Pin the growth constant of the largest quasi-Sidon subset of $\{1,\ldots,N\}$ (Erdős #840)

Let $f(N)$ be the size of the largest quasi-Sidon subset $A\subset\{1,\ldots,N\}$, where $A$ is quasi-Sidon if $$\lvert A+A\rvert=(1+o(1))\binom{\lvert A\rvert}{2},$$ i.e. all but a vanishing proportion of the pairwise sums $a+b$ are distinct (a genuine Sidon set has $\lvert A+A\rvert=\binom{\lvert A\rvert}{2}+\lvert A\rvert$, so every Sidon set is quasi-Sidon). How does $f(N)$ grow? In particular, determine the constant $c$ such that $f(N)=(c+o(1))N^{1/2}$.

Acceptance. FULLY RESOLVES: a proof that $f(N)=(c+o(1))N^{1/2}$ for an explicit constant $c$, i.e. matching upper and lower bounds — a complete written proof (machine-checkable preferred); the o(1) in the defining property makes this proof-shaped, and no finite computation alone can close it. ADVANCES: (a) a lower bound $f(N)\geq (c'+o(1))N^{1/2}$ with $c'$ strictly greater than the $2/\sqrt{3}$ constant stated in the background, via an explicit construction with proof; (b) an upper bound with constant strictly smaller than Pikhurko's constant stated in the background, with proof; (c) a reproducible finite-$N$ computational study: exact optima of the relaxed problem (max $|A|$ with $|A+A|\geq(1-\delta)\binom{|A|}{2}$) for a grid of $N$ and $\delta$, with code, exhaustiveness certificates for exact instances, and the extremal sets — together with any new construction pattern it reveals; (d) resolution of the analogous constant for a natural intermediate family (e.g. $|A+A|\geq(1-\epsilon)\binom{|A|}{2}$ for fixed $\epsilon$). Deliver the proof file, or the construction plus verification, or the optimization code with tables and certificates.