witness diameter X 4.1231 · Z 4.0 (Euclidean support diameter of the stored distance witnesses in the layout; an upper bound on the exhibited logicals’ spread, not a minimum over all logicals)
d_circ ≤ 5 (min over bases; penalty-only, clamped to ≤ d)
d_circ^X 5 · fault-set witness of 5 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 5)
[4, 5, 15, 23, 33]
d_circ^Z 5 · fault-set witness of 5 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 5)
[0, 7, 179, 307, 446]
no measured logical error rate yet; d_circ is a floor, and the measured tier records the prefactor it cannot see
Verified 2D layout
as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair
X checkZ checkqubit site (126)dashed: the pair setting the interaction radiushover a check to isolate its qubits; click to pin — repeated clicks cycle through overlapping checks; click empty space to release
routing cost 0 nearest-neighbor SWAPs per round in total, at most 0 for one check (heuristic: MST lower bound on the layout, with one lattice step = the minimum qubit spacing 1; not a rank)
notes Same multi-band construction as the board's [[202,10,5]], [[278,14,5]] and the open [[676,36,5]], at the minimum size: k = 4*2-2 = 6. Not equivalent to any board entry: n and k both differ. Pareto scope stated honestly: frontier in the 8 layout-restricted cells; in the 4 unrestricted cells [[112,8,5]] (weight 4, no layout) dominates it. Distance is a witness-backed upper bound from 20000 RIS trials per side, not a certified exact distance.
family other (a tag, not a ranking)
locality 2D-local single (computed from the layout)
[[126,6,5]] — multi-band dense packing, the smallest working instance
Direction & hypothesis
Target cell: local-2d-single x weight-4. This entry is the smallest working instance of the multi-band packing — 4 bands of 2-and-1 patches — submitted as the family's low-n anchor.
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, which works only above a measured 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. 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 = 2 patches per even band and 1 per odd band, at vertical pitch 6 and horizontal patch pitch 2d + 2 = 12, d = 5. The logical count is the full patch count:
k = rows * m - floor(rows / 2) = 4 * 2 - 2 = 6, n = 126, w = 4
single layer, interaction radius sqrt(2).
**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 layout-restricted cell — dominates it. Where a layout is required, nothing dominates this code: before this family landed, nothing layout-restricted reached k >= 6 at d >= 5 below n = 368.
The threshold. At the published vertical pitch d - 1, adding bands adds qubits and no logicals at all. Above a threshold pitch, k becomes the full patch count with distance preserved. Measured thresholds: pitch_min = 6 at d = 5, 10 at d = 7, 12 at d = 9. Below the threshold 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. Like the board's other multi-band entries this code sits at pitch_min(5) = 6 exactly.
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 120 checks — one component covering all 126 qubits. Load-bearing here more than anywhere: with odd bands carrying a single patch, a fusion failure would leave an isolated distance-5 surface code and an inherited distance. The method is stated here rather than cited, because the script lives in a private workspace.
The board's [[202,10,5]], [[278,14,5]], [[676,36,5]] (same construction, more bands or patches) and the two-band ladder rungs trade n against k and are all mutually non-dominated with this entry.
Dead ends
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.
pitch >= 2d: bands disconnect, distance 1.
Odd pitch: breaks CSS outright.
Efficiency: a Pareto result, not a density record. Geometric efficiency is
k d^2 / n = 1.190 here — the family's lowest, the price of the small-n anchor position. 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, rows = 4, m = 2, pitch = 6:
Four bands at vertical pitch 6, horizontal patch pitch 2d + 2 = 12.
Even-indexed bands carry m = 2 rotated surface-code patches of distance 5;
odd-indexed bands carry 1, offset half a horizontal pitch (6) to the right. Total 6 patches, and k = 6.
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).