User blog:Rgetar/Uncountable regular cardinals and ε numbers in booster-base

Today I came up with the rules how to find out is a booster-base (BB) expression an uncountable regular cardinal.

Designations
Let Bn = [Cn] (see Next steps section of my Ordinal Explorer v6.0 and Ordinal Explorer v6.1 blog), where n ≥ 1, that is
 * B1 = [C] = Ω
 * B2 = [C2] = L
 * B3 = [C3] = R
 * B4 = [C4] = S

Bn = [Bn + 1]:
 * Ω = [L]
 * L = [R]
 * R = [S]
 * S = [B5]

Next Bn number above β is [next Bn + 1 number above β]β.

(Here we need, while next Bn + 1 number above β > booster(β), replace β with base(β), because in [a][b][c][d]e should be a ≤ b ≤ c ≤ d).

If β < Bn then next Bn number above β isBn.

Rules
[X]β is uncountable regular cardinal iff
 * next Bn number above β ≤ cof(X)

where
 * next Bn number above β ≤ X < next Bn + 1 number above β

and
 * n ≥ 2

If we remove "n ≥ 2" condition, then, I think, we get rule for ε numbers.

What to do next
Formulate rules for not regular [X]β. I already made up them, but I need to formulate them, and then try to test them in a program. Anyway, I suspect that in Ordinal Explorer v6.1 I did something wrong, because rules there are slightly different from rules I made up today, and actually the system there may work, but be not so powerful.