, 221, 275, 276] [222, 223, 277] [224, 225, 278, 279] [226, 227, 280, 281] [228, 229, 282] [230, 231, 283, 284] [232, 233, 285, 286] [234, 235, 287] [236, 237, 288, 289] [238, 239, 290, 291] [240, 241, 292] [242, 243, 293, 294] [244, 245, 295, 296] [246, 247, 297] [248, 249, 298, 299] [250, 251, 300, 301] [252, 253, 302] [254, 255, 303, 304] [256, 257, 305, 306] [258, 259, 307] [260, 261, 308, 309] [262, 263, 310, 311] [264, 265, 312] [266, 267] [313, 314] [268, 315, 316] [269, 270, 317, 318] [271, 272, 319, 320] [273, 321, 322] [274, 275, 323, 324] [276, 277, 325, 326] [278, 327, 328] [279, 280, 329, 330] [281, 282, 331, 332] [283, 333, 334] [284, 285, 335, 336] [286, 287, 337, 338] [288, 339, 340] [289, 290, 341, 342] [291, 292, 343, 344] [293, 345, 346] [294, 295, 347, 348] [296, 297, 349, 350] [298, 351, 352] [299, 300, 353, 354] [301, 302, 355, 356] [303, 357, 358] [304, 305, 359, 360] [306, 307, 361, 362] [308, 363, 364] [309, 310, 365, 366] [311, 312, 367, 368] [369, 370] [314, 315, 373, 374] [316, 317, 375, 376] [318, 319, 377, 378] [320, 321, 379, 380] [322, 323, 381, 382] [324, 325, 383, 384] [326, 327, 385, 386] [328, 329, 387, 388] [330, 331, 389, 390] [332, 333, 391, 392] [334, 335, 393, 394] [336, 337, 395, 396] [338, 339, 397, 398] [340, 341, 399, 400] [342, 343, 401, 402] [344, 345, 403, 404] [346, 347, 405, 406] [348, 349, 407, 408] [350, 351, 409, 410] [352, 353, 411, 412] [354, 355, 413, 414] [356, 357, 415, 416] [358, 359, 417, 418] [360, 361, 419, 420] [362, 363, 421, 422] [364, 365, 423, 424] [366, 367, 425, 426] [368, 369, 427, 428] [370, 371, 429, 430] [372, 373, 431, 432] [374, 375, 433, 434] [378, 379, 436, 437] [380, 381, 438, 439] [384, 385, 441, 442] [386, 387, 443, 444] [390, 391, 446, 447] [392, 393, 448, 449] [396, 397, 451, 452] [398, 399, 453, 454] [402, 403, 456, 457] [404, 405, 458, 459] [408, 409, 461, 462] [410, 411, 463, 464] [414, 415, 466, 467] [416, 417, 468, 469] [420, 421, 471, 472] [422, 423, 473, 474] [426, 427, 476, 477] [428, 429, 478, 479] [432, 433, 482, 483] [434, 435, 484, 485] [437, 438, 488, 489] [439, 440, 490, 491] [442, 443, 494, 495] [444, 445, 496, 497] [447, 448, 500, 501] [449, 450, 502, 503] [452, 453, 506, 507] [454, 455, 508, 509] [457, 458, 512, 513] [459, 460, 514, 515] [462, 463, 518, 519] [464, 465, 520, 521] [467, 468, 524, 525] [469, 470, 526, 527] [472, 473, 530, 531] [474, 475, 532, 533] [477, 478, 536, 537] [479, 480, 538, 539] [481, 482, 540, 541] [483, 484, 542, 543] [485, 486, 544, 545] [487, 488, 546, 547] [489, 490, 548, 549] [491, 492, 550, 551] [493, 494, 552, 553] [495, 496, 554, 555] [497, 498, 556, 557] [499, 500, 558, 559] [501, 502, 560, 561] [503, 504, 562, 563] [505, 506, 564, 565] [507, 508, 566, 567] [509, 510, 568, 569] [511, 512, 570, 571] [513, 514, 572, 573] [515, 516, 574, 575] [517, 518, 576, 577] [519, 520, 578, 579] [521, 522, 580, 581] [523, 524, 582, 583] [525, 526, 584, 585] [527, 528, 586, 587] [529, 530, 588, 589] [531, 532, 590, 591] [533, 534, 592, 593] [535, 536, 594, 595] [537, 538, 596, 597] [541, 542] [543, 544, 599, 600] [545, 546, 601, 602] [547, 548, 