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Claim. For every integer $N\ge1$, every Sidon set $A\subseteq\{1,\ldots,N\}$ and every $\epsilon>0$, there are arbitrarily large integers $M$ and a set $B\subseteq\{N+1,\ldots,M\}$ such that $A\cup B$ is a Sidon set of size at least

$$\left(\frac1{\sqrt2}-\epsilon\right)M^{1/2}.$$

Here a Sidon set is one whose sums $a+b$ over unordered pairs, repeated elements included, are pairwise distinct.

Covers. Problem 44 asks for size at least $(1-\epsilon)M^{1/2}$ for every $\epsilon>0$. The claim gives the coefficient $1/\sqrt2$ in place of $1$, and so answers the question yes, for every $N$ and $A$, at each $\epsilon>1-1/\sqrt2$. For $0<\epsilon\le1-1/\sqrt2$, that is for the coefficient $1$ itself, the question remains open.

Credit. The extension specializes Singer's prime-power difference-set theorem, consumed as an external premise at statement level; its proof is not reconstructed. Deleting finitely many conflicting differences from a Singer set and separating the three kinds of pair sums are Plasma AI's. No novelty is claimed.

Standing. L31 is author-recorded at tier 0, without independent examination or a Lean proof. Catalog acceptance is pending.

Depends on. L31 (tier 0), which consumes no other native claim; its one external premise is Singer's theorem, credited above.