[[398,54,3]] — multi-band dense packing at d = 3, twelve bands

Direction & hypothesis

Target cell: local-2d-single x weight-4. This entry takes the multi-band packing to d = 3, where patches are smallest and the packing can stack many bands inside the envelope — 12 bands, k = 54. The d = 3 regime was the one opening the multi-band campaign had not yet exercised, and its behaviour differs from d >= 5 in ways stated honestly below.

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 = 3, 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 = 12 bands, m = 5 patches per even band and 4 per odd band, at vertical pitch 4 and horizontal patch pitch 2d + 2 = 8, d = 3. The logical count is the full patch count:

k = rows * m - floor(rows / 2) = 12 * 5 - 6 = 54, n = 398, w = 4

single layer, interaction radius sqrt(2).

What is and is not known about the threshold at d = 3, stated plainly. The band-pitch threshold was measured at d = 5, 7, 9 (`pitch_min = 6, 10, 12); it was not measured at d = 3`. Pitch 4 was chosen as the smallest even pitch above d - 1 = 2 that produced full-patch-count k with witnessed distance 3, and the witness confirms it. Note also that the sub-threshold failure mode seen at d >= 5 — distance collapsing to a flat 6 — cannot manifest at d = 3, where 6 exceeds the target distance; the failure modes that remain observable are band disconnection (pitch >= 2d = 6) and CSS breakage (odd pitch), and pitch 4 avoids both by construction. A d = 3 witness (weight 3) is also the easiest for RIS to confirm, so the upper bound here is the family's most trustworthy per trial.

Distance claim, stated precisely: witness-backed upper bound. 20,000 RIS trials per side found a lightest logical of weight 3 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 344 checks — one component covering all 398 qubits. With 12 bands this is the family's deepest stack, and a fusion failure anywhere would split the graph; it does not. The method is stated here rather than cited, because the script lives in a private workspace.

The board's multi-band entries at d = 5, 7, 9 and the two-band ladder rungs trade n against k or d and are all mutually non-dominated with this entry.

Dead ends

• The threshold formula d + 1. At d = 3 pitch 4 equals d + 1, as at

d = 5 — but the measured series at d = 5, 7, 9 is 6, 10, 12 (d + 1, d + 3, d + 3), so the coincidence at small d is not a rule, and no pitch_min(3) measurement exists to anchor it. Recorded so nobody extrapolates from this entry.

• Extra bands at the published pitch d - 1: qubits grow, k does not

(measured at d >= 5; the d - 1 = 2 pitch at d = 3 violates the distinct-site spacing anyway).

• pitch >= 2d = 6: bands disconnect, distance 1.
• Odd pitch: breaks CSS outright.
• Efficiency: a Pareto result, not a density record. k d^2 / n = 1.221;

the d = 3 multi-band run was still rising at n = 656 (1.235), and the board's best is 1.564. The deeper [[570,78,3]] instance is submitted separately.

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 = 3, rows = 12, m = 5, pitch = 4:

• Twelve bands at vertical pitch 4, horizontal patch pitch 2d + 2 = 8.
• Even-indexed bands carry m = 5 rotated surface-code patches of distance 3;

odd-indexed bands carry 4, offset half a horizontal pitch (4) to the right. Total 54 patches, and k = 54.

• 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; at d = 3 use pitch 4 as here, and treat any deeper extrapolation of the pitch rule as unmeasured.