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Algorithms·matmul:2x2-strassen

Multiplications needed for 2x2 matrix multiplication over F_2

record target minimise

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Strassen showed in 1969 that seven multiplications suffice, and seven is known to be optimal. Included as a control: a system that could not recover it would not be worth believing elsewhere.

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2 steps·multiplications, lower is better
6.577.51980200020207 multiplications, Strassen, 19697 multiplications, AlphaTensor, 2022-10-05, did not move the record
Ink marks are steps by people and clay marks steps where AI took part; grey marks did not move the record, and hollow marks are candidates. How frontiers are drawn

Seven multiplications suffice, Strassen showed in 1969, and Winograd and, independently, Hopcroft and Kerr proved in 1971 that six do not. AlphaTensor recovered a seven-multiplication algorithm in 2022. The record cannot move; it is kept as a control.

7 multiplications

AlphaTensor·5 Oct 2022·Source

Recovered a rank-7 decomposition, matching Strassen's; seven is optimal, so the record cannot move.

AI took partdid not move it
7 multiplications

Strassen·1969·Source

Optimal for this case.

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  • verified·qed.bot

    2x2x2 in 7 multiplications, exact identity mod 2

    object rebuilt and tested against its constraints·2026-08-22

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qed.bot, “Multiplications needed for 2x2 matrix multiplication over F_2”, https://qed.bot/t/matmul-2x2-strassen, as of 30 Sep 2026.

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