Estimate $g_k(N)$: the surplus forcing all pairwise sums of some $k$ integers into $A$ (Erdős #866)

Let $k\geq 3$, and let $g_k(N)$ be minimal such that if $A\subseteq\{1,\ldots,2N\}$ has $\lvert A\rvert\geq N+g_k(N)$ then there exist integers $b_1,\ldots,b_k$ such that all $\binom{k}{2}$ pairwise sums $b_i+b_j$ ($i<j$) lie in $A$ (the $b_i$ themselves need not lie in $A$). Estimate $g_k(N)$.

Acceptance. FULLY RESOLVES: determine the growth order of $g_k(N)$ for every $k\geq 3$ — i.e. exhibit, with complete proof, the correct exponents $\theta_k$ with $g_k(N)=N^{\theta_k+o(1)}$ for each $k\geq 7$ (closing the gap between $N^{1-2^{-k}}$ and $N^{1-\epsilon}$), together with the exact eventual value of $g_4(N)$; machine-checkable (Lean/Coq) preferred, otherwise a full written proof. ADVANCES: (a) determine $g_4$ exactly, or strictly improve the explicit bound $g_4(N)\leq 2032$ stated in the background, with proof; (b) determine the growth order of $g_k(N)$ for any single $k\geq 7$, or strictly improve the general upper bound $N^{1-2^{-k}}$ or the large-$k$ lower bound $N^{1-\epsilon}$; (c) sharpen $g_5$ or $g_6$ to asymptotics with explicit constants; (d) a verified table of exact values $g_k(N)$ for small $N$ and $k\in\{3,4,5,6\}$, each entry certified by an extremal set of size $N+g_k(N)-1$ with no valid $b_1,\ldots,b_k$ plus a reproducible exhaustive-search certificate that every larger set works. Deliver the proof file, or the code plus certificates and tables.