Remark: Family 109. Integer multiplication below n n

Multiplies two $n$-bit integers exactly at every input length in deterministic worst-case time $O(n(\log n{)}^{1-\kappa })$, with $\kappa =2^{-182}$, on one fixed finite-alphabet Turing machine with finitely many one-dimensional tapes. This disproves the Schönhage–Strassen $n\log n$ optimality conjecture in the ordinary multitape bit model.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.