[[271,7,7]] — dense-packed rotated surface patches at d = 7, the m = 4 rung

Direction & hypothesis

Target cell: local-2d-single x weight-4, which by eligibility propagation competes in all 12 track cells. The cell's moderate-k, moderate-d interior is empty — nothing on the board reaches k >= 6 at d >= 5 below n = 368.

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. This entry frees m and sits at m = 4, d = 7 — a different distance regime from the d = 5 rungs, which is the point of submitting it.

What was searched

A survey of the 338 board codes across the 12 cells to locate 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.

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 = 7, m = 4: n = (148 * 4 - 50) / 2 = 271, k = 7.

The parameterization reproduces the board's entire existing m = 3 column exactly — [[101,5,5]], [[197,5,7]], [[325,5,9]], [[485,5,11]], [[677,5,13]] — including the d = 7 member [[197,5,7]], which is the direct m = 3 neighbour of this code. That exact reproduction 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 7 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. Worth noting that RIS screening gets harder as d grows, so a d = 7 bound is weaker evidence per trial than a d = 5 bound at the same trial count.

Connectivity: the Tanner graph is a single component, checked by union-find over qubits joined by sharing any check on either side, scanning all 264 checks — one component covering all 271 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.

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 rows * m - floor(rows / 2) with distance preserved, but only above a threshold measured at pitch_min = 6 for d = 5, 10 for d = 7, and 12 for d = 9. Below the threshold the distance collapses to a flat 6 regardless of d and n — which at d = 7 is a genuine loss, and is easy to mistake for a working code if only d = 5 is examined. At pitch >= 2d the bands stop sharing checks and the distance collapses to 1. Odd pitch breaks the CSS condition outright.

The working pitches are not d + 1. That relation holds at d = 5 (pitch 6) and fails at d = 7 (pitch 10) and d = 9 (pitch 12). The working pitch is the measured pitch_min, not an offset from d. Recording this because it is the trap this round actually fell into before catching it.

This family is a Pareto result, not a density record. Geometric efficiency is exactly k d^2 / n here, and this ladder's ceiling is 4 d^2 / (3 d^2 + 1) -> 4/3. This code sits at 1.266; 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

For d = 7, m = 4, two bands:

• Two bands at vertical pitch d - 1 = 6, horizontal patch pitch 2d + 2 = 16.
• Lower band carries m = 4 rotated surface-code patches of distance d = 7;

upper band carries m - 1 = 3, offset by half the horizontal pitch, making 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: generalize its patch counts from (3, 2) to (m, m - 1) and the rest of the mask logic is unchanged.