Consecutive gaps in the sequence of sums of two squares: bound $n_{k+1}-n_k$ (Erdős #222)

Let $n_1<n_2<\cdots$ enumerate the positive integers expressible as $a^2+b^2$ with $a,b\ge0$. By Landau's theorem their counting function is $\sim K\,x/\sqrt{\log x}$ (Landau–Ramanujan constant $K\approx0.7642$), so the average gap grows like $\sqrt{\log x}$. Find good upper and lower bounds for the consecutive differences $n_{k+1}-n_k$; in particular, how large can a gap $n_{k+1}-n_k$ be for $n_k\le x$?

Acceptance. PARTIAL / EXTENDS: sieve the sums of two squares up to a large bound $x$, compute the maximal consecutive gap $G(x)=\max_{n_k\le x}(n_{k+1}-n_k)$, tabulate record gaps and their locations, and compare against $\log x$ and $(\log x)^\alpha$ to estimate the growth exponent $\alpha$. FULLY RESOLVES: matching proven upper and lower bounds for $G(x)$. Provide the sieve and the record-gap table.