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[[85,40,5]] d =
n
85
k
40
d
5
kd²/n
11.765
w
24
X/Z
1

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Distance

X/Z asymmetry 1 · d_X = 5, d_Z = 5 · w_X = 24, w_Z = 24 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 5 · witness weight 5 (claimed upper_bound)
witness operator (support, 5 qubits)
[14, 22, 34, 41, 77]
d_Z 5 · witness weight 5 (claimed upper_bound)
witness operator (support, 5 qubits)
[5, 17, 24, 60, 82]
certificate exact, d = 5 · CryptoMiniSat 5.14.7 SAT
X: no logical < 5 exists; Z: no logical < 5 exists

Diagnostics

computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking
girth H_X 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 24 · H_Z 24
qubit degrees H_X 1–11 (mean 5.929) · H_Z 1–11 (mean 6.776)
trapping sets H_X (1,1)×7 (2,1)×5 (3,1)×14 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 7 (1,2): 8 (1,3): 6 (1,4): 4 (1,5): 6 (1,6): 14 (1,7): 9 (1,8): 17 (1,9): 7 (1,10): 6 (1,11): 1 (2,1): 5 (2,2): 17 (2,3): 42 (2,4): 58 (2,5): 92 (2,6): 175 (2,7): 315 (2,8): 443 (2,9): 428 (2,10): 415 (2,11): 305 (2,12): 238 (2,13): 96 (2,14): 34 (2,15): 10 (2,16): 1 (3,1): 14 (3,2): 83 (3,3): 273 (3,4): 639 (3,5): 1502 (3,6): 3203 (3,7): 6249 (3,8): 9557 (3,9): 12217 (3,10): 13126 (3,11): 11790 (3,12): 8646 (3,13): 5158 (3,14): 2703 (3,15): 1103 (3,16): 413 (3,17): 77 (3,18): 7 (3,19): 1
trapping sets H_Z (1,1)×7 (2,1)×3 (3,1)×6 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 7 (1,2): 3 (1,3): 8 (1,4): 6 (1,5): 2 (1,6): 7 (1,7): 10 (1,8): 10 (1,9): 12 (1,10): 15 (1,11): 5 (2,1): 3 (2,2): 13 (2,3): 23 (2,4): 30 (2,5): 60 (2,6): 83 (2,7): 147 (2,8): 313 (2,9): 384 (2,10): 409 (2,11): 424 (2,12): 347 (2,13): 302 (2,14): 164 (2,15): 58 (2,16): 15 (2,17): 2 (2,18): 2 (3,1): 6 (3,2): 49 (3,3): 111 (3,4): 263 (3,5): 575 (3,6): 1333 (3,7): 2911 (3,8): 5182 (3,9): 8051 (3,10): 10900 (3,11): 12810 (3,12): 12834 (3,13): 10354 (3,14): 6934 (3,15): 4076 (3,16): 2038 (3,17): 913 (3,18): 325 (3,19): 51 (3,20): 2

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction Greedy check-deletion chain from board code codes/85-37-5.json (reconstructed from qecdb.org record 68e107e6). Deleted 3 X-checks [7, 8, 9] over 3 greedy steps, each priced by the check-deletion theorem (unpublished theorem by @mathysrennela, claim 12.1). k' = 37 + 3 = 40 exact by rank; the deleted X rows weigh 24, above the inherited d <= 5 tier (term 1), so the packaged witnesses are gate-confirmed weight-5 logicals on both sides.
model GLM 5.3 Flash (claimed, not verified)
date 2026-09-09
notes Derived code, honest provenance: check-deletion child of its source entry codes/85-37-5.json (k 37 -> 40 at equal n, distance tier, and weight class; different check sets, not equivalent to the source under any code symmetry). Deleted 3 X-checks [7, 8, 9]; the deleted X rows are the embedded per-side X witnesses, the source's Z witness is inherited unchanged (term-1 of the check-deletion theorem, an unpublished theorem by @mathysrennela). Parameter novelty vs the literature unverified.
family other (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight > 8 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[85,40,5]] — check-deletion chain of codes/85-37-5.json

Direction & hypothesis

Check-deletion move (an unpublished check-deletion theorem by @mathysrennela, PROVEN by the author; numbered claim 12.1 in the author's taxonomy): deleting r independent checks gives k' = k + r exact and `d' = min(d, min deleted-check weight, min exposed-logical weight)`. The board-wide wave-1 census (529 sources, 39,818 moves, 348 survivors at 31 distinct points) promoted [[700,212,28]] (PR #934, merged). Wave 2 asked whether *greedy chains* keep going: delete one check per step, re-price, repeat. Hypothesis for this source: three weight-24 X-checks carry slack that converts to logical qubits without disturbing the d = 5 tier.

What was searched

Greedy chain (driver kept as local staging; the method here is complete): at each step, enumerate candidate deletions, price each by the theorem's free bound (term 2: the deleted row's weight), prune on combined-Tanner connectivity (issue #921 semantics, pre-filtered as a gate-in-waiting), qubit coverage (no qubit unchecked on either side), and board potential; delete the best candidate; recompute k exactly by GF(2) rank; repeat to a depth cap of 10. Endpoint screening: 2500 RIS trials/side, then the trusted gate (validate_candidate: verify + 8000-trial refutation + dedup + novelty).

