I made some of the structure systems to represent BEAF spaces beyond tetrational arrays (X^^X).

Fast-climbing structure system

I decide that the structure X{X}X should correspond to the small Veblen ordinal (SVO, ψ(Ω^ω)) in the fast-growing hierarchy (FGH). Madore's OCF is used here. Structures from X^^X to X^^^X are pretty much the same as the original climbing method, but from X^^^X onwards, it needs some extra climbing attributes to reach X^^^^X.

• X^^^X has ordinal level Γ0 = φ(1, 0, 0) = ψ(Ω^Ω)

• X^^X|(X^^^X+1) has ordinal level ε(Γ0 + 1) = ψ(Ω^Ω + 1)

• X^^X^2|(X^^^X+1) has ordinal level ζ(Γ0 + 1) = ψ(Ω^Ω + Ω)

• X^^X^X|(X^^^X+1) has ordinal level φ(ω, Γ0 + 1) = ψ(Ω^Ω + Ω^ω)

• X^^(X^^^X)|2 has ordinal level φ(Γ0, 1) = ψ(Ω^Ω + Ω^ψ(Ω^Ω))

• X^^X^(X^^^X + 1) has ordinal level φ(Γ0 + 1, 0) = ψ(Ω^Ω + Ω^(ψ(Ω^Ω) + 1))

• X^^X^X^(X^^^X + 1) has ordinal level φ(Γ0·ω, 0) = ψ(Ω^Ω + Ω^(ψ(Ω^Ω)·ω))

• X^^(X^^X|(X^^^X+1)) has ordinal level φ(ε(Γ0 + 1), 0) = ψ(Ω^Ω + Ω^ψ(Ω^Ω + 1))

• X^^(X^^X^2|(X^^^X+1)) has ordinal level φ(ζ(Γ0 + 1), 0) = ψ(Ω^Ω + Ω^ψ(Ω^Ω + Ω))

• X^^X^^(X^^(X^^^X)|2) has ordinal level φ(φ(Γ0, 1), 0) = ψ(Ω^Ω + Ω^ψ(Ω^Ω + Ω^ψ(Ω^Ω)))

• X^^^X|2 has ordinal level Γ1 = φ(1, 0, 1) = ψ(Ω^Ω·2)

• X^^^X|3 has ordinal level Γ2 = φ(1, 0, 2) = ψ(Ω^Ω·3)

• X^^^X|X = X^^^(X+1) has ordinal level Γ(ω) = φ(1, 0, ω) = ψ(Ω^Ω·ω)

• X^^^X|X^^X has ordinal level Γ(ε0) = φ(1, 0, φ(1, 0)) = ψ(Ω^Ω·ψ(0))

• X^^^X|X^^X^2 has ordinal level Γ(ζ0) = φ(1, 0, φ(2, 0)) = ψ(Ω^Ω·ψ(Ω))

• X^^^X|X^^X^X has ordinal level Γ(φ(ω, 0)) = φ(1, 0, φ(ω, 0)) = ψ(Ω^Ω·ψ(Ω^ω))

• X^^^X|X^^^X = X^^^X>2 has ordinal level Γ(Γ0) = φ(1, 0, φ(1, 0, 0)) = ψ(Ω^Ω·ψ(Ω^Ω))

• X^^^X|X^^^X|X^^^X = X^^^X>3 has ordinal level Γ(Γ(Γ0)) = φ(1, 0, φ(1, 0, φ(1, 0, 0))) = ψ(Ω^Ω·ψ(Ω^Ω·ψ(Ω^Ω)))

And now, there is a big liftoff...

