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[[9,7,1]] d =
n
9
k
7
d
1
kd²/n
0.777778
w
6
X/Z
2
g
0.0311
r
3.1623
layers
1
swaps
3

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Distance

X/Z asymmetry 2 · d_X = 1, d_Z = 2 · w_X = 6, w_Z = 0 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 1 · witness weight 1 (claimed upper_bound)
witness operator (support, 1 qubits)
[0]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[0, 2]
certificate exact, d = 1 · graphlike_A (source sha256 66a15ffb) and graphlike_B (56d04349), two independent exact programs for graph-like check matrices; github.com/TimoKellerMath/quantum-error-correction @ ae52261bfdbd, board-certificates-2026-11 · level solver: a solver verdict, no artifact a third party can check without re-running the encoding
X: graphlike_A and graphlike_B: d_X = 1, the length of a shortest homologically nontrivial cycle of the graph of H_Z (no nontrivial X-logical of weight <= 0); words of weight 1 rechecked; Z: graphlike_A and graphlike_B: d_Z = 2, the length of a shortest homologically nontrivial cycle of the graph of H_X (no nontrivial Z-logical of weight <= 1); words of weight 2 rechecked

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 acyclic (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 6 · H_Z 0
qubit degrees H_X 1–2 (mean 1.333) · H_Z 0
trapping sets H_X (1,1)×6 (2,0)×9 (3,0)×27 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 6 (1,2): 3 (2,0): 9 (2,1): 18 (3,0): 27 (3,1): 20 (3,2): 19
trapping sets H_Z (1,0)×9 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,0): 9
witness diameter X 0.0 · Z 1.0 (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 (9)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 3 nearest-neighbor SWAPs per round in total, at most 2 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 CSS parent 12-2-4, 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 ≤ 6 (computed)

How this code was found

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

[[9,7,1]] — priced-calculus deletion/peel chain from a 12-qubit CSS parent

Direction & hypothesis

Weight-6 local-2d 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 single-check deletion (C-DELETE-CHECK) and peel (E-PEEL) moves on a d>=4 parent lands a d>=1 survivor at n=9.

What was searched

Parents: CSS entries with n<=14, d>=4. 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 runs already yielded 11-3-3, 11-8-1, 10-7-1. Depth-3 added 10-8-1 and 9-7-1 as new points; other paths collapsed to duplicates or dominated codes.

Tools

Model: Beam 1.0. Harness: research/kit, machine/priced_calculus.py and machine/planner.py from github.com/MathysRennela/code-calculus-dev @ 5084747d90dee5fd3927f66bca33bbe3198ba67d. Gate: verify/validate_candidate.py.

Reproduction

Parent: codes/12-2-4.json (or 12-2-4-b.json). Depth-3 chain of C-DELETE-CHECK / E-PEEL per the priced calculus consumer. Full matrices in codes/9-7-1.json.

Parity checks

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