[[253,13,5]] — dense-packed rotated surface patches, the m = 7 rung of the five-logical family

Direction & hypothesis

Target cell: local-2d-single x weight-4. Eligibility propagates from stricter classes to looser ones on both axes, so a single-layer weight-4 code competes in all 12 cells. That cell's moderate-k, moderate-d interior is empty: before this family's rungs landed, nothing on the board reached k >= 9 at d >= 5.

Hypothesis: the five-logical dense packing of arXiv:2511.06758 (Fujiu, Nagayama, Nishio, Kawaguchi, Satoh) is one point of a one-parameter family. The paper fixes m = 3 patches on the lower band and m - 1 = 2 on the upper; nothing in the construction requires m = 3. This entry is m = 7, continuing the ladder whose m = 5 and m = 6 rungs ([[177,9,5]], [[215,11,5]]) are already on the board.

What was searched

A survey of the 338 board codes over the 12 track cells to find non-dominated openings, then a scan of the patch-count ladder m at d = 5, 7, 9, 11, 13, screened with the kit's RIS distance surrogate. This code's witness search ran 20,000 RIS trials per CSS side.

No search was needed to *find* the code once the family was parameterized — the work was in identifying the free parameter, then confirming distance at each rung.

Evidence trail

Freeing m at the published band pitch gives a closed form:

n = ((3d^2 + 1) m - (d^2 + 1)) / 2, k = 2m - 1, w = 4

single layer, interaction radius sqrt(2). At d = 5, m = 7: n = (76 * 7 - 26) / 2 = 253, k = 13.

The parameterization reproduces the board's original m = 3 column exactly — [[101,5,5]], [[197,5,7]], [[325,5,9]], [[485,5,11]], [[677,5,13]]. That is the main evidence this generalizes the published construction rather than resembling it. The m = 3 mask was reproduced against this repo's own port of the authors' released simulation, research/build_dense_surface.py.

Distance claim, stated precisely: witness-backed upper bound. 20,000 RIS trials per side found a lightest logical of weight 5 on both the X and Z sides and nothing lighter, so both sides carry confidence: upper_bound. No certificate accompanies this and no exact claim is made.

Connectivity: the Tanner graph is a single component, checked by union-find over qubits joined by sharing any check on either side, scanning all 240 checks — one component covering all 253 qubits. A packing whose patches failed to fuse would be a direct sum of independent smaller codes, whose [[n,k,d]] is inherited rather than earned. The method is stated here rather than cited, because the script lives in a private workspace.

Rungs of this ladder at equal d and different m are mutually non-dominated (each trades n against k), so this entry does not supersede [[177,9,5]] or [[215,11,5]], nor they it. The marginal cost of each additional logical pair is (3d^2 + 1) / 2 = 38 qubits, constant in m — 19.0 qubits per logical.

Dead ends

Extra bands at the published pitch buy nothing. Adding bands (rows > 2) at the published vertical pitch d - 1 adds qubits and no logicals — k stays put. Raising the pitch makes k the full patch count with distance preserved, but only above a threshold (pitch_min = 6 at d = 5, 10 at d = 7, 12 at d = 9); below it the distance collapses to a flat 6 regardless of d and n, at pitch >= 2d the bands disconnect, and odd pitch breaks CSS outright.

This family is a Pareto result, not a density record. With r = sqrt(2) and unit density the geometric efficiency is exactly k d^2 / n; this ladder's ceiling is 4 d^2 / (3 d^2 + 1) -> 4/3, approached from below as m grows. This rung sits at 1.285; the board's best is 1.564.

Tools

Claude Opus 5 (matching provenance.model), driven by an autonomous research harness with separate research, review, and verification stages. Repo tooling: research/kit/submit.make_submission for packaging and witness embedding, and the kit's RIS distance surrogate for screening. Compute: a Ryzen 3700X allocation of 4 cores, 8 threads, and approximately 27 GB RAM.

Reproduction

Build the site mask directly; no search is needed once the parameters are fixed.

For d = 5, m = 7, two bands:

• Two bands at vertical pitch d - 1 = 4, horizontal patch pitch 2d + 2 = 12.
• Lower band carries m = 7 rotated surface-code patches of distance d = 5;

upper band carries m - 1 = 6, offset by half the horizontal pitch, which is what makes the packing brick-staggered rather than a grid.

• Data qubits occupy the odd/odd sites of the mask.
• Of the remaining occupied sites, those with (x + y) mod 4 == 2 measure

X-checks and the rest measure Z-checks.

• Every check acts on its four diagonal data neighbours, giving w = 4

throughout and interaction radius sqrt(2) on a single layer.

The m = 3 case of this rule is exactly research/build_dense_surface.py in this repo, which is the recommended starting point: generalize its patch counts from (3, 2) to (m, m - 1) and the rest of the mask logic is unchanged.