In a mail on February 17, 2022, Shawn gave the following machine M with BBB(M) > 1.7 * 10^502:
1RB 1LC 1RC 0RD 0LB 0RC 0RE 1RD 1LE 1LA
In a mail on February 22, 2022, he gave an analysis of the machine, using configurations
g(x,n,m) = ...0 bin(x) 1^n <E 1^m 0...
I give here a more detailed analysis, using these configurations.
Let g(x,0,m) = ...0 bin(x) (E0) 1^m 0...,
and, if n > 0, g(x,n,m) = ...0 bin(x) 1^(n - 1) (E1) 1^m 0...,
where bin(x) is the number x written in binary notation, and x is even if n > 0, because g(2x + 1,n,m) = g(x,n + 1,m).
Then we have:
(a) ...0(A0)0... --(8)--> g(0,1,3)
(b) if k > 0 or m > 0, g(2x,3k + 1,m) --(5k^2 + 2km + 15k + m + 6)--> g(4x + 2,5k + m,2)
(c) g(4x,3k + 2,m) --(5k^2 + 2km + 15k + m + 14) --> g(8x + 2,5k + m + 1,2)
(d) g(4x + 2,3k + 2,m) --(5k^2 + 2km + 15k + m + 12)--> g(8x + 2,5k + m + 1,2)
(e) if k > 0, g(2x,3k,m) --(5k^2 + 2km + 10k)--> g(x,0,5k + m)
(f) g(2x + 1,0,m) --(1)--> g(x,1,m + 1)
(g) if x > 0, g(2x,0,m)--(1)--> g(x,0,m + 1)
(h) if k > 0, g(0,3k,m) --(5k^2 + 2km + 5k - m - 1)--> ...0(D0)0... last state D
Then ...0(A0)0... --> last state D using 1971 transitions with configurations g(x,n,m).
Number of transitions of type
(a) 1
(b) 622
(c) 57
(d) 451
(e) 387
(f) 387
(g) 65
(h) 1
Note: a transition of type (e) is always followed by some (possibly none) transitions of type (g) and then a transition of type (f)..
Pascal