• X^^^X>X = X^^^(X+1) has ordinal level φ(1, 1, 0) = ψ(Ω^(Ω + 1))

• X^^^X|(X^^^X>X+1) has ordinal level φ(1, 0, φ(1, 1, 0) + 1) = ψ(Ω^(Ω + 1) + Ω^Ω)

• X^^^(X>X)|2 has ordinal level φ(1, 1, 1) = ψ(Ω^(Ω + 1)·2)

• X^^^(X>X)|X = X^^^X>(X+1) has ordinal level φ(1, 1, ω) = ψ(Ω^(Ω + 1)·ω)

• X^^^(X>X)|X^^^(X>X) = X^^^X>(X·2) has ordinal level φ(1, 1, φ(1, 1, 0)) = ψ(Ω^(Ω + 1)·ψ(Ω^(Ω + 1)))

• X^^^X>X^2 has ordinal level φ(1, 2, 0) = ψ(Ω^(Ω + 2))

• X^^^(X>X^2)|X has ordinal level φ(1, 2, ω) = ψ(Ω^(Ω + 2)·ω)

• X^^^X>X^3 has ordinal level φ(1, 3, 0) = ψ(Ω^(Ω + 3))

• X^^^X>X^4 has ordinal level φ(1, 4, 0) = ψ(Ω^(Ω + 4))

• X^^^X>X^X has ordinal level φ(1, ω, 0) = ψ(Ω^(Ω + ω))

• X^^^X>X^^X = X^^^(X+2) has ordinal level φ(1, φ(1, 0), 0) = ψ(Ω^(Ω + ψ(0)))

• X^^^X>X^^X^2 has ordinal level φ(1, φ(2, 0), 0) = ψ(Ω^(Ω + ψ(Ω)))

• X^^^X>X^^^X = X^^^(X·2) has ordinal level φ(1, φ(1, 0, 0), 0) = ψ(Ω^(Ω + ψ(Ω^Ω)))

• X^^^X>X^^^X>X = X^^^(X·2+1) has ordinal level φ(1, φ(1, 1, 0), 0) = ψ(Ω^(Ω + ψ(Ω^(Ω + 1))))

• X^^^X>X^^^X>X^^^X = X^^^(X·3) has ordinal level φ(1, φ(1, φ(1, 0, 0), 0), 0) = ψ(Ω^(Ω + ψ(Ω^(Ω + ψ(Ω^Ω)))))

How about X^^^X^2?

• X^^^X^2 has ordinal level φ(2, 0, 0) = ψ(Ω^(Ω·2))

• X^^^(X^2)|2 has ordinal level φ(2, 0, 1) = ψ(Ω^(Ω·2)·2)

• X^^^(X^2)|X has ordinal level φ(2, 0, ω) = ψ(Ω^(Ω·2)·ω)

• X^^^(X^2)>X = X^^^(X^2+1) has ordinal level φ(2, 1, 0) = ψ(Ω^(Ω·2 + 1))

• X^^^(X^2)>X^2 has ordinal level φ(2, 2, 0) = ψ(Ω^(Ω·2 + 2))

• X^^^(X^2)>X^X has ordinal level φ(2, ω, 0) = ψ(Ω^(Ω·2 + ω))

• X^^^(X^2)>X^^^(X^2) = X^^^(X^2·2) has ordinal level φ(2, φ(2, 0, 0), 0) = ψ(Ω^(Ω·2 + ψ(Ω^(Ω·2))))

• X^^^X^3 has ordinal level φ(3, 0, 0) = ψ(Ω^(Ω·3))

• X^^^(X^3)>X has ordinal level φ(3, 1, 0) = ψ(Ω^(Ω·3 + 1))

• X^^^X^4 has ordinal level φ(4, 0, 0) = ψ(Ω^(Ω·4))

• X^^^X^5 has ordinal level φ(5, 0, 0) = ψ(Ω^(Ω·5))

• X^^^X^X has ordinal level φ(ω, 0, 0) = ψ(Ω^(Ω·ω))

• X^^^X^^X has ordinal level φ(φ(1, 0), 0, 0) = ψ(Ω^(Ω·ψ(0)))

• X^^^X^^X^2 has ordinal level φ(φ(2, 0), 0, 0) = ψ(Ω^(Ω·ψ(Ω)))

• X^^^X^^^X has ordinal level φ(φ(1, 0, 0), 0, 0) = ψ(Ω^(Ω·ψ(Ω^Ω)))

• X^^^X^^^X^^^X has ordinal level φ(φ(φ(1, 0, 0), 0, 0), 0, 0) = ψ(Ω^(Ω·ψ(Ω^(Ω·ψ(Ω^Ω)))))

And finally, we reached the hexational structure X^^^^X (well, you need the double > symbol).

