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[[819,43,5]] d ≤
n
819
k
43
d
5
kd²/n
1.313
w
4
X/Z
1
g
1.31
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)
[604, 657, 705, 749, 794]
d_Z 5 · witness weight 5 (claimed upper_bound)
witness operator (support, 5 qubits)
[474, 528, 572, 620, 672]
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 3.691) · H_Z 2–4 (mean 3.691)
qubit degrees H_X 1–2 (mean 1.748) · H_Z 1–2 (mean 1.748)
trapping sets H_X (1,1)×206 (2,0)×44 (3,0)×32 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 206 (1,2): 613 (2,0): 44 (2,1): 486 (2,2): 1482 (3,0): 32 (3,1): 1274 (3,2): 3615 (3,3): 376 (3,4): 816
trapping sets H_Z (1,1)×206 (2,0)×44 (3,0)×32 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 206 (1,2): 613 (2,0): 44 (2,1): 486 (2,2): 1482 (3,0): 32 (3,1): 1274 (3,2): 3615 (3,3): 376 (3,4): 816
witness diameter X 4.4721 · 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 (819)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=9, 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

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