Attack log — Linear covering codes of radius two

2026-08-16 — q1 (recovered from Codex session 01a009f7-5a24-71e1-bf4c-9667f4d2cbc4)

• Targets tried: $(r,n)=(8,25)$ (documented 26) — best anneal missed 3 of 256 syndromes; CP-SAT/Z3 no model. $(9,38)$ (documented 39) — CP-SAT UNKNOWN; Gabidulin 39-set deletions missed 8.
• Click: seed from the 51-column Kaikkonen–Rosendahl matrix at $r=10$. Fixed-cardinality targeted simulated annealing, xorshift64 run 0, hit 0 uncovered at proposal 3,600,281.
• Witness: 50 distinct nonzero 10-bit columns. F$_2$-rank 10. Exhaustive pair-XOR: 1024/1024. Density $319/256=1.24609375$.
• Commit remembered in Math-agent memory as 5b93f34. Watcher (Grok, 03:32 PT) independently rebuilt columns from H_r10_n50.txt (LSB = row 1) and confirmed 1024/1024.
• $n=49$ stochastic run left 7 uncovered syndromes — not a lower bound.

2026-08-16 — q2 (recovered from Codex session 01a00a33-832e-7d13-bf0e-59aeab97ec82)

• Push 50→49 using H_r10_n50.txt as seed. Search residue still 7 uncovered. Files: search_n49.c, search_n49_lifted.c, build_q2_residue.py, verify_q2_residue.py.
• Do not claim $f(2)$. Holes $(8,25)$ and $(9,38)$ remain secondary.

2026-08-16 afternoon — q3 (Claude, covering-result branch)

• Built result/ as a self-contained, independently reproducible artifact for the whole result. Run it with cd problems/covering/result && ./run_all.sh — 74 checks, exit 0, byte-identical output across runs.
• Two verifiers from scratch, two languages, opposite algorithms: result/verify/verify.py is pair-driven, result/verify/verify.rs is syndrome-driven. run_all.sh diffs their fact dumps. Both re-derive everything from the matrix text; neither reads a stored certificate. A third, quarantined oracle was run only afterwards and agreed on every value; it is not in the tree.
• The missing piece the overnight run never computed: a valid $(2,0)$-partition of $H$ into 10 blocks, against the paper's computer-searched $p(H_{KR})=11$ (arXiv:2511.02542 Thm 5.2). All 973 syndromes that need a pair have a cross-block one; 821 of them have a unique pair, so the partition is tightly forced.
• Minimality: deleting any single column leaves $\ge 9$ uncovered syndromes (best deletions 381, 479, 927 leave exactly 9). So the 50-set is a minimal 1-saturating set in $PG(9,2)$ — an LO code. Not padded.
• $d=3$, exactly 10 dependent triples; exactly one, $(491,734,821)$, spans three distinct blocks (6/4/9). Analogue of Thm 5.2(ii), which is what QM$_5^2$ needs at $r=28$.
• Propagation: Construction QM$_2^2$ (Thm 4.1, eq. (4.2), (4.4)) implemented from the paper. $n_0\ge 2^m\ge p(H_0)$ permits exactly $m=4,5$, asserted not assumed. Results exhaustively verified: $r=18$, $n=815$, 262144/262144; $r=20$, $n=1631$, 1048576/1048576. Published values 831 and 1663.
• These matrices differ from the alternate allocations in result/data/alt/ (the indicator allocation is a free choice) and both verify — two distinct witnesses at each $r$, not one.
• Family: $n=51\cdot 2^{r/2-5}-1$, $\bar\mu(2)\le 2601/2048\approx 1.27002$, against the paper's $2704/2048\approx 1.32031$ and the pre-2025 $729/512\approx 1.42383$. Even $r\le 64$ reachable by QM$_2^2$ from this seed: 10, 18, 20, 30–40, 46–64. Not reachable: 12, 14, 16, 22, 24, 26, 28, 42, 44 — the QM$_3^2$/QM$_5^2$ analogues are not implemented and nothing is claimed there.
• Soft spot, stated in the note: only $m=4,5$ are exhaustively verified. The rest of the family inherits the paper's $p(H_C)\le 2^{m+1}+1$ instead of computing partitions. That is the first thing a referee will push on.
• One number in the brief lost to the code: $2601/2048$ is $1.27002$, not $1.26999$. Fraction right, decimal wrong; no bound changes.
• Priority unresolved — Table 5.1 is "best as far as the authors know". Checklist and a drafted email to Davydov/Marcugini/Pambianco in result/PRIORITY.md. Nothing goes out before a human works that list.