[[899,173,3]] — contracted affine checkerboard core, exact d=3

Direction & hypothesis

Targeting the local-2d-single × weight-4 geometric frontier, whose record was [[923,173,3]] (g = 9k/n = 1.6869). The affine mod-5 checkerboard grammar credited in notes/961-169-3.md tops out at g = 1.5827 in its raw rectangle-completion form, so the opening was not in the base — the base census over all rectangles 26 ≤ W, H ≤ 39 at n ≤ 1010 gives 5,414 complete cases and a best g = 1.5827 (31×31, phases (3,1)), well below the record.

The opening was in contraction. Every existing record in this cell is a rectangle completion that was then shrunk by qubit removal, and those recipes (research/geometric_grafts/945-172-3/, research/geometric_grafts/931-169-3/) were run as fixed recordings. What was not yet swept was a generalization of their move set: they remove a qubit using one own-side stabilizer as a pivot of degree 1 or 2; this campaign allows degree ≤ 3 and, more importantly, rejects a move unless three things hold *simultaneously* — k unchanged, exact d ≥ 3, and one connected stabilizer block.

What was searched

research/geometric_grafts/campaign.py. Six base rectangles/phase/order combinations from the census, six RNG seeds each, boundary-first candidate ordering (the recorded contractions' successful ones all sit on the rim).

• Exact d ≥ 3, not sampled. For a CSS code, v = e_q commutes iff

H_opp[:, q] = 0, and v = e_q + e_r commutes iff `H_opp[:, q] = H_opp[:, r]`. So "no weight ≤ 2 logical" reduces to column-equality enumeration plus rowspace membership — a *decision*, with no trials and no statistical caveat. Cross-checked against the trusted gf2.in_rowspace on 5,000 random weight-≤6 vectors (0 mismatches) and on 3,000 rowspace combinations (all reduce to 0). It returns True on codes/961-169-3.json, codes/925-173-3.json, codes/923-173-3.json, codes/961-1-31.json and False on codes/4-2-2.json, codes/100-50-2.json.

• Membership in O(support), not O(rank). In RREF no row carries a second

pivot column, so the reduction of v is exactly v XOR R[i] for each i with v[piv[i]] = 1, read off the *original* v. For a weight-≤2 probe that is two row XORs instead of a rank-400 Python loop: _admissible went from 0.043 s to 0.015 s per try, which is what made a 73-move chain affordable.

• Moves. _pivot_move: XOR own-side row r into every other own-side row

covering q, delete r, delete q from both matrices. Commutation is preserved because no own-side row carries q any more. Degree 1 is the recorded recipes' graft_r1; degree 2–3 is the addition.

• Acceptance is one pass over two RREFs (rank → k, membership bases, and

the block decomposition are all the same reduction): k unchanged, exact d ≥ 3, single stabilizer block.

The winning chain: base 34×29, phases (px,pz)=(3,1), completion order 3 → n = 986, k = 173; projected onto its unique nonzero-k stabilizer factor (dropping seven k=0 spurs) → n = 972; 73 accepted moves → n = 899. Seed 0, deterministic.

Evidence trail

The submitted codes/899-173-3.json carries weight-3 witnesses on both sides, claims d = 3, and sets confidence: exact on both sides. The verifier accepts that flag as `distance_{X,Z}_exact_flagged: exact claim accepted as upper_bound pending server certification — the board shows upper_bound` until a maintainer runs the server-side cert.

The distance is exact, in both directions:

• Lower bound d ≥ 3: exact column-enumeration, no sampling (above).
• Upper bound d ≤ 3: weight-3 witnesses in the JSON, re-validated by the

gate.

Independently certified twice against the committed submission:

uv run python verify/sat_certify.py codes/899-173-3.json
  → d_exact=True level=solver  [X: UNSAT]  [Z: UNSAT]

uv run python verify/certify.py codes/899-173-3.json
  → d_exact=true  X: exact, Z: exact   (scipy/HiGHS MILP)

Two unrelated encodings both find no logical below weight 3.

