The APSP hypothesis: are all-pairs shortest paths computable in truly subcubic time?

Yes: deterministic $O(n^{2.9995})$ algorithms for APSP and for the (min,+)-product with polynomially bounded integer weights on the word RAM (the Lean development states $n^{2.99942}$), and Exact Triangle in $O(n^{2.9983})$, refuting the APSP and Exact Triangle hypotheses; a Las Vegas $O(n^{2.998})$ version works for real weights. The core is a new thin-matrix-product algorithm solving Lopsided All-Edges Sparse Triangle, with known reductions giving the rest. The same paper refutes the 3SUM hypothesis, catalogued separately. The constants are enormous; the impact is on the theory of subcubic equivalences, not on practice. SETH and Orthogonal Vectors are unaffected.

Yes: deterministic $O(n^{2.9995})$ algorithms for APSP and for the (min,+)-product with polynomially bounded integer weights on the word RAM (the Lean development states $n^{2.99942}$), and Exact Triangle in $O(n^{2.9983})$, refuting the APSP and Exact Triangle hypotheses; a Las Vegas $O(n^{2.998})$ version works for real weights. The core is a new thin-matrix-product algorithm solving Lopsided All-Edges Sparse Triangle, with known reductions giving the rest. The same paper refutes the 3SUM hypothesis, catalogued separately. The constants are enormous; the impact is on the theory of subcubic equivalences, not on practice. SETH and Orthogonal Vectors are unaffected.