603] [549, 550, 604, 605] [551, 552, 606, 607] [553, 554, 608] [555, 556, 609, 610] [557, 558, 611, 612] [559, 560, 613] [561, 562, 614, 615] [563, 564, 616, 617] [565, 566, 618] [567, 568, 619, 620] [569, 570, 621, 622] [571, 572, 623] [573, 574, 624, 625] [575, 576, 626, 627] [577, 578, 628] [579, 580, 629, 630] [581, 582, 631, 632] [583, 584, 633] [585, 586, 634, 635] [587, 588, 636, 637] [589, 590, 638] [591, 592, 639, 640] [593, 594, 641, 642] [595, 596, 643] [597, 598] [644, 645] [599, 646, 647] [600, 601, 648, 649] [602, 603, 650, 651] [604, 652, 653] [605, 606, 654, 655] [607, 608, 656, 657] [609, 658, 659] [610, 611, 660, 661] [612, 613, 662, 663] [614, 664, 665] [615, 616, 666, 667] [617, 618, 668, 669] [619, 670, 671] [620, 621, 672, 673] [622, 623, 674, 675] [624, 676, 677] [625, 626, 678, 679] [627, 628, 680, 681] [629, 682, 683] [630, 631, 684, 685] [632, 633, 686, 687] [634, 688, 689] [635, 636, 690, 691] [637, 638, 692, 693] [639, 694, 695] [640, 641, 696, 697] [642, 643, 698, 699] [700, 701] [645, 646, 704, 705] [647, 648, 706, 707] [649, 650, 708, 709] [651, 652, 710, 711] [653, 654, 712, 713] [655, 656, 714, 715] [657, 658, 716, 717] [659, 660, 718, 719] [661, 662, 720, 721] [663, 664, 722, 723] [665, 666, 724, 725] [667, 668, 726, 727] [669, 670, 728, 729] [671, 672, 730, 731] [673, 674, 732, 733] [675, 676, 734, 735] [677, 678, 736, 737] [679, 680, 738, 739] [681, 682, 740, 741] [683, 684, 742, 743] [685, 686, 744, 745] [687, 688, 746, 747] [689, 690, 748, 749] [691, 692, 750, 751] [693, 694, 752, 753] [695, 696, 754, 755] [697, 698, 756, 757] [699, 700, 758, 759] [701, 702, 760, 761] [703, 704, 762, 763] [705, 706, 764, 765] [709, 710, 767, 768] [711, 712, 769, 770] [715, 716, 772, 773] [717, 718, 774, 775] [721, 722, 777, 778] [723, 724, 779, 780] [727, 728, 782, 783] [729, 730, 784, 785] [733, 734, 787, 788] [735, 736, 789, 790] [739, 740, 792, 793] [741, 742, 794, 795] [745, 746, 797, 798] [747, 748, 799, 800] [751, 752, 802, 803] [753, 754, 804, 805] [757, 758, 807, 808] [759, 760, 809, 810] [763, 764, 813, 814] [765, 766, 815, 816] [768, 769, 818, 819] [770, 771, 820, 821] [773, 774, 823, 824] [775, 776, 825, 826] [778, 779, 828, 829] [780, 781, 830, 831] [783, 784, 833, 834] [785, 786, 835, 836] [788, 789, 838, 839] [790, 791, 840, 841] [793, 794, 843, 844] [795, 796, 845, 846] [798, 799, 848, 849] [800, 801, 850, 851] [803, 804, 853, 854] [805, 806, 855, 856] [808, 809, 858, 859] [810, 811, 860, 861] [812, 813, 862, 863] [814, 815, 864, 865] [817, 818, 867, 868] [819, 820, 869, 870] [822, 823, 872, 873] [824, 825, 874, 875] [827, 828, 877, 878] [829, 830, 879, 880] [832, 833, 882, 883] [834, 835, 884, 885] [837, 838, 887, 888] [839, 840, 889, 890] [842, 843, 892, 893] [844, 845, 894, 895] [847, 848, 897, 898] [849, 850, 899, 900] [852, 853, 902, 903] [854, 855, 904, 905] [857, 858, 907, 908] [859, 860, 909, 910] [863, 864] [865, 866] [868, 869] [870, 871] [873, 874] [875, 876] [878, 879] [880, 881] [883, 884] [885, 886] [888, 889] [890, 891] [893, 894] [895, 896] [898, 899] [900, 901] [903, 904] [905, 906] [908, 909] [910, 911]
