A Latin square critical set is a subset of the filled cells of a Latin square that has a unique completion
to that square and is minimal under deletion of cells. Minimality is with respect
to containment, not cardinality. The least size over all Latin
squares of order 
is denoted 
.
Bean (2026) reports the exact value
 |
where 
is the floor function. The proof asserts that every
partial Latin square with fewer than 
filled cells has an even number of completions. Together with a critical set of this
size in the cyclic Latin square, this gives the claimed
minimum. The general parity argument has not been independently verified here.
Bean (2026) states that Codex and Claude substantially generated the proof, with human direction and review. Small examples and intermediate identities were checked computationally, but those checks alone do not establish the all-orders theorem.