• X^^^^X has ordinal level φ(1, 0, 0, 0) = ψ(Ω^Ω^2)

• X^^^X|(X^^^^X+1) has ordinal level φ(1, 0, φ(1, 0, 0, 0) + 1) = ψ(Ω^Ω^2 + Ω^Ω)

• X^^^(X>X)|(X^^^^X+1) has ordinal level φ(1, 1, φ(1, 0, 0, 0) + 1) = ψ(Ω^Ω^2 + Ω^(Ω + 1))

• X^^^(X^2)|(X^^^^X+1) has ordinal level φ(2, 0, φ(1, 0, 0, 0) + 1) = ψ(Ω^Ω^2 + Ω^(Ω·2))

• X^^^(X^X)|(X^^^^X+1) has ordinal level φ(ω, 0, φ(1, 0, 0, 0) + 1) = ψ(Ω^Ω^2 + Ω^(Ω·ω))

• X^^^(X^^^^X)|2 has ordinal level φ(φ(1, 0, 0, 0), 0, 1) = ψ(Ω^Ω^2 + Ω^(Ω·ψ(Ω^Ω^2)))

• X^^^^X|2 has ordinal level φ(1, 0, 0, 1) = ψ(Ω^Ω^2·2)

• X^^^^X|X has ordinal level φ(1, 0, 0, ω) = ψ(Ω^Ω^2·ω)

• X^^^^X|X^^^^X has ordinal level φ(1, 0, 0, φ(1, 0, 0, 0)) = ψ(Ω^Ω^2·ψ(Ω^Ω^2))

• X^^^^X>X has ordinal level φ(1, 0, 1, 0) = ψ(Ω^(Ω^2 + 1))

• X^^^^X>X^2 has ordinal level φ(1, 0, 2, 0) = ψ(Ω^(Ω^2 + 2))

• X^^^^X>X^X has ordinal level φ(1, 0, ω, 0) = ψ(Ω^(Ω^2 + ω))

• X^^^^X>X^^^^X has ordinal level φ(1, 0, φ(1, 0, 0, 0), 0) = ψ(Ω^(Ω^2 + ψ(Ω^(Ω^2))))

• X^^^^X>>X has ordinal level φ(1, 1, 0, 0) = ψ(Ω^(Ω^2 + Ω))

• X^^^^X>>X>X has ordinal level φ(1, 1, 1, 0) = ψ(Ω^(Ω^2 + Ω + 1))

• X^^^^X>>X>X^2 has ordinal level φ(1, 1, 2, 0) = ψ(Ω^(Ω^2 + Ω + 2))

• X^^^^X>>X>X^X has ordinal level φ(1, 1, ω, 0) = ψ(Ω^(Ω^2 + Ω + ω))

• X^^^^X>>X>X^^^^X>>X has ordinal level φ(1, 1, φ(1, 1, 0, 0), 0) = ψ(Ω^(Ω^2 + Ω + ψ(Ω^(Ω^2 + Ω))))

• X^^^^X>>X^2 has ordinal level φ(1, 2, 0, 0) = ψ(Ω^(Ω^2 + Ω·2))

• X^^^^X>>X^3 has ordinal level φ(1, 3, 0, 0) = ψ(Ω^(Ω^2 + Ω·3))

• X^^^^X>>X^X has ordinal level φ(1, ω, 0, 0) = ψ(Ω^(Ω^2 + Ω·ω))

• X^^^^X>>X^^^^X has ordinal level φ(1, φ(1, 1, 0, 0), 0, 0) = ψ(Ω^(Ω^2 + Ω·ψ(Ω^Ω^2)))

• X^^^^X^2 has ordinal level φ(2, 0, 0, 0) = ψ(Ω^(Ω^2·2))

• X^^^^(X^2)>X has ordinal level φ(2, 0, 1, 0) = ψ(Ω^(Ω^2·2 + 1))

