Exact (2,0)-partition numbers for two binary covering codes of covering radius two

H is 10 x 50 over F₂ with 50 distinct nonzero columns below 2¹0 (the only CS1-bound theorem, Hₛhape, proves length / distinctness / nonzero / < 2¹0; the Appendix-A bit-matrix vs Table-1 agreement is an out-of-Lean check, referee 2026-09-03)

The 1276 sums of at most two columns cover all 1024 syndromes, with multiplicity histogram 1:859, 2:129, 4:24, 5:9, 6:3

Exactly 973 syndromes need a pair (so the covering radius is exactly 2); their pair-multiplicity histogram is 1:821, 2:123, 4:19, 5:8, 6:2

H has exactly 10 dependent column triples (so d = 3), and exactly one, (491,734,821), meets three distinct blocks of the published partition

Every single-column deletion leaves at least 9 of the 1024 syndromes uncovered, with the minimum 9 attained exactly at columns 381, 479, 927

Table 2 of the source is a valid (2,0)-partition of H into ten nonempty blocks

The Kaikkonen-Rosendahl 10 x 51 matrix, reconstructed from the MSB-first hex of arXiv:2511.02542 Thm 4.3: 51 distinct nonzero columns, coverage 1024/1024, the guard h5+h27+h29 = 0, and Psi_KR of its Thm 5.2 is a valid (2,0)-partition into 11 blocks with that triple in three distinct blocks

p(H) = 10: no (2,0)-partition of the 50 columns of H into nine (hence into any k <= 9) blocks exists, so the ten-block partition of Table 2 is minimal

p(H_KR) = 10: a (2,0)-partition of the Kaikkonen-Rosendahl 51-column matrix into ten blocks exists and nine is impossible – whereas Theorem 5.2 of arXiv:2511.02542 exhibits an eleven-block partition (its p(H_KR; Psi_KR) = 11 is the cardinality of the exhibited partition, not a minimum – a different quantity, paper Remark 4.3; wording fixed 2026-09-03)

The forced-edge graph of H (821 edges, the syndromes with a unique covering pair) has chromatic number exactly 6: a 6-clique and a proper 6-colouring. So p(H) = 10 is not a chromatic-number bound; the four extra blocks come from the 152 multi-pair syndromes

l₂(10,2) <= 50 as a statement about a binary linear code: the 10 x 50 matrix over ZMod 2 has a surjective syndrome map, every word of F₂⁵0 is within hammingDist 2 of its kernel, and some word is not within 1 – so the code is [50,40]₂ of covering radius exactly 2

Negative controls: nine blocks and one block both refuted; eleven refuted as a minimum; merging the two singleton blocks of Table 2 leaves exactly one unsplit syndrome; deleting column 381 breaks coverage with exactly 9 uncovered; 404 of the 1225 column pairs are not forced edges; and the LRAT replay itself is non-vacuous – verifyCert returns false for the nine-block certificate against the satisfiable ten-block CNF and across the two matrices in both directions

The MSB-first misreading of M_KR still covers all 1024 syndromes, so the covering check cannot detect the convention error that Section 4.3 of arXiv:2608.27494v1 says it detects; the dependent-triple guard (48, not 0) and Psi_KR (27 unsplit syndromes) do detect it

Price of the proved symmetry break: on the nine-block CNF for H, cadical with 0 pinned columns produced a 9.09 GB LRAT in 28 minutes without terminating, with 3 pinned 13.2 s / 63.5 MB, with 4 pinned 1.44 s / 10.1 MB, with 5 pinned 0.47 s / 2.6 MB, with 6 pinned 0.10 s / 1.13 MB

Neither seed admits a (2,0)-partition into 2³ = 8 blocks, so the hypothesis n₀ >= 2ᵐ >= p(H₀) of Construction QM₂² fails at m = 3 for both – closing the route that would have given n = 407 (from H) or 415 (from H_KR) at r = 16, against the published 431