How large is the biggest Sidon subset of the squares $\{1^2,\ldots,N^2\}$? Is it $N^{1-o(1)}$? (Erdős #773)

A set $A$ of integers is a Sidon set (a $B_2$ set) if all pairwise sums $a+a'$ ($a\le a'$) are distinct — equivalently all pairwise differences are distinct. Let $S(N)$ be the size of the largest Sidon subset $A\subseteq\{1^2,2^2,\ldots,N^2\}$ (the first $N$ perfect squares). Erdős (and Alon–Erdős) asked how large $S(N)$ can be, and specifically whether $S(N)=N^{1-o(1)}$. Computing $S(N)$ for fixed $N$ is a finite combinatorial optimization: build the graph/constraint 'no two disjoint pairs share a sum' and find a maximum Sidon subset (max independent set in the associated conflict hypergraph, encodable as ILP/SAT/clique). GOAL (computational): compute or lower-bound $S(N)$ for $N$ as large as feasible, and fit $\log S(N)/\log N$ to test the $N^{1-o(1)}$ hypothesis.

Acceptance. PARTIAL / PRIMARY (finite, machine-checkable): exact values (or certified lower bounds via explicit witnesses) of $S(N)$ for $N$ as large as feasible (state $N$), delivered with (i) an explicit maximum Sidon subset $A\subseteq\{1,\ldots,N^2\text{ squares}\}$ for each $N$, (ii) a verifier confirming all pairwise sums are distinct, and (iii) for exact values, an optimality certificate (ILP/SAT UNSAT bound), plus a fit of $\log S(N)/\log N$ toward $1$. Extending the table of $S(N)$ (cf. A390813) is itself a result. FULLY RESOLVES: a proof determining the order of $S(N)$ (e.g. $S(N)=N^{1-o(1)}$ or $S(N)=O(N^{1-c})$). State $N$ and the solver used.