• X^^^^(X^2)>>X has ordinal level φ(2, 1, 0, 0) = ψ(Ω^(Ω^2·2 + Ω))

• X^^^^X^3 has ordinal level φ(3, 0, 0, 0) = ψ(Ω^(Ω^2·3))

• X^^^^X^4 has ordinal level φ(4, 0, 0, 0) = ψ(Ω^(Ω^2·4))

• X^^^^X^X has ordinal level φ(ω, 0, 0, 0) = ψ(Ω^(Ω^2·ω))

• X^^^^X^^^^X has ordinal level φ(φ(1, 0, 0, 0), 0, 0, 0) = ψ(Ω^(Ω^2·ψ(Ω^Ω^2)))

Moving on X^^^^^X...

• X^^^^^X has ordinal level φ(1, 0, 0, 0, 0) = ψ(Ω^Ω^3)

• X^^^^^X>X has ordinal level φ(1, 0, 0, 1, 0) = ψ(Ω^(Ω^3 + 1))

• X^^^^^X>>X has ordinal level φ(1, 0, 1, 0, 0) = ψ(Ω^(Ω^3 + Ω))

• X^^^^^X>>>X has ordinal level φ(1, 1, 0, 0, 0) = ψ(Ω^(Ω^3 + Ω^2))

• X^^^^^X>>>X^X has ordinal level φ(1, ω, 0, 0, 0) = ψ(Ω^(Ω^3 + Ω^2·ω))

• X^^^^^X^2 has ordinal level φ(2, 0, 0, 0, 0) = ψ(Ω^(Ω^3·2))

• X^^^^^X^X has ordinal level φ(ω, 0, 0, 0, 0) = ψ(Ω^(Ω^3·ω))

• X^^^^^X^^^^^X has ordinal level φ(φ(1, 0, 0, 0, 0), 0, 0, 0, 0) = ψ(Ω^(Ω^3·ψ(Ω^Ω^3)))

• X^^^^^^X has ordinal level φ(1, 0, 0, 0, 0, 0) = ψ(Ω^Ω^4)

• X^^^^^^X>>>>X has ordinal level φ(1, 1, 0, 0, 0, 0) = ψ(Ω^(Ω^4 + Ω^3))

• X^^^^^^X^2 has ordinal level φ(2, 0, 0, 0, 0, 0) = ψ(Ω^(Ω^4·2))

• X^^^^^^X^X has ordinal level φ(ω, 0, 0, 0, 0, 0) = ψ(Ω^(Ω^4·ω))

• X^^^^^^X^^^^^^X has ordinal level φ(φ(1, 0, 0, 0, 0, 0), 0, 0, 0, 0, 0) = ψ(Ω^(Ω^4·ψ(Ω^Ω^4)))

• X^^^^^^^X has ordinal level φ(1, 0, 0, 0, 0, 0, 0) = ψ(Ω^Ω^5)

• X^^^^^^^^X has ordinal level φ(1, 0, 0, 0, 0, 0, 0, 0) = ψ(Ω^Ω^6)

• X^^^^^^^^^X has ordinal level φ(1, 0, 0, 0, 0, 0, 0, 0, 0) = ψ(Ω^Ω^7)

• X^^^^^^^^^^X has ordinal level φ(1, 0, 0, 0, 0, 0, 0, 0, 0, 0) = ψ(Ω^Ω^8)

• ...

So, X{N+2}X^2 is X{N+2}X[N]>X{N+2}X[N]>X{N+2}X[N]>X{N+2}X[N]>...

