  As one of the three main pillars of fine-grained complexity theory, the 3SUM
problem explains the hardness of many diverse polynomial-time problems via
fine-grained reductions. Many of these reductions are either directly based on
or heavily inspired by P\u{a}tra\c{s}cu's framework involving additive hashing
and are thus randomized. Some selected reductions were derandomized in previous
work [Chan, He; SOSA'20], but the current techniques are limited and a major
fraction of the reductions remains randomized.
  In this work we gather a toolkit aimed to derandomize reductions based on
additive hashing. Using this toolkit, we manage to derandomize almost all known
3SUM-hardness reductions. As technical highlights we derandomize the hardness
reductions to (offline) Set Disjointness, (offline) Set Intersection and
Triangle Listing -- these questions were explicitly left open in previous work
[Kopelowitz, Pettie, Porat; SODA'16]. The few exceptions to our work fall into
a special category of recent reductions based on structure-versus-randomness
dichotomies.
  We expect that our toolkit can be readily applied to derandomize future
reductions as well. As a conceptual innovation, our work thereby promotes the
theory of deterministic 3SUM-hardness.
  As our second contribution, we prove that there is a deterministic universe
reduction for 3SUM. Specifically, using additive hashing it is a standard trick
to assume that the numbers in 3SUM have size at most $n^3$. We prove that this
assumption is similarly valid for deterministic algorithms.
