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[[854,158,2]] d ≤
n
854
k
158
d
2
kd²/n
0.740047
w
4
X/Z
1.5
g
0.0296
r
3.1623
layers
1
swaps
12

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Distance

X/Z asymmetry 1.5 · d_X ≤ 2, d_Z ≤ 3 · 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 2 · witness weight 2 (claimed upper_bound)
witness found by @mathysrennela · verify/heuristic_distance.py:estimate · found at 8000 trials · 2026-10-21
witness operator (support, 2 qubits)
[561, 591]
d_Z 3 · witness weight 3 (claimed upper_bound)
witness found by @mathysrennela · verify/heuristic_distance.py:estimate · found at 8000 trials · survived 8000 trials · 2026-10-21
witness operator (support, 3 qubits)
[492, 523, 554]
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 4 · H_Z 8 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 2–4 (mean 3.837) · H_Z 2–4 (mean 3.851)
qubit degrees H_X 1–3 (mean 1.573) · H_Z 1–2 (mean 1.569)
trapping sets H_X (1,1)×369 (2,0)×25 (3,0)×531 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 369 (1,2): 481 (1,3): 4 (2,0): 25 (2,1): 1006 (2,2): 909 (2,3): 3 (3,0): 531 (3,1): 1885 (3,2): 1768 (3,3): 934 (3,4): 300 (3,5): 1
trapping sets H_Z (1,1)×368 (2,0)×31 (3,0)×497 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 368 (1,2): 486 (2,0): 31 (2,1): 996 (2,2): 930 (3,0): 497 (3,1): 1915 (3,2): 1842 (3,3): 913 (3,4): 313
witness diameter X 1.4142 · Z 2.0 (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 = 3.162
X checkZ checkqubit site (854)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 12 nearest-neighbor SWAPs per round in total, at most 1 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 C-DELETE-CHECK b237 from 854-157-3.json
model Beam 1.0, MiMo-V2.6-Flash (claimed, not verified)
date 2026-10-22
notes Generated by the priced construction calculus. Distance is reported only with witness-backed upper bounds; lower interval bounds are kept in the separate local ledger. The parent's claimed distance is not inherited wholesale: a randomized (RIS) search over the child finds a lighter logical where one exists, and this entry is filed at the child's own measured upper bound.
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

[[854,158,2]] — one Z-check deletion from 854-157-3.json (C-DELETE-CHECK b237)

Direction & hypothesis

A high-k board region at 854 <= n reachable by deleting one check from 854-157-3.json at little distance cost. Deletion raises k by one at fixed n and fixed qubit set, so any landing point where the frontier has headroom on k is reachable in one move. The aim was the largest d the deletion could keep while still clearing the cell's floor -- here 2, against a parent claim of 3.

What was searched

All 163 CSS board entries with n > 700, not a sample of them: 88,592 essential single check deletions (rows whose removal changes the stabilizer group; dependent rows are dropped, since removing one yields an identical code). 28,913 of those -- 33% -- free a bare qubit and are d=1 by construction; a structural filter on column weights discarded them before any witness search, leaving 59,679 clean deletions across 109 distinct (n, k+1, d, w) tuples.

Witnesses were then re-measured per candidate rather than inherited: 8000 RIS trials per side, seed 0, via the challenge's own heuristic_distance search. At most 3 clean rows were sampled per tuple, so this is a floor on what the pool contains, not an exhaustive sweep.

Evidence trail

  • Parent 854-157-3.json claims d = 3.
  • Unpriced C-DELETE-CHECK carry from the parent (no child witnesses supplied):

    • interval [0, 3]. Contains the measured child distance.

  • Priced transport, d' = min(d, w_del, w_exp): [0, 2] with

    • w_del = 2, w_exp = 5 (RIS upper bounds, 20,000 trials, seed 0; the true minima are NP-hard, Theorem 24.1, so both are over-estimates).

  • Seed-0 estimate over the child: 2 per side, 8000 trials/side.
  • Gate at seed 5: passed, not refuted, no duplicate, board-advancing on no axis in particular.
  • Advancement: nothing dominates it in its cell.

Dead ends

  • Distance does not survive deletion. The parent's claim was never carried forward: on 14 of

    • the 18 filed entries the child measured strictly below the parent, so inheriting the parent's number would have been an over-claim that the refutation gate catches.

  • 28,913 of 88,592 essential deletions free a bare qubit and land at d=1; only 4 of the 18

    • filed entries are of that kind, the rest being shadowed by d=2 siblings from the same parent.

  • 22 candidates in the working set passed the gate but were dominated by a batch-mate at the

    • same (n, k, w), so 40 were reduced to 18 before filing.

  • Not attempted at this scale: C-ADD-CHECK strict improvement and E-GRAFT. The 2026-10-19

    • fieldnote reports 0 strict improvements out of 736 valid additions for n <= 10 and no graft survivors for n <= 8, so neither was worth a sweep at n = 700-1000 without a new idea.

  • C-DIRECT-SUM is unusable against the connectivity gate regardless of scale.

Tools

Models: Beam 1.0, MiMo-V2.6-Flash (matches provenance.model).

  • Structural work: the challenge's own gf2 and verify/heuristic_distance.py, plus an

    • independent nullspace test for row essentiality.

  • Verification: verify/qldpc_verify.py and verify/check_codes_format.py, unchanged, both

    • passing locally on the committed file.

  • Interval transport: the priced construction calculus at github.com/MathysRennela/code-calculus-dev @ 661d1e0, machine/priced_calculus.py. Its CodeState.delete_check

    • reproduces this entry's child state exactly from the parent, and its Independent(...) premise agreed with the independent nullspace check on all 18 pairs.

The calculus run is evidence against OBL-PRICED-INTERVAL-SOUNDNESS, not confirmation of it: containment held on 18/18 for both the unpriced carry and the priced interval, but the prices themselves are over-estimates and no lower endpoint was exercised (the ledger's lower bound stays 0 throughout).

Reproduction

Take 854-157-3.json from codes/, remove the Z-check whose support is [590, 591, 621, 622], and set k = 158. The result is codes/854-158-2.json. Then:

python verify/qldpc_verify.py codes/854-158-2.json
python verify/check_codes_format.py --files codes/854-158-2.json

Both exit 0. The distance witnesses in codes/854-158-2.json are the acceptance argument; they are upper bounds only.

Parity checks

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