[[95,22,7]] · QEC Challenge

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[[95,22,7]] d ≤

kd²/n

11.3474

X/Z

1.14

0.0071

6.3246

layers

swaps

460

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Distance

X/Z asymmetry 1.14 · d_X ≤ 8, d_Z ≤ 7 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 8 · witness weight 8 (claimed upper_bound)

witness found by @vprusso · research/kit/surrogate.lightest_logical · found at 2000 trials · survived 2000 trials · 2026-11-03

witness operator (support, 8 qubits)

[20, 36, 42, 46, 49, 58, 59, 74]

d_Z 7 · witness weight 7 (claimed upper_bound)

witness found by @vprusso · research/kit/surrogate.lightest_logical · found at 2000 trials · survived 2000 trials · 2026-11-03

witness operator (support, 7 qubits)

[3, 17, 24, 50, 64, 88, 89]

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 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 8 · H_Z 7–8 (mean 7.973)

qubit degrees H_X 1–4 (mean 3.032) · H_Z 1–4 (mean 3.105)

trapping sets H_X (1,1)×3 (2,1)×3 (3,1)×11 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,1): 3 (1,2): 23 (1,3): 37 (1,4): 32 (2,1): 3 (2,2): 45 (2,3): 151 (2,4): 306 (2,5): 285 (2,6): 166 (3,1): 11 (3,2): 112 (3,3): 536 (3,4): 1658 (3,5): 3307 (3,6): 4115 (3,7): 3053 (3,8): 1377 (3,9): 214 (3,10): 106

trapping sets H_Z (1,1)×3 (2,1)×2 (3,1)×5 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,1): 3 (1,2): 19 (1,3): 38 (1,4): 35 (2,1): 2 (2,2): 37 (2,3): 135 (2,4): 281 (2,5): 331 (2,6): 188 (3,1): 5 (3,2): 91 (3,3): 444 (3,4): 1392 (3,5): 3109 (3,6): 4377 (3,7): 3613 (3,8): 1582 (3,9): 288 (3,10): 117

witness diameter X 5.831 · Z 6.7082 (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

X checkZ checkqubit site (49)2 qubits stacked (2 layers)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 460 nearest-neighbor SWAPs per round in total, at most 10 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 @vprusso

provenance submitted through the challenge

novelty novelty not audited

construction Priced construction calculus E-PEEL of codes/96-22-8.json ([[96,22,8]], max check weight 8; a two-block group-algebra code over SmallGroup(48,24) from arXiv:2610.13880): delete qubit 43 (0-based index in the parent file) together with its unique X check a6, and relabel the surviving qubits in increasing order of their parent indices. Consumer: github.com/unitaryfoundation/code-calculus @ 1b9a942, machine/priced_calculus.py.

model Claude Fable 5.1 (claimed, not verified)

builds on arXiv:2610.13880

date 2026-11-04

notes Single-step priced-calculus E-PEEL child of board entry 96-22-8. verify/validate_candidate.py found no board entry sharing this code's fingerprint or Weisfeiler-Leman signature, local-Hadamard images included.

family 2bga-nonabelian (a tag, not a ranking)

locality 2D-local bilayer (computed from the layout)

weight class weight ≤ 8 (computed)

How this code was found

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

[[95,22,7]] priced calculus E-PEEL of codes/96-22-8.json

Direction & hypothesis

Unrestricted weight-8 CSS cell. The priced construction calculus was applied one move deep to every CSS entry on the board, and this code is the E-PEEL of codes/96-22-8.json ([[96,22,8]], max check weight 8; a two-block group-algebra code over SmallGroup(48,24) from arXiv:2610.13880). Peeling qubit 43 with the one X check that touches it keeps k = 22 and lowers the witnessed distance bound by one, from 8 to 7, while dropping one qubit. Max check weight stays 8. The layout is the parent's with the peeled qubit removed; the verifier re-measured its radius and layer count.

What was searched

Every C-DELETE-CHECK, E-PEEL, and E-PUNCTURE move on each of the 2,783 CSS board entries as of 2026-11-02, 1,684,787 moves in all, with the consumer at github.com/unitaryfoundation/code-calculus @ 1b9a942. Children with a bare qubit, a disconnected Tanner graph, a Weisfeiler-Leman signature equal to a board entry or to an earlier child, or parameters already dominated inside the parent's cell were dropped before any search, leaving 334,191 distinct children. Children with n <= 60 and an inherited distance bound of at least 2, and children with 61 <= n <= 100 and a bound of at least 5, were screened with research/kit/surrogate.lightest_logical at 2,000 RIS trials per side; the survivors the kit's novelty pre-check called board-advancing were staged with research/kit/submit.py and gated with verify/validate_candidate.py, one representative per distinct (n, k, d, max weight, locality) point. This code is that representative for [[95,22,7]] at max check weight 8. The screen distance is a witness-backed upper bound; the rungs below are the independent confirmation runs on the staged file.

Evidence trail

Confirmation ladder for the submitted code, each rung a fresh seed unless marked otherwise, reading the lightest logical found:

| budget (trials/side) | lightest logical | | ---: | ---: | | 2,000 (staging screen, seeds 1 and 2) | 7 | | 1,000,000 (seed 101) | 7 | | 1,000,000 (seed 102) | 7 | | 10,000,000 (seed 103) | 7 |

4 fresh-seed rung(s) agree at the best bound, the deepest of them at 10,000,000 trials per side. The submitted witness reproduces it: X: d <= 8 (upper_bound), Z: d <= 7 (upper_bound).

Dead ends

None recorded for this candidate.

