[[177,9,5]] — dense-packed rotated surface patches, the five-logical packing generalized by patch count

Direction & hypothesis

Target cell: local-2d-single x weight-4. Eligibility propagates from stricter classes to looser ones on both axes, so a code that is single-layer and weight-4 competes in all 12 cells while an unrestricted any-weight code competes in one. That cell is also the thinnest on the board, and its moderate-k, moderate-d interior is empty: nothing on the board reaches k >= 6 at d >= 5 below n = 368, and nothing reaches k >= 9 at d >= 5 at all.

The structural hypothesis: the five-logical dense packing of arXiv:2511.06758 (Fujiu, Nagayama, Nishio, Kawaguchi, Satoh) is not a single code but one point of a family. The paper fixes a brick lattice of rotated surface-code patches at m = 3 patches on the lower band and m - 1 = 2 on the upper. Nothing in the construction requires m = 3, so m was freed.

What was searched

A survey of the 338 board codes over the 12 track cells, to locate non-dominated openings rather than absolute records. Then a scan over the patch-count ladder m at d = 5, 7, 9, 11, 13, and a second scan over multi-band packings parameterized as rows x m x pitch.

Distance screening used the kit's RIS surrogate. The submitted 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). The submitted code is d = 5, m = 5: n = (76 * 5 - 26) / 2 = 177, k = 9.

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]]. That is independent evidence this is the published construction generalized, and not a lookalike that happens to land nearby. 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 the submission carries confidence: upper_bound on both sides. It is not an exact claim and no certificate accompanies it.

Connectivity: the Tanner graph is a single component. Checked by union-find over qubits joined by sharing any check on either side, scanning all 168 checks — one component covering all 177 qubits. This matters because a packing of patches that 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.

The paper's "approximately three-fourths" space overhead is the m -> infinity limit (3d^2 + 1) / (4d^2) -> 3/4; the published m = 3 instance sits at 0.808, and the ladder approaches the limit from above as m grows.

Dead ends

The band pitch is the parameter that matters, and it has a threshold. Adding bands (rows > 2) at the *published* vertical pitch d - 1 adds qubits and no logicals at all — 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 **6, regardless of d and n** — a flat floor that is easy to mistake for a valid code if only one d is examined. At pitch >= 2d the bands stop sharing checks entirely and the distance collapses to 1. Odd pitch breaks the CSS condition outright.

This family is a Pareto result, not an efficiency record. With r = sqrt(2) and unit density the geometric efficiency is exactly k d^2 / n, and the two-band ladder's ceiling is 4 d^2 / (3 d^2 + 1) -> 4/3. The submitted code sits at 1.271. The board's best is 1.564. The claim here is a frontier position in a thin cell, not a best-in-class density.

A false-relation trap worth recording. While describing multi-band variants it is tempting to report the working pitches as d + 1. That holds at d = 5 (pitch 6) and fails at d = 7 (pitch 10) and d = 9 (pitch 12), where the working pitch is the measured pitch_min, not any simple offset from d. Three staged documents carried that false relation in their construction prose before it was caught; they are excluded from this submission.

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 = 5, two bands:

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

upper band carries m - 1 = 4, 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.