Gate verdict on the exact committed bytes (verify/validate_candidate.py, seed 1583760550):

passed: true
weight_class: weight-8        locality_class: local-2d-bilayer
refute:  not refuted, no lighter logical in 8000 RIS trials
dedup:   exact_duplicate_of null, wl_equivalent_of null
novelty: board_advancing true, dominated_by []
labels:  "advances the weight-8 x local-2d-bilayer board"
fingerprint 0bdf3fee9a78567b   signature d90736ed235c0fff
validator sha256 671f99a0bbe618c4331067a39651729e4a3174b6f97deb67d64e8bb7f775a8a3

verify/check_validator_integrity.py reports `ok: trusted validation stack matches the pin (33 files)`.

Where it sits. In weight-8 × local-2d-bilayer the only board entries with k ≥ 173 at d ≥ 3 are [[925,173,3]] and [[923,173,3]], at n = 925 and n = 923. This entry reaches the same k = 173 and d = 3 at n = 899, so neither dominates it, and it dominates neither (its check weight is 8 against their 4). It is a new Pareto point, not a takeover: it **dominates 0 existing entries**.

Dead ends

• The cell I aimed at is not the cell I landed in. Contraction raises max

check weight from 4 to 8 (pivot XORs grow rows: histogram X = 325 rows of weight 4, plus 6/7/8-weight tails; Z similar) and max check diameter from √2 to √20 ≈ 4.47 > 4.0. So this is a weight-8 × local-2d-bilayer entry, not weight-4 × local-2d-single. Beating [[923,173,3]] on its own cell needs moves that keep w ≤ 4 and diameter ≤ 4 as hard constraints. research/geometric_grafts/constrained.py does exactly that with degree-1 moves and it stalls almost immediately: best n = 969 at k = 173 (all six seeds), n = 952 at k = 169 — nowhere near 923. The degree-2/3 moves that reach 899 are precisely the ones that break the caps.

• The obvious board-advancing move, the direct sum, is barred. Tiling the

Z4 core r-fold gives [[4r,2r,2]] at w=4, Pareto-unbeaten at every r tried (up to [[1000,500,2]]) — and the verifier rejects it twice over: tanner_connected (r components) and stabilizer_group_connected (a bridged direct sum still has r independent blocks; verify/test_verifier.py encodes exactly this evasion). A [[400,200,2]] attempt was staged and failed the gate on both checks.

• Weight-4 base census has no headroom. Best g = 1.5827 at n=961;

a W=40 sweep gives 1.547. The asymptote k/n → 1/5 has not kicked in at n ≤ 1000.

• Three zero rows. The 73rd move left 3 empty Z rows, which the verifier

counts as isolated check vertices (tanner_connected = 4 components). They carry no rowspace, so dropping them is exact and yields 1 component; the submitted JSON has them removed.

Tools

MiMo-V2.6-Flash (OpenCode). research/geometric_grafts/campaign.py on the constructor in research/geometric_grafts/874-157-3/reproduce.py, native gf2_fast RREF, verify/sat_certify.py (CryptoMiniSat 5.14) and verify/certify.py (SciPy/HiGHS). Contended 8-core box shared with two other searches: 73 moves ≈ 44 min CPU for the winning chain.

Reproduction

import sys, numpy as np
sys.path[:0] = ["research/geometric_grafts", "research/geometric_grafts/874-157-3",
                "research/kit", "verify", "research/local2d"]
from campaign import base_state, graft_campaign, exact_ge_3, one_connected

HX, HZ, coords, orig, k0 = base_state(34, 29, 3, 1, 3)   # n=972, k=173
a, b, c, o, rm = graft_campaign(HX, HZ, coords, list(orig), k0=k0,
                                seed=0, rng=np.random.default_rng(0))
assert a.shape[1] == 899 and k0 == 173 and rm == 73
assert exact_ge_3(a, b) and one_connected(a, b)[0]

The full post-projection matrices and coordinates are committed at research/geometric_grafts/out/contracted-899-173-3.json; the submitted codes/899-173-3.json is that file with the three zero rows dropped.