[[139,7,5]] — dense-packed rotated surface patches, the m = 4 rung

Direction & hypothesis

Target cell: local-2d-single x weight-4. This entry is the smallest new rung of the patch-count ladder — the first step past the published m = 3 packing — and fills the gap between the board's [[101,5,5]] and the ladder's [[177,9,5]].

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. This entry frees m to 4 at d = 5.

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 = 5, m = 4: n = (76 * 4 - 26) / 2 = 139, k = 7.

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 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.

**Frontier scope, stated honestly: this entry holds a Pareto position in the 8 local-2d-single and local-2d-bilayer cells, not all 12.** In the 4 unrestricted cells the board's [[112,8,5]] — weight 4 but carrying no layout, hence absent from every local-2d-* cell — dominates it (n and k both better at equal d and w). Where a layout is required, nothing dominates this code.

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 132 checks — one component covering all 139 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.

Board rungs [[177,9,5]], [[215,11,5]], [[253,13,5]], [[271,7,7]], [[345,9,7]], [[447,7,9]], [[367,19,5]] and the open [[569,9,9]], [[667,7,11]] trade n against k or d and are all mutually non-dominated with this entry.

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; at pitch >= 2d the bands disconnect (distance 1); odd pitch breaks CSS outright.

This family is a Pareto result, not a density record. Geometric efficiency is exactly k d^2 / n here; the ladder's ceiling is 4/3 at d = 5. This rung sits at 1.259; 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 = 5, m = 4, two bands:

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

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.