A polynomial whose pairwise sums are all distinct (a polynomial Sidon set): does one exist? (Erdős #324)

Does there exist a polynomial $f(x)\in\mathbb{Z}[x]$ such that all the sums $f(a)+f(b)$ with $0\leq a<b$ (nonnegative integers) are distinct? Equivalently, is there a polynomial $f$ for which the image $\{f(n):n\geq 0\}$ is a Sidon set — a set all of whose pairwise sums are distinct?

Acceptance. FULLY RESOLVES (existence direction): a specific polynomial $f\in\mathbb{Z}[x]$ together with a proof that the sums $f(a)+f(b)$, $0\leq a<b$, are pairwise distinct (equivalently that the associated equal-pair-sum equation has only trivial solutions) — machine-checkable proof preferred. FULLY RESOLVES (nonexistence direction): a proof that no polynomial $f\in\mathbb{Z}[x]$ has this property. ADVANCES: (a) settle the degree-$5$ monomial case — prove $f(x)=x^5$ is a Sidon set (i.e. $x^5+y^5=z^5+w^5$ has only trivial nonnegative solutions), or exhibit an explicit collision refuting it; (b) extend the collision-free verified range for $x^5$ (or another explicit candidate) to a new record $N$, supplying the search code and a certificate that no nontrivial equal pair-sum occurs below $N$; (c) prove a new nonexistence result for a degree beyond those already excluded — e.g. that no general polynomial of some degree $d\geq 4$ works (Dubickas-Novikas handled general degree $3$; $x^4$ is excluded only as a monomial) — with proof; (d) construct, in the style of Ruzsa, an explicit polynomial-image Sidon set of a new degree, with proof. Deliver the polynomial plus proof, the nonexistence proof, or the collision-search code plus certified range / found collision.