H_Z (432 checks, sparse supports)
[0, 1, 50, 51] [2, 3, 52, 53] [4, 54] [5, 6, 55, 56] [7, 8, 57, 58] [9, 59] [10, 11, 60, 61] [12, 13, 62, 63] [14, 64] [15, 16, 65, 66] [17, 18, 67, 68] [19, 69] [20, 21, 70, 71] [22, 23, 72, 73] [24, 74] [25, 26, 75, 76] [27, 28, 77, 78] [29, 79] [30, 31, 80, 81] [32, 33, 82, 83] [34, 84] [35, 36, 85, 86] [37, 38, 87, 88] [39, 89] [40, 41, 90, 91] [42, 43, 92, 93] [44, 94] [45, 46, 95, 96] [47, 48, 97, 98] [49, 99] [50, 100] [51, 52, 101, 102] [53, 54, 103, 104] [55, 105] [56, 57, 106, 107] [58, 59, 108, 109] [60, 110] [61, 62, 111, 112] [63, 64, 113, 114] [65, 115] [66, 67, 116, 117] [68, 69, 118, 119] [70, 120] [71, 72, 121, 122] [73, 74, 123, 124] [75, 125] [76, 77, 126, 127] [78, 79, 128, 129] [80, 130] [81, 82, 131, 132] [83, 84, 133, 134] [85, 135] [86, 87, 136, 137] [88, 89, 138, 139] [90, 140] [91, 92, 141, 142] [93, 94, 143, 144] [95, 145] [96, 97, 146, 147] [98, 99, 148, 149] [100, 101, 150, 151] [102, 103, 152, 153] [104, 154, 155] [105, 106, 156, 157] [107, 108, 158, 159] [109, 160, 161] [110, 111, 162, 163] [112, 113, 164, 165] [114, 166, 167] [115, 116, 168, 169] [117, 118, 170, 171] [119, 172, 173] [120, 121, 174, 175] [122, 123, 176, 177] [124, 178, 179] [125, 126, 180, 181] [127, 128, 182, 183] [129, 184, 185] [130, 131, 186, 187] [132, 133, 188, 189] [134, 190, 191] [135, 136, 192, 193] [137, 138, 194, 195] [139, 196, 197] [140, 141, 198, 199] [142, 143, 200, 201] [144, 202, 203] [145, 146, 204, 205] [147, 148, 206, 207] [149, 208] [150, 209] [151, 152, 210, 211] [153, 154, 212, 213] [155, 156, 214, 215] [157, 158, 216, 217] [159, 160, 218, 219] [161, 162, 220, 221] [163, 164, 222, 223] [165, 166, 224, 225] [167, 168, 226, 227] [169, 170, 228, 229] [171, 172, 230, 231] [173, 174, 232, 233] [175, 176, 234, 235] [177, 178, 236, 237] [179, 180, 238, 239] [181, 182, 240, 241] [183, 184, 242, 243] [185, 186, 244, 245] [187, 188, 246, 247] [189, 190, 248, 249] [191, 192, 250, 251] [193, 194, 252, 253] [195, 196, 254, 255] [197, 198, 256, 257] [199, 200, 258, 259] [201, 202, 260, 261] [203, 204, 262, 263] [205, 206, 264, 265] [207, 208, 266, 267] [213, 214, 269, 270] [215, 216, 271, 272] [219, 220, 274, 275] [221, 222, 276, 277] [225, 226, 279, 280] [227, 228, 281, 282] [231, 232, 284, 285] [233, 234, 286, 287] [237, 238, 289, 290] [239, 240, 291, 292] [243, 244, 294, 295] [245, 246, 296, 297] [249, 250, 299, 300] [251, 252, 301, 302] [255, 256, 304, 305] [257, 258, 306, 307] [261, 262, 309, 310] [263, 264, 311, 312] [268, 269, 316, 317] [270, 271, 318, 319] [273, 274, 322, 323] [275, 276, 324, 325] [278, 279, 328, 329] [280, 281, 330, 331] [283, 284, 334, 335] [285, 286, 336, 337] [288, 289, 340, 341] [290, 291, 342, 343] [293, 294, 346, 347] [295, 296, 348, 349] [298, 299, 352, 353] [300, 301, 354, 355] [303, 304, 358, 359] [305, 306, 360, 361] [308, 309, 364, 365] [310, 311, 366, 367] [313, 314, 372, 373] [315, 316, 374, 375] [317, 318, 376, 377] [319, 320, 378, 379] [321, 322, 380, 381] [323, 324, 382, 383] [325, 326, 384, 385] 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Code ID 912-48-5 · download JSON · raw on GitHub