And we catch up with the real edge at the small Veblen ordinal:

• X{X}X has ordinal level ψ(Ω^Ω^ω)

• X^^^X|(X{X}X+1) has ordinal level ψ(Ω^Ω^ω + Ω^Ω)

• X^^^^X|(X{X}X+1) has ordinal level ψ(Ω^Ω^ω + Ω^Ω^2)

• X{X}X|2 has ordinal level ψ(Ω^Ω^ω·2)

• X{X}X|3 has ordinal level ψ(Ω^Ω^ω·3)

• X{X}X|X has ordinal level ψ(Ω^Ω^ω·ω)

• X{X}X|X{X}X has ordinal level ψ(Ω^Ω^ω·ψ(Ω^Ω^ω))

• X{X}X>X has ordinal level ψ(Ω^(Ω^ω + 1))

• X{X}X>X^2 has ordinal level ψ(Ω^(Ω^ω + 2))

• X{X}X>X^X has ordinal level ψ(Ω^(Ω^ω + ω))

• X{X}X>X{X}X has ordinal level ψ(Ω^(Ω^ω + ψ(Ω^Ω^ω)))

• X{X}X>>X has ordinal level ψ(Ω^(Ω^ω + Ω))

• X{X}X>>>X has ordinal level ψ(Ω^(Ω^ω + Ω^2))

• X{X}X>>>>X has ordinal level ψ(Ω^(Ω^ω + Ω^3))

• X{X}X^2 has ordinal level ψ(Ω^(Ω^ω·2))

• X{X}X^3 has ordinal level ψ(Ω^(Ω^ω·3))

• X{X}X^X has ordinal level ψ(Ω^(Ω^ω·ω))

• X{X}X{X}X has ordinal level ψ(Ω^(Ω^ω·ψ(Ω^Ω^ω)))

• X{X+1}X has ordinal level ψ(Ω^Ω^(ω + 1))

• X{X+1}X>X has ordinal level ψ(Ω^(Ω^(ω + 1) + 1))

• X{X+1}X>>X has ordinal level ψ(Ω^(Ω^(ω + 1) + Ω))

• X{X+1}X[X]>X has ordinal level ψ(Ω^(Ω^(ω + 1) + Ω^ω))

• X{X+1}X^2 has ordinal level ψ(Ω^(Ω^(ω + 1)·2))

• X{X+1}X^X has ordinal level ψ(Ω^(Ω^(ω + 1)·ω))

• X{X+1}X{X+1}X has ordinal level ψ(Ω^(Ω^(ω + 1)·ψ(Ω^Ω^(ω + 1))))

• X{X+2}X has ordinal level ψ(Ω^Ω^(ω + 2))

• X{X+3}X has ordinal level ψ(Ω^Ω^(ω + 3))

• X{X·2}X has ordinal level ψ(Ω^Ω^(ω·2))

• X{X·2+1}X has ordinal level ψ(Ω^Ω^(ω·2 + 1))

• X{X·3}X has ordinal level ψ(Ω^Ω^(ω·3))

• X{X·4}X has ordinal level ψ(Ω^Ω^(ω·4))

• X{X^2}X has ordinal level ψ(Ω^Ω^ω^2)

• X{X^3}X has ordinal level ψ(Ω^Ω^ω^3)

• X{X^X}X has ordinal level ψ(Ω^Ω^ω^ω)

• X{X^X^X}X has ordinal level ψ(Ω^Ω^ω^ω^ω)

• X{X^^X}X has ordinal level ψ(Ω^Ω^ψ(0))

• X{X^^X^2}X has ordinal level ψ(Ω^Ω^ψ(Ω))

• X{X^^X^X}X has ordinal level ψ(Ω^Ω^ψ(Ω^ω))

• X{X^^^X}X has ordinal level ψ(Ω^Ω^ψ(Ω^Ω))

• X{X^^^^X}X has ordinal level ψ(Ω^Ω^ψ(Ω^Ω^2))

• X{X{X}X}X has ordinal level ψ(Ω^Ω^ψ(Ω^Ω^ω))

• X{X{X^^^X}X}X has ordinal level ψ(Ω^Ω^ψ(Ω^Ω^ψ(Ω^Ω)))

• X{X{X{X}X}X}X has ordinal level ψ(Ω^Ω^ψ(Ω^Ω^ψ(Ω^Ω^ω)))

• X{X{X{X{X}X}X}X}X has ordinal level ψ(Ω^Ω^ψ(Ω^Ω^ψ(Ω^Ω^ψ(Ω^Ω^ω))))

• ...

And finally, the limit of X{X{X{X{...}X}X}X}X in the fast-climbing structure system has the large Veblen