The 3SUM hypothesis: is 3SUM solvable in truly subquadratic time?

Yes: a deterministic $O(n^{1.9992})$ algorithm for 3SUM on polynomially bounded integers on the word RAM, refuting the hypothesis, and a Las Vegas $O(n^{1.998})$ algorithm for real inputs using only additions, subtractions and comparisons. The core is a new algorithm for wanted entries of thin matrix products (a modified Coppersmith rectangular algorithm with a Schönhage identity), which solves Lopsided All-Edges Sparse Triangle; known reductions give 3SUM. The same paper refutes the APSP hypothesis, catalogued separately. The constants are enormous, so the impact is on the theory: the conditional lower bounds built on 3SUM no longer stand as stated. SETH and Orthogonal Vectors are unaffected.

Yes: a deterministic $O(n^{1.9992})$ algorithm for 3SUM on polynomially bounded integers on the word RAM, refuting the hypothesis, and a Las Vegas $O(n^{1.998})$ algorithm for real inputs using only additions, subtractions and comparisons. The core is a new algorithm for wanted entries of thin matrix products (a modified Coppersmith rectangular algorithm with a Schönhage identity), which solves Lopsided All-Edges Sparse Triangle; known reductions give 3SUM. The same paper refutes the APSP hypothesis, catalogued separately. The constants are enormous, so the impact is on the theory: the conditional lower bounds built on 3SUM no longer stand as stated. SETH and Orthogonal Vectors are unaffected.