Evidence trail

  • k: 37 -> 40, exact by GF(2) rank arithmetic (never bookkeeping).
  • Claim: d <= 5 per side, upper_bound confidence. The deleted X rows weigh 24 — above the tier — so the claim is inherited (term 1). The packaged witnesses are gate-confirmed weight-5 logicals on both sides (the verifier checks each: commutes with the kept opposite checks, outside the kept own rowspace, weight 5).
  • Trusted verifier passed locally (verify/qldpc_verify.py codes/85-40-5.json,

    • earned_distance block).

  • Board position: strictly dominates its source entry codes/85-37-5.json on k (37 -> 40) at equal n, distance tier, and weight class. Cell: weight-9plus x unrestricted.

Dead ends

  • The chain stopped after 3 accepted steps: the step-guard found no further deletion passing the free-bound/coverage/kill criteria.
  • Wave-1 lesson applied: the claim is priced by the theorem's terms, never by

    • the randomized search's output — at large n that search is far from tight (measured: weight-84 proposals where weight-28 witnesses exist).

Tools

GLM 5.3 Flash (matches provenance.model). Repo kit: GF(2) rank core, RIS surrogate, trusted verifier. The greedy-chain driver is fully specified above (~50 lines against the kit) and is deliberately not committed with this PR (one concern per PR).

Reproduction

1. Load codes/85-37-5.json from this PR's tree. 2. Delete the listed X-checks by 0-based index; keep every other check verbatim (indices refer to the parent's checks.X ordering). 3. k' = n - rank(H'_X) - rank(H_Z) = 40. 4. Verify: uv run python verify/qldpc_verify.py codes/85-40-5.json.

