[[276,5,13]] — reduction of the board's [[299,5,13]]

What is new here

The source, @FarLab and @vprusso's codes/299-5-13.json, carries **ten qubits that are not entangled with anything**. They are removable by an argument, with no distance search of any kind, and the argument is what this note is mainly about. Twenty-three qubits come out in total; the other thirteen come from the graft moves used in my earlier reductions on this board.

Move 0: fixed qubits

Call a qubit *q* fixed when the unit vector e_q lies in the row space of H_Z — that is, when some element of the stabilizer group acts on *q* alone. It does not have to be one of the published generators.

Three consequences, in order:

1. Commutation forces H_X to have a zero column at *q*. An X-check meeting *q* would anticommute with a weight-1 Z element supported there. 2. So *q* carries a product state, and n → n-1 leaves *k* unchanged: the rank of H_Z drops by exactly one and the rank of H_X cannot move. 3. The distance is unchanged exactly. Every logical representative can be multiplied by that weight-1 stabilizer until its support at *q* is empty, so the minimum logical weight cannot move in either direction.

The repository already implements this idea, in boundary_engine._cleanup (arXiv:2504.08887 Sec. III D step 4) — but only for a literal weight-1 generator row. Membership in the row *space* is strictly weaker, and it is what the board actually contains. One RREF settles every qubit at once: *q* is fixed exactly when every null-space vector of H_Z vanishes at *q*, and the RREF row with pivot *q* is then e_q itself.

The deletion is arranged so that no check can grow. Take the RREF with its row-operation record; exactly one generator of the subset summing to e_q is replaced by e_q, which leaves the row space unchanged because that generator equals e_q plus the rest of the subset; e_q then clears column *q* from the other rows, which can only lower their weights and their support diameters; the row and the column are then dropped. Maximum check weight stays 6 and the measured interaction radius stays 3.1623 across this stage.

In codes/299-5-13.json the fixed qubits are 25, 38, 51, 64, 77, 90, 103, 116, 129 and 142 — a stride-13 tail, each lying in zero X-checks. Deleting them gives [[289,5,13]] with the distance still exactly 13, inheriting the source's MILP-certified value rather than re-asserting it.

Sweeping the whole board, thirteen entries carry fixed qubits. Most of them are codes I have already reduced further by other means; codes/299-5-13.json is the one where the fixed qubits are the interesting part of the story.

Moves 1–3: the graft set

Applied after move 0, to a fixpoint, on the source's own coordinates.

1. Graft (Liang, Eberhardt, Chen, arXiv:2504.08887 Sec. III E): remove a qubit lying in exactly one stabilizer of some type, together with that stabilizer. CSS commutation survives automatically, because no other same-type row touches the qubit. 2. Weight-1 cleanup (Sec. III D step 4): the literal form of move 0, which grafting creates opportunities for. Also exact, also search-free. 3. Capped merge-graft — mine, introduced with [[454,8,17]] (#935). If a qubit lies in exactly *r* ≤ 3 stabilizers of one type, pick a pivot row R_p and replace every other R_i by R_p + R_i. Those are row operations on the generators: the code is unchanged, only the generating set moves, and the qubit then sits in R_p alone, where move 1 applies. Every pivot choice is tried.

Moves 1 and 2 can only shrink a check; move 3 can widen one, so every merged row is accepted only if it still has weight ≤ 6 and a support diameter within the bilayer cap, and the whole layout radius is re-measured after each accepted move. Here it moved: 3.1623 before, 4.0000 after, still far inside the bilayer cap of 7.0 and in fact still inside the single-layer cap of 4.0, so the code stays in the cell it came from. Twelve merge-grafts and one cleanup took 289 → 276.

Unlike move 0, move 3 has no distance argument. Each accepted merge-graft was screened and then confirmed twice with independent seeds — a filter, not evidence.

Verification

• Qubit positions are the source's. Every surviving qubit keeps the coordinate it

has in codes/299-5-13.json; the reduction only deletes. Layers unchanged at 2.

• The fixed-qubit claim was re-derived outside the reducer: for each of the ten,

rank([H_Z; e_q]) == rank(H_Z) and the H_X column is empty; deleting the ten columns from the *original* matrices gives k = 5, commuting checks, and the same two row spaces as the packaged output.

• Fresh-seed RIS ladder on the reduced code: 13 at 20k → 200k → 1M → 5M → 20M

trials per rung, independent seeds, both sides searched jointly. Nothing lighter than 13 appeared. The lightest valid logical found is the witness in this file.

• verify/validate_candidate.py passes: board_advancing: true, no duplicate and

no WL-equivalent entry, cell weight-6 × local-2d-bilayer.

Distance is reported as an upper bound. Twenty million trials is the deepest null result I have; on this project a [[682,10,38]] candidate survived three fresh seeds to five million trials and then fell at twenty, so a flat ladder is not a proof.

What it costs and what it buys

[[276,5,13]] dominates codes/299-5-13.json on (n, k, d, w) and is dominated by nothing inside weight-6 × local-2d-bilayer. kd²/n goes 2.826 → 3.062. The source is not superseded as a construction — it is the construction; this is a reduction of it, and the k=5 open-boundary family is @FarLab and @vprusso's.