[[418,10,7]] — multi-band dense packing at d = 7, where the pitch threshold parts from d + 1

Direction & hypothesis

Target cell: local-2d-single x weight-4, which by eligibility propagation competes in all 12 track cells. This is the multi-band packing's first entry beyond d = 5 — the instance that shows the band-pitch threshold is a measured quantity, not a formula in d.

The dense packing of arXiv:2511.06758 (Fujiu, Nagayama, Nishio, Kawaguchi, Satoh) is a pair of bands of rotated surface-code patches. Freeing the band count extends the packing into a second dimension, above a band-pitch threshold.

What was searched

A survey of the 338 board codes across the 12 cells, then a multi-band scan parameterized as rows x m x pitch over band counts, patch counts, and vertical band pitch, at d = 5, 7, 9. Screening used the kit's RIS distance surrogate; this code's witness search ran 20,000 RIS trials per CSS side.

Evidence trail

This code is rows = 4 bands, m = 3 patches per even band and 2 per odd band, at vertical pitch 10 and horizontal patch pitch 2d + 2 = 16, d = 7. The logical count is the full patch count:

k = rows * m - floor(rows / 2) = 4 * 3 - 2 = 10, n = 418, w = 4

single layer, interaction radius sqrt(2).

**The pitch is 10 because that is where the threshold was measured, not because of any formula in d.** Measured thresholds: pitch_min = 6 at d = 5, 10 at d = 7, 12 at d = 9. At d = 5 the threshold happens to equal d + 1; at d = 7 and d = 9 it is d + 3. An earlier staged draft of this very code described its pitch as d + 1, which is false at d = 7 — the error was caught in review before submission, and is recorded here because it is exactly the kind of plausible-looking relation a later searcher might assume. Below the threshold the distance collapses to a flat 6 regardless of d and n (at d = 7 a real loss); at pitch >= 2d = 14 the bands disconnect (distance 1); odd pitch breaks CSS outright.

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.

Connectivity: the Tanner graph is a single component, checked by union-find over qubits joined by sharing any check on either side, scanning all 408 checks — one component covering all 418 qubits. Load-bearing for the multi-band family: a code whose bands failed to fuse would be a direct sum of independent smaller codes with an inherited [[n,k,d]]. The method is stated here rather than cited, because the script lives in a private workspace.

The board's d = 5 multi-band entries ([[126,6,5]], [[202,10,5]], [[278,14,5]]), the two-band ladder rungs, and the d = 7 ladder rungs ([[271,7,7]], [[345,9,7]]) trade n against k or d and are all mutually non-dominated with this entry.

Dead ends

• Describing the working pitch as d + 1. True at d = 5, false at

d = 7 and d = 9. The threshold is measured, and this entry is the counterexample to the formula.

• Extra bands at the published pitch d - 1: qubits grow, k does not.
• Sub-threshold pitches: distance collapses to a flat 6, independent of d

and n — the collapsed code still verifies as valid, which is the trap.

• pitch >= 2d: bands disconnect, distance 1.
• Odd pitch: breaks CSS outright.
• Efficiency: a Pareto result, not a density record. k d^2 / n = 1.172

here — the raised pitch costs qubits, which is the price of the second dimension at d = 7. 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, rows = 4, m = 3, pitch = 10:

• Four bands at vertical pitch 10, horizontal patch pitch 2d + 2 = 16.
• Even-indexed bands carry m = 3 rotated surface-code patches of distance 7;

odd-indexed bands carry 2, offset half a horizontal pitch (8) to the right. Total 10 patches, and k = 10.

• 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. Even bands carry (x + y) mod 4 == 0 checks on vertical patch edges and == 2 on horizontal edges; odd bands the reverse.

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

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

The two-band, pitch d - 1 case of this rule is the published packing and is exactly research/build_dense_surface.py in this repo. Generalize its band count and patch counts, then raise the vertical pitch to at least pitch_min(d) — measured, not assumed.