The Schonhage-Strassen conjecture that n log n is optimal for integer multiplication on multitape Turing machines

Theorem 1.1: one fixed deterministic multitape Turing machine multiplies two $n$ -bit integers exactly in worst-case time $O(n(\lg n)^{1-\kappa})$ with $\kappa=2^{-182}$ , for every $n$ , so $T(n)/(n\lg n)\to0$ and the $n\log n$ lower bound fails in this model. Corollaries: exact division and integer square root in the same time, and (with Harvey-van der Hoeven's reduction) transposition of $k\times k$ binary matrices in $O(k^2(\lg k)^{1-\kappa})$ . Not shown: any practical algorithm (constants are astronomically large), or anything about Boolean circuit size or other machine models; the true complexity of multiplication remains open.

Exact FrontierDelta

Prior state unknown→disproved

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Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The Schonhage-Strassen conjecture that n log n is optimal for integer multiplication on multitape Turing machines · Schonhage-Strassen n log n conjecture

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Registry verification: unreviewed · announcement · candidate

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Unreleased internal OpenAI model
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VibeMathed record: The Schonhage-Strassen conjecture that n log n is optimal for integer multiplication on multitape Turing machines

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No independent mathematician has checked this yet. Checked here: abstract, introduction (Theorem 1.1) and the history section's lower-bound scope subsection of the TeX source, read against the conjecture as the manuscript cites it. The manuscript makes the model explicit: one deterministic Turing machine with a fixed finite alphabet and a fixed number of one-dimensional tapes, exact for every input length, worst-case time. It notes that Schonhage's linear-time storage-modification-machine result and Groff's unit-cost RAM result are in other models. The algorithm and its 11 sections of tape and precision analysis were not refereed. No Lean formalization. The README cautions that unformalized results could have issues.

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The Schonhage-Strassen conjecture that n log n is optimal for integer multiplication on multitape Turing machines

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Schonhage-Strassen 1971, Schnelle Multiplikation grosser Zahlen

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Schonhage-Strassen 1971, Schnelle Multiplikation grosser Zahlen evidence for this event

VibeMathed record: The Schonhage-Strassen conjecture that n log n is optimal for integer multiplication on multitape Turing machines evidence for this event

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