Friedman's "Long Finite Sequences'': The End of the Busy Beaver Contest

Abstract: Harvey Friedman gives a comparatively short description of an ``unimaginably large'' number $n(3)$ , beyond, e.g. the values $$ A(7,184)&lt; A({7198},158386) &lt; n(3)$$ of Ackermann's function - but finite. We implement Friedman's combinatorial problem about subwords of words over a 3-letter alphabet on a family of Turing machines, which, starting on empty tape, run (more than) $n(3)$ steps, and then halt. Examples include a (44,8) (symbol,state count) machine as well as a (276,2) and a (2,1840) one. In total, there are at most 37022 non-trivial pairs $(n,m)$ with Busy Beaver values ${\tt BB(n,m)} &lt; A(7198,158386).$ We give algorithms to map any $(|Q|,|E|)$ TM to another, where we can choose freely either $|Q&#39;|\geq 2$ or $|E&#39;|\geq 2$ (the case $|Q&#39;|=2$ for empty initial tape is the tricky one). Given the size of $n(3)$ and the fact that these TMs are not {\it holdouts}, but assured to stop, Friedman's combinatorial problem provides a definite upper bound on what might ever be possible to achieve in the Busy Beaver contest. We also treat $n(4)&gt; A<sup>{(A(187196))}(1)$.