A 10-state Turing machine, called n1, has been identified that surpasses the current BB(10) champion. The champion's published bound is f_ω(f_ω(25)), at level ω of the fast-growing hierarchy. n1 moves the BB(10) record to level ω+1. The machine is built from an 8-state base machine that agrees with the BB(8) record on states A–E and runs the same digit-list computation as the record, which is the BB(7) champion's program with 36 digits.
The researchers extended the base machine by sending its halting transition to a new state I and adding states I and J. The machine halts from the blank tape. Its exact score is proved in Coq, a machine-checked proof assistant, with no axioms. The score lies between two values of f_(ω+1), placing it at a higher level of the hierarchy than the previous champion.
The proofs establish that the number of 1s left on the tape exceeds Graham's number, and that it is greater than F applied N = (6T − 19)/7 ≈ 0.86·T times to 33, where T = 2↑^37 3 is the value the BB(8) record computes. The standard score is the number of ones plus one, because the halting transition writes a 1.
The analysis follows Ligocki's notation for the BB(7) champion, in which tape contents are encoded as blocks of alternating digits and zeros. Each clearing step on a token changes the accumulator value by a tower of exponentials whose height depends on the token's depth. The result is a machine-checked lower bound that dwarfs earlier records.
The machine was written up on 2026-10-08 and is documented in a public repository by jeff87654. The result is significant for the busy beaver problem, which asks for the maximum number of steps or ones a halting machine of a given size can produce.
Source: Hacker News · Summarized by HeadlinesBriefing