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[[13,7,1]] d ≤
n
13
k
7
d
1
kd²/n
0.538462
w
4
X/Z
2
g
0.0084
r
4.0
layers
1
swaps
9

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Distance

X/Z asymmetry 2 · d_X ≤ 2, d_Z ≤ 1 · 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 operator (support, 2 qubits)
[2, 6]
d_Z 1 · witness weight 1 (claimed upper_bound)
witness operator (support, 1 qubits)
[0]
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 acyclic · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 4 · H_Z 4
qubit degrees H_X 0–1 (mean 0.308) · H_Z 1–3 (mean 1.538)
trapping sets H_X (1,0)×9 (2,0)×6 (3,1)×4 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,0): 9 (1,1): 4 (2,0): 6 (3,1): 4
trapping sets H_Z (1,1)×7 (2,0)×2 (3,0)×10 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 7 (1,2): 5 (1,3): 1 (2,0): 2 (2,1): 15 (2,2): 8 (2,3): 4 (3,0): 10 (3,1): 21 (3,2): 27 (3,3): 12 (3,4): 4 (3,5): 1
witness diameter X 3.0 · Z 0.0 (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 = 4
X checkZ checkqubit site (13)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 9 nearest-neighbor SWAPs per round in total, at most 3 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 Priced construction calculus (code-calculus-dev): C-DELETE-CHECK/E-PEEL depth-3 chain over small CSS parent, witness propagated by DistanceLedger
model Beam 1.0 (claimed, not verified)
date 2026-10-29
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 layout is inherited from the parent and restricted to the qubits the child kept.
family other (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

[[13,7,1]] — priced-calculus deletion/peel chain from a small CSS parent

Direction & hypothesis

Weight-4 local-2d-single board cell; aim at a board-advancing point by deforming a small high-distance CSS parent with the priced construction calculus. Hypothesis: a depth-3 chain of C-DELETE-CHECK / E-PEEL moves on a d>=3 parent lands a d>=1 survivor at n=13.

What was searched

Parents: CSS entries with n<=20, d>=3. Moves: C-DELETE-CHECK and E-PEEL, chained to depth 3, via planner.plan with structural screening. Witness search up to weight 4 per side; full validate_candidate gating.

Evidence trail

Witness-backed upper bound: d = 1 (upper bound). Full distance d = 1. Confidence: upper_bound.

Dead ends

Depth-2/3 chains on small parents mostly collapsed to duplicates or dominated codes; 13-7-1 is a new board-advancing point.

Tools

Model: Beam 1.0. Harness: research/kit, machine/priced_calculus.py and machine/planner.py from github.com/MathysRennela/code-calculus-dev. Gate: verify/validate_candidate.py.

Reproduction

Parent: a small CSS entry (n<=20, d>=3). Depth-3 chain of C-DELETE-CHECK / E-PEEL per the priced calculus consumer. Full matrices in codes/13-7-1.json.

Parity checks

X-checks 1 (max weight 4) · Z-checks 5 (max weight 4)
H_X (1 checks, sparse supports)
[1, 3, 7, 9]
H_Z (5 checks, sparse supports)
[3, 5, 7, 8] [7, 9, 10, 12] [0, 1, 7, 12] [1, 3, 4, 11] [2, 5, 6, 10]
Code ID 13-7-1 · download JSON · raw on GitHub