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[[735,39,5]] d ≤
n
735
k
39
d
5
kd²/n
1.327
w
4
X/Z
1
g
1.33
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)
[448, 489, 529, 563, 604]
d_Z 5 · witness weight 5 (claimed upper_bound)
witness operator (support, 5 qubits)
[7, 8, 40, 41, 44]
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 3.707) · H_Z 2–4 (mean 3.707)
qubit degrees H_X 1–2 (mean 1.755) · H_Z 1–2 (mean 1.755)
trapping sets H_X (1,1)×180 (2,0)×36 (3,0)×30 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 180 (1,2): 555 (2,0): 36 (2,1): 432 (2,2): 1350 (3,0): 30 (3,1): 1134 (3,2): 3309 (3,3): 342 (3,4): 744
trapping sets H_Z (1,1)×180 (2,0)×36 (3,0)×30 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 180 (1,2): 555 (2,0): 36 (2,1): 432 (2,2): 1350 (3,0): 30 (3,1): 1134 (3,2): 3309 (3,3): 342 (3,4): 744
witness diameter X 4.4721 · Z 4.0 (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 (735)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

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=6, m=7, pitch=6.
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

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