Tools

Model: Claude Fable 5.1 (Claude Code). Repo tooling: research/kit/coordination.py, research/kit/submit.py, research/kit/surrogate.py, research/audits/leader_audit.py. The local verdict is verify/validate_candidate.py; the bundle was assembled by research/kit/promote.py.

Reproduction

Load codes/96-22-8.json, delete qubit 43 (0-based index in that file) together with its unique X check a6, where the X rows are numbered a1, a2, ... in file order after dropping rows dependent on earlier ones, as CodeState.from_challenge in machine/priced_calculus.py labels them, and relabel the surviving qubits in increasing order of their parent indices; in github.com/unitaryfoundation/code-calculus @ 1b9a942 this is apply_operation(state, {'op': 'E-PEEL', 'qubit': 43, 'sector': 'X', 'row_id': 'a6'}). Then run research/audits/leader_audit.py ladder on the result with --ladder 1000000:101,102 to reproduce the confirmation rungs; the witnesses in the entry are the ones those runs validated.

Parity checks

X-checks 36 (max weight 8) · Z-checks 37 (max weight 8)

H_X (36 checks, sparse supports)

[0, 16, 31, 44, 47, 63, 71, 75] [1, 13, 23, 45, 48, 70, 78, 82] [2, 20, 27, 46, 49, 60, 74, 85] [3, 6, 23, 30, 49, 50, 51, 77] [0, 4, 20, 42, 47, 51, 76, 85] [6, 24, 34, 41, 53, 67, 81, 89] [0, 7, 27, 37, 53, 54, 55, 84] [1, 8, 24, 28, 48, 55, 83, 89] [8, 9, 29, 38, 55, 56, 90, 92] [1, 10, 34, 40, 47, 57, 58, 87] [2, 11, 31, 35, 49, 58, 75, 86] [11, 12, 36, 42, 58, 59, 76, 93] [13, 17, 37, 41, 57, 60, 61, 88] [1, 3, 14, 18, 50, 52, 58, 61] [8, 14, 15, 19, 59, 61, 62, 63] [5, 16, 36, 46, 52, 60, 63, 93] [2, 16, 17, 43, 48, 64, 65, 91] [6, 13, 18, 38, 53, 65, 82, 90] [18, 19, 28, 39, 65, 66, 83, 94] [3, 20, 40, 44, 64, 67, 68, 92] [2, 4, 7, 21, 54, 56, 65, 68] [5, 11, 21, 22, 66, 68, 69, 70] [9, 23, 39, 41, 56, 67, 70, 94] [7, 24, 43, 45, 50, 71, 72, 93] [6, 8, 10, 25, 51, 57, 59, 72] [9, 18, 25, 26, 52, 72, 73, 74] [7, 13, 28, 32, 60, 62, 72, 75] [21, 28, 29, 33, 73, 75, 76, 77] [9, 15, 30, 34, 47, 62, 74, 77] [10, 30, 31, 46, 54, 78, 79, 94] [0, 11, 17, 32, 55, 64, 66, 79] [4, 12, 32, 33, 56, 79, 80, 81] [10, 14, 20, 35, 67, 69, 79, 82] [15, 25, 35, 36, 80, 82, 83, 84] [17, 21, 24, 38, 61, 71, 73, 85] [22, 32, 38, 39, 62, 85, 86, 87]

H_Z (37 checks, sparse supports)

[0, 4, 10, 30, 47, 51, 54, 79] [1, 8, 17, 37, 48, 55, 57, 61] [2, 3, 11, 40, 49, 58, 64, 68] [3, 14, 24, 41, 50, 61, 67, 89] [3, 4, 5, 25, 51, 52, 68, 80] [5, 14, 16, 26, 52, 63, 69, 91] [6, 7, 18, 43, 50, 53, 65, 72] [7, 21, 31, 44, 54, 68, 71, 75] [7, 8, 9, 32, 55, 56, 62, 72] [9, 21, 23, 33, 56, 70, 73, 77] [10, 13, 25, 45, 57, 72, 78, 82] [10, 11, 12, 14, 58, 59, 69, 79] [12, 15, 25, 27, 59, 74, 80, 84] [2, 13, 16, 28, 48, 60, 65, 75] [13, 14, 15, 38, 61, 62, 82, 90] [15, 28, 30, 39, 62, 77, 83, 94] [0, 15, 16, 40, 47, 63, 64, 84] [17, 20, 32, 46, 60, 64, 79, 85] [17, 18, 19, 21, 61, 65, 66, 73] [19, 22, 32, 34, 62, 66, 81, 87] [6, 20, 23, 35, 49, 51, 67, 82] [20, 21, 22, 42, 68, 69, 76, 85] [22, 35, 37, 69, 84, 86, 88] [1, 22, 23, 43, 48, 50, 70, 87] [0, 24, 27, 38, 53, 55, 71, 85] [24, 25, 26, 28, 72, 73, 83, 89] [26, 29, 38, 40, 73, 87, 90, 92] [0, 11, 28, 29, 55, 66, 75, 76] [4, 12, 29, 41, 56, 76, 81, 88] [3, 29, 30, 45, 50, 77, 78, 92] [1, 31, 34, 42, 47, 58, 78, 89] [31, 32, 33, 35, 75, 79, 80, 86] [33, 36, 42, 43, 76, 80, 91, 93] [1, 18, 35, 36, 52, 58, 82, 83] [8, 19, 36, 44, 59, 63, 83, 92] [2, 4, 38, 39, 56, 65, 85, 86] [5, 11, 39, 45, 66, 70, 86, 93]

Code ID 95-22-7 · download JSON · raw on GitHub