Parity checks

X-checks 21 (max weight 24) · Z-checks 24 (max weight 24)
H_X (21 checks, sparse supports)
[0, 2, 3, 9, 12, 13, 18, 21, 22, 27, 30, 31, 32, 34, 36, 38, 43, 46, 48, 49, 52, 54, 56, 61] [1, 3, 4, 10, 13, 14, 19, 22, 23, 28, 31, 32, 33, 35, 37, 39, 44, 47, 49, 50, 53, 55, 57, 62] [2, 4, 5, 11, 14, 15, 20, 23, 24, 29, 32, 33, 34, 36, 38, 40, 45, 48, 50, 51, 54, 56, 58, 63] [3, 5, 6, 12, 15, 16, 21, 24, 25, 30, 33, 34, 35, 37, 39, 41, 46, 49, 51, 52, 55, 57, 59, 64] [4, 6, 7, 13, 16, 17, 22, 25, 26, 31, 34, 35, 36, 38, 40, 42, 47, 50, 52, 53, 56, 58, 60, 65] [5, 7, 8, 14, 17, 18, 23, 26, 27, 32, 35, 36, 37, 39, 41, 43, 48, 51, 53, 54, 57, 59, 61, 66] [6, 8, 9, 15, 18, 19, 24, 27, 28, 33, 36, 37, 38, 40, 42, 44, 49, 52, 54, 55, 58, 60, 62, 67] [10, 12, 13, 19, 22, 23, 28, 31, 32, 37, 40, 41, 42, 44, 46, 48, 53, 56, 58, 59, 62, 64, 66, 71] [11, 13, 14, 20, 23, 24, 29, 32, 33, 38, 41, 42, 43, 45, 47, 49, 54, 57, 59, 60, 63, 65, 67, 72] [12, 14, 15, 21, 24, 25, 30, 33, 34, 39, 42, 43, 44, 46, 48, 50, 55, 58, 60, 61, 64, 66, 68, 73] [13, 15, 16, 22, 25, 26, 31, 34, 35, 40, 43, 44, 45, 47, 49, 51, 56, 59, 61, 62, 65, 67, 69, 74] [14, 16, 17, 23, 26, 27, 32, 35, 36, 41, 44, 45, 46, 48, 50, 52, 57, 60, 62, 63, 66, 68, 70, 75] [15, 17, 18, 24, 27, 28, 33, 36, 37, 42, 45, 46, 47, 49, 51, 53, 58, 61, 63, 64, 67, 69, 71, 76] [16, 18, 19, 25, 28, 29, 34, 37, 38, 43, 46, 47, 48, 50, 52, 54, 59, 62, 64, 65, 68, 70, 72, 77] [17, 19, 20, 26, 29, 30, 35, 38, 39, 44, 47, 48, 49, 51, 53, 55, 60, 63, 65, 66, 69, 71, 73, 78] [18, 20, 21, 27, 30, 31, 36, 39, 40, 45, 48, 49, 50, 52, 54, 56, 61, 64, 66, 67, 70, 72, 74, 79] [19, 21, 22, 28, 31, 32, 37, 40, 41, 46, 49, 50, 51, 53, 55, 57, 62, 65, 67, 68, 71, 73, 75, 80] [20, 22, 23, 29, 32, 33, 38, 41, 42, 47, 50, 51, 52, 54, 56, 58, 63, 66, 68, 69, 72, 74, 76, 81] [21, 23, 24, 30, 33, 34, 39, 42, 43, 48, 51, 52, 53, 55, 57, 59, 64, 67, 69, 70, 73, 75, 77, 82] [22, 24, 25, 31, 34, 35, 40, 43, 44, 49, 52, 53, 54, 56, 58, 60, 65, 68, 70, 71, 74, 76, 78, 83] [23, 25, 26, 32, 35, 36, 41, 44, 45, 50, 53, 54, 55, 57, 59, 61, 66, 69, 71, 72, 75, 77, 79, 84]
H_Z (24 checks, sparse supports)
[0, 2, 3, 9, 12, 13, 18, 21, 22, 27, 30, 31, 32, 34, 36, 38, 43, 46, 48, 49, 52, 54, 56, 61] [1, 3, 4, 10, 13, 14, 19, 22, 23, 28, 31, 32, 33, 35, 37, 39, 44, 47, 49, 50, 53, 55, 57, 62] [2, 4, 5, 11, 14, 15, 20, 23, 24, 29, 32, 33, 34, 36, 38, 40, 45, 48, 50, 51, 54, 56, 58, 63] [3, 5, 6, 12, 15, 16, 21, 24, 25, 30, 33, 34, 35, 37, 39, 41, 46, 49, 51, 52, 55, 57, 59, 64] [4, 6, 7, 13, 16, 17, 22, 25, 26, 31, 34, 35, 36, 38, 40, 42, 47, 50, 52, 53, 56, 58, 60, 65] [5, 7, 8, 14, 17, 18, 23, 26, 27, 32, 35, 36, 37, 39, 41, 43, 48, 51, 53, 54, 57, 59, 61, 66] [6, 8, 9, 15, 18, 19, 24, 27, 28, 33, 36, 37, 38, 40, 42, 44, 49, 52, 54, 55, 58, 60, 62, 67] [7, 9, 10, 16, 19, 20, 25, 28, 29, 34, 37, 38, 39, 41, 43, 45, 50, 53, 55, 56, 59, 61, 63, 68] [8, 10, 11, 17, 20, 21, 26, 29, 30, 35, 38, 39, 40, 42, 44, 46, 51, 54, 56, 57, 60, 62, 64, 69] [9, 11, 12, 18, 21, 22, 27, 30, 31, 36, 39, 40, 41, 43, 45, 47, 52, 55, 57, 58, 61, 63, 65, 70] [10, 12, 13, 19, 22, 23, 28, 31, 32, 37, 40, 41, 42, 44, 46, 48, 53, 56, 58, 59, 62, 64, 66, 71] [11, 13, 14, 20, 23, 24, 29, 32, 33, 38, 41, 42, 43, 45, 47, 49, 54, 57, 59, 60, 63, 65, 67, 72] [12, 14, 15, 21, 24, 25, 30, 33, 34, 39, 42, 43, 44, 46, 48, 50, 55, 58, 60, 61, 64, 66, 68, 73] [13, 15, 16, 22, 25, 26, 31, 34, 35, 40, 43, 44, 45, 47, 49, 51, 56, 59, 61, 62, 65, 67, 69, 74] [14, 16, 17, 23, 26, 27, 32, 35, 36, 41, 44, 45, 46, 48, 50, 52, 57, 60, 62, 63, 66, 68, 70, 75] [15, 17, 18, 24, 27, 28, 33, 36, 37, 42, 45, 46, 47, 49, 51, 53, 58, 61, 63, 64, 67, 69, 71, 76] [16, 18, 19, 25, 28, 29, 34, 37, 38, 43, 46, 47, 48, 50, 52, 54, 59, 62, 64, 65, 68, 70, 72, 77] [17, 19, 20, 26, 29, 30, 35, 38, 39, 44, 47, 48, 49, 51, 53, 55, 60, 63, 65, 66, 69, 71, 73, 78] [18, 20, 21, 27, 30, 31, 36, 39, 40, 45, 48, 49, 50, 52, 54, 56, 61, 64, 66, 67, 70, 72, 74, 79] [19, 21, 22, 28, 31, 32, 37, 40, 41, 46, 49, 50, 51, 53, 55, 57, 62, 65, 67, 68, 71, 73, 75, 80] [20, 22, 23, 29, 32, 33, 38, 41, 42, 47, 50, 51, 52, 54, 56, 58, 63, 66, 68, 69, 72, 74, 76, 81] [21, 23, 24, 30, 33, 34, 39, 42, 43, 48, 51, 52, 53, 55, 57, 59, 64, 67, 69, 70, 73, 75, 77, 82] [22, 24, 25, 31, 34, 35, 40, 43, 44, 49, 52, 53, 54, 56, 58, 60, 65, 68, 70, 71, 74, 76, 78, 83] [23, 25, 26, 32, 35, 36, 41, 44, 45, 50, 53, 54, 55, 57, 59, 61, 66, 69, 71, 72, 75, 77, 79, 84]
Code ID 85-40-5 · download JSON · raw on GitHub