Minimal additive 2-basis for $\{0,\ldots,n\}$: pin the constant in $g(n)^2\sim cn$ (Erdős #791)

Let $g(n)$ be minimal such that there exists $A\subseteq \{0,\ldots,n\}$ of size $g(n)$ with $\{0,\ldots,n\}\subseteq A+A$ (such an $A$ is called a finite additive $2$-basis for $\{0,\ldots,n\}$: every integer in $\{0,\ldots,n\}$ is a sum of two elements of $A$). Estimate $g(n)$. In particular is it true that $g(n)\sim 2n^{1/2}$?

Acceptance. FULLY RESOLVES: determine the asymptotic behavior of $g(n)$ — a proof that $g(n)^2/n$ converges together with the exact value of the limit (with matching construction and lower-bound proof). Machine-checkable (Lean/Coq) proof preferred; otherwise a complete written proof with all steps. ADVANCES: (a) an explicit 2-basis construction achieving $g(n)^2\le (c'+o(1))n$ with $c'$ strictly smaller than the best upper-bound constant stated in the background, with proof or machine-verifiable certificate of the covering property; (b) a lower bound with constant strictly larger than the best lower-bound constant stated in the background, with full proof; (c) new exact values of $g(n)$ extending OEIS A066063 beyond its current range, with reproducible search code and an exhaustiveness certificate; (d) a proof that the limit $\lim g(n)^2/n$ exists. Deliver the construction + verification code, or the proof file, or the search code + extended table of exact values.