q16: covering exploration

Started 2026-09-23. Three finite handles, each outside the settled q9–q12 searches:

1. Inherited partition merge, $r=41$, radius 3. The q13 817-column seed has a certified 34-block $(3,0)$-partition. Merging any two blocks gives 33, the field-size threshold for QM$_4^3$ with $m=5$. This would give $n=32(817+1)-1=26175$, shorter than the notebook's 26206 and the published 26238. Test all 561 unordered merges using exact counts of distinct-block three-column representations. This is the first handle.
2. New small radius-3 seed at $r=12$. q12 excluded $p\le17$ only for the specific OK37 matrix. A different $n\le37$ radius-3 seed with such a partition would permit QM$_4^3$ at $m=4$ and $r=24$. This requires a matrix search, not a repeat of q12's fixed-matrix SAT proof.
3. Two quotient-block replacement at $r=10$. q9 replaced one block and left 49 open. Simultaneously changing two blocks is a distinct move; its feasibility depends on a constrained pair-cover search, rather than a deletion or one-block repair.

The first handle is the smallest exact finite test and has a 31-column prize.

Outcome

The exact merge sweep tested all 561 pairs. Two preserve the $(3,0)$ partition: (7,31) and (15,24). The first is committed as partition_r26_n817_p33.txt. The independent full seed sweep and lift identity check prove the inequality in CLAIM.md. Replay with sh compute/q16/run_all.sh from the problem folder.

A follow-up exact deletion screen counts every representation by at most three distinct seed columns. Every one of the 817 columns participates in a unique representation of at least one syndrome. This holds with the 33-block restriction (59,867,904 uniquely represented syndromes) and without it (59,734,976). Thus no one-column deletion of this particular seed can retain radius 3, even if the partition is discarded. This does not exclude a different 816-column matrix.

The search code is search_merges.c; run it from the repository root after compiling with cc -O3 -std=c11 and pass the 817-column seed and q13 34-block partition. The scan is exact and uses about 600 MiB.