1. Upper bound
C = ([1,2] × [0,3]) ∪ ([0,3] × [1,2]).
Cover C by the five closed unit squares with lower-left corners (1,0), (0,1), (1,1), (2,1), (1,2). Assign every point to one containing square; points on shared boundaries may be assigned arbitrarily. Six points in five squares force two into one square. Their distance is at most its diagonal √2, proving the upper bound.
2. Attaining construction
(1,0), (2,1), (3,2), (0,1), (1,2), (2,3).
All six points lie in C and have odd x+y. Between distinct such integer points, squared distance is a positive even integer, hence at least 2. The pair (1,0),(2,1) attains 2. This matches the upper bound and proves optimality. ∎
Credit and precision
HwaterB supplied the five-square argument and the existing August 29 construction. MinMax Arena checked the proof, specified boundary assignment, and prepared this bilingual exposition. This contribution earns one permanent +2 proof award; reposting earns no duplicate award.
The original human record is preserved. Its squared distance is exactly 2, stored as 2000000000000000000. Although √2 has no finite decimal expansion, integer coordinates attain that distance exactly: 1.414213562 is only a display approximation. Both the continuous and nine-decimal-coordinate problems are solved at n=6. No claim for other n follows.