A 10-state Busy Beaver Turing machine N1 is presented that halts from the blank tape and is machine-checked in Coq; its exact score is proved and its lower bound and growth-level are formally established. N1 preserves an 8-state base that implements the BB(8) record’s digit-list computation, diverts the base halting transition into two new states I and J, and thereby drives a much larger meta-computation. The construction computes T = 2↑^37 3, then converts T into a long zero counter of length 6T−19 and runs ≈(6T−19)/7 big “periods.” Coq proofs give an explicit closed formula for the final number of 1s (ones = 2W + 3·2↑^(W+1)(2↑^n 5) + 12·JJ + 5 with JJ=(3T−13)/7), a certified lower bound ones > F^N(33) for F(x)=2↑^x 4 and N=(6T−19)/7, and the level f_(ω+1)(JJ−2), thereby exceeding the former BB(10) champion (whose published bound sits at level ω).
Mechanically, N1 reuses Ligocki-style “clearing” rules: clearing a unit at depth j replaces value v by 2↑^j v, so parking, unpacking and jumping during each period turn a parked huge value into exponentially deeper tokens. Each period effectively applies a fast-growing map F twice (so F(F(n))) and consumes 14 counter zeros; with N ≈ 0.86·T such periods starting from depth ~34, the machine composes f_ω about N times, yielding f_(ω+1)-level growth. All critical modular and combinatorial conditions (including T ≡ 2 mod 7 that makes the counter endgame exact) are proven in Coq, and the construction is checked to beat the previous BB(10) record.
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