3SUM in $O(n^{1.999112})$ word-RAM steps

For every $\kappa\in\mathbb{N}$, there exist a finite program $P$, a natural number $b$, a function $T:\mathbb{N}\to\mathbb{N}$, and a natural number $K$ such that $T(n)^{125000}\le K n^{249889}$ for every natural number $n\ge2$, using the exact rational exponent $1.999112=249889/125000$, and the following holds for every natural number $n$, every indexed family of integers $(x_0,\ldots,x_{n-1})$ satisfying $|x_i|\le n^\kappa$ at every index, and every natural word width $W\ge b(\operatorname{log}_2(n)+1)$, where $\operatorname{log}_2(n)=\lfloor\log_2 n\rfloor$ for positive $n$ and $\operatorname{log}_2(0)=0$: starting at program position $0$, $P$ halts within $T(n)$ instructions, with the halting instruction counted, and accepts if and only if there exist three pairwise distinct indices $i,j,k\in\{0,\ldots,n-1\}$ with $x_i+x_j+x_k=0$. The program, $b$, $T$, and $K$ may depend on $\kappa$ but are shared by all these $n$, inputs, and widths. Memory is an integer-addressed collection of $W$-bit words, initially containing the residue of $n$ modulo $2^W$ in cell $0$, the residues of $x_0,\ldots,x_{n-1}$ in cells $1,\ldots,n$, and zero in every other cell, including all negative-address cells. Each program instruction is one of the following, with its directly named cell addresses arbitrary fixed integers and its jump target a fixed natural-number program position: write the word $1$ to a named cell; add, subtract, or multiply the words in two named cells and write the result modulo $2^W$ to a named cell; load into a named cell from the address given by the signed interpretation of a word in a named cell; store the word in a named cell at the address given by the signed interpretation of a word in a named cell; jump to a named program position if a named cell's signed value is negative; accept; or reject. Signed interpretation is two's complement; for $W=0$ the sole word has signed value $0$. A write or a failed conditional jump advances the program position by one, and a successful conditional jump changes it to the specified position; accessing a program position past the list behaves as a reject instruction. Every instruction costs one step, and zero available steps never produces a halting verdict. Acceptance and rejection are the only required output: final memory is unrestricted. The correctness requirement includes $n=0$ and $n=1$, for which no asymptotic bound on $T(n)$ is imposed, and also $n=2$; all instances with fewer than three positions must reject. The integer $\kappa$ may be $0$, with $n^0=1$ also at $n=0$ and the empty input's size bound vacuous; no positivity is required of $b$ or $K$, and width $0$ is included whenever it satisfies the stated width inequality.