[[345,9,7]] — dense-packed rotated surface patches at d = 7, the m = 5 rung

Direction & hypothesis

Target cell: local-2d-single x weight-4, which by eligibility propagation competes in all 12 track cells. Before this family's rungs began landing, nothing on the board reached k >= 6 at d >= 5 below n = 368, and nothing reached k >= 9 at d >= 5 at all. This entry reaches k = 9 at d = 7.

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 to 5 at d = 7.

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 = 5: n = (148 * 5 - 50) / 2 = 345, k = 9.

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]] — including the d = 7 member [[197,5,7]], this code's m = 3 neighbour. 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. 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 336 checks — one component covering all 345 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), as are rungs at different d: the d = 7, m = 4 rung [[271,7,7]] was submitted separately and neither supersedes the other. The marginal cost of each additional logical at d = 7 is (3d^2 + 1)/4 = 37 qubits, constant in m.

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 measured at pitch_min = 6 for d = 5, 10 for d = 7, 12 for d = 9. Below it the distance collapses to a flat 6 regardless of d and n — a genuine loss at d = 7. At pitch >= 2d the bands disconnect (distance 1); odd pitch breaks CSS outright.

The working multi-band pitches are not d + 1. That holds at d = 5 (pitch 6) and fails at d = 7, where the working pitch is 10 (d + 3). The threshold is measured, not an offset from d.

This family is a Pareto result, not a density record. Geometric efficiency is exactly k d^2 / n here; the ladder's ceiling is 4 d^2 / (3 d^2 + 1), which at d = 7 is 196/148 = 1.324. This rung sits at 1.278; 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 = 5, two bands:

• Two bands at vertical pitch d - 1 = 6, horizontal patch pitch 2d + 2 = 16.
• Lower band carries m = 5 rotated surface-code patches of distance d = 7;

upper band carries m - 1 = 4, 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.