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[[54,12,6]] d ≤
n
54
k
12
d
6
kd²/n
8.0
w
7
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 6, d_Z ≤ 6 · w_X = 7, w_Z = 7 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 6 · witness weight 6 (claimed upper_bound)
witness found by @vprusso · research/kit/surrogate.lightest_logical · found at 3000 trials · survived 3000 trials · 2026-11-03
witness operator (support, 6 qubits)
[2, 8, 11, 27, 36, 46]
d_Z 6 · witness weight 6 (claimed upper_bound)
witness found by @vprusso · research/kit/surrogate.lightest_logical · found at 3000 trials · survived 3000 trials · 2026-11-03
witness operator (support, 6 qubits)
[19, 24, 30, 31, 48, 49]
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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 6–7 (mean 6.81) · H_Z 6–7 (mean 6.905)
qubit degrees H_X 1–4 (mean 2.648) · H_Z 1–4 (mean 2.685)
trapping sets H_X (1,1)×2 (2,1)×2 (3,1)×18 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 2 (1,2): 22 (1,3): 23 (1,4): 7 (2,1): 2 (2,2): 38 (2,3): 132 (2,4): 156 (2,5): 80 (2,6): 9 (3,1): 18 (3,2): 117 (3,3): 516 (3,4): 1197 (3,5): 1416 (3,6): 910 (3,7): 402 (3,8): 124 (3,9): 18 (3,10): 1
trapping sets H_Z (1,1)×2 (2,1)×3 (3,1)×16 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 2 (1,2): 20 (1,3): 25 (1,4): 7 (2,1): 3 (2,2): 33 (2,3): 131 (2,4): 163 (2,5): 89 (2,6): 10 (3,1): 16 (3,2): 101 (3,3): 500 (3,4): 1217 (3,5): 1492 (3,6): 1029 (3,7): 430 (3,8): 146 (3,9): 25 (3,10): 1

Construction & provenance

authors @vprusso
provenance submitted through the challenge
novelty novelty not audited
construction Priced construction calculus E-PEEL of codes/55-12-6.json ([[55,12,6]], max check weight 7; itself a priced-calculus E-PEEL of codes/56-12-7.json, a two-block group-algebra code over SmallGroup(28,4) from arXiv:2610.13880): delete qubit 20 (0-based index in the parent file) together with its unique Z check b3, 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 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 55-12-6. verify/validate_candidate.py found no board entry sharing this code's fingerprint or Weisfeiler-Leman signature, local-Hadamard images included.
family other (a tag, not a ranking)
locality unrestricted (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

[[54,12,6]] priced calculus E-PEEL of codes/55-12-6.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/55-12-6.json ([[55,12,6]], max check weight 7; itself a priced-calculus E-PEEL of codes/56-12-7.json, a two-block group-algebra code over SmallGroup(28,4) from arXiv:2610.13880). Peeling qubit 20 with the one Z check that touches it keeps k = 12 and the witnessed distance bound d <= 6 while dropping one qubit, so it sits one qubit to the left of its parent on the frontier. Max check weight stays 7.

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 3,000 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 [[54,12,6]] at max check weight 7. 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 | | ---: | ---: | | 3,000 (staging screen, seeds 1 and 2) | 6 | | 1,000,000 (seed 101) | 6 | | 1,000,000 (seed 102) | 6 | | 10,000,000 (seed 103) | 6 |

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 <= 6 (upper_bound), Z: d <= 6 (upper_bound).

Matched-depth peer audit (research/audits/leader_audit.py pair): credible: both claims held at matched depth, the gain survives; candidate [[54,12,6]] w7 reads d <= 6 and peer codes/54-12-5.json reads d <= 5 (matched depth: 2,000,000 trials, seeds 51, 52)

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/55-12-6.json, delete qubit 20 (0-based index in that file) together with its unique Z check b3, where the Z rows are numbered b1, b2, ... 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': 20, 'sector': 'Z', 'row_id': 'b3'}). 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 21 (max weight 7) · Z-checks 21 (max weight 7)
H_X (21 checks, sparse supports)
[0, 16, 20, 23, 26, 38, 52] [1, 14, 21, 22, 27, 36, 53] [2, 13, 19, 25, 28, 35, 49] [4, 11, 24, 30, 33, 51] [0, 5, 18, 25, 30, 31, 40] [4, 6, 17, 21, 26, 32, 39] [2, 5, 7, 22, 32, 33, 46] [2, 8, 15, 23, 27, 34, 37] [3, 4, 9, 20, 34, 35, 44] [0, 8, 10, 29, 36, 43] [6, 9, 11, 25, 36, 37, 50] [1, 6, 12, 19, 31, 38, 41] [7, 8, 13, 24, 38, 39, 48] [3, 12, 14, 23, 33, 40, 47] [4, 10, 13, 15, 40, 41, 53] [5, 10, 16, 21, 35, 42, 45] [2, 11, 12, 17, 42, 43, 52] [1, 7, 16, 18, 37, 44, 51] [8, 14, 17, 19, 30, 44, 45] [6, 15, 16, 28, 46, 47] [12, 18, 21, 34, 48, 49]
H_Z (21 checks, sparse supports)
[0, 6, 22, 26, 31, 36, 46] [1, 8, 20, 27, 29, 38, 44] [3, 10, 25, 29, 35, 40, 50] [4, 5, 19, 30, 32, 35, 41] [5, 12, 24, 31, 33, 42, 48] [6, 7, 23, 32, 37, 38, 47] [4, 7, 14, 33, 39, 44, 53] [8, 9, 21, 34, 36, 39, 45] [2, 9, 16, 35, 37, 46, 52] [1, 10, 11, 36, 41, 42, 51] [8, 11, 18, 30, 37, 43, 48] [0, 12, 13, 38, 40, 43, 49] [6, 13, 28, 39, 41, 50] [5, 14, 15, 27, 40, 45, 46] [12, 15, 20, 34, 41, 47, 52] [3, 16, 17, 26, 42, 44, 47] [10, 17, 22, 32, 43, 45, 53] [9, 18, 19, 31, 44, 49, 50] [16, 19, 24, 28, 38, 45, 51] [13, 20, 21, 26, 35, 48, 53] [2, 21, 27, 32, 42, 49]
Code ID 54-12-6-c · download JSON · raw on GitHub