Claude refutes 3SUM, APSP hypotheses

This is a landmark for algorithmic theory, though the tiny exponent savings are unlikely to change production workloads soon.

• –The breakthrough comes from computing sparse entries of thin matrix products faster than full multiplication.
• –Known reductions transfer that improvement to 3SUM, APSP, Exact Triangle, and related problems.
• –Claude reportedly generated the core algorithm autonomously during a 16-million-token research session.
• –Lean formalization substantially strengthens confidence in the proofs and makes the result unusually reproducible.
• –Balanced sparse-triangle problems and many other hardness conjectures remain unresolved, so this is a major crack—not the end of fine-grained complexity.