For every fixed $\kappa\in\mathbb N$, there exist a finite deterministic word-RAM program $P$, a natural constant $b$, and a step-bound function $T$ that work for every input size $n$ and every word width $W\ge b(\operatorname{Nat.log2}(n)+1)$. Given $n$ integers $x_0,\ldots,x_{n-1}$ with $|x_i|\le n^\kappa$, the program halts within $T(n)$ steps and accepts if and only if there are pairwise distinct indices $i,j,k$ such that $x_i+x_j+x_k=0$.

The time bound is $T(n)=O(n^{1.9992})$: precisely, there is a natural constant $K$ such that $T(n)^{1250}\le K n^{2499}$ for every $n\ge2$. The program must also halt and reject when $n<3$. Repeated integer values at different positions are allowed; no witness triple needs to be output.

This is exactly EndStatement.Theorem_22_3SUM from the accompanying formalization. It is the integer 3SUM part of Theorem 22, using Corollary 26, presented here as an open proof obligation.