The integer 3-SUM Hypothesis excludes an $O(n^{2-\varepsilon})$ word-RAM algorithm for every constant $\varepsilon>0$, on arrays of $n$ distinct integers in $[-n^3,n^3]$. Words have $O(\log n)$ bits.

$Deterministic$ uses deterministic algorithms; $Randomized$ also excludes randomized algorithms with two-sided error at most $1/3$ and a worst-case time bound. The latter is the randomized convention supplied by this submission. We do not identify it here with an expected-time or a zero-error formulation.

These are propositions. The bound concerns a fixed improvement in the exponent; it permits savings by logarithmic or other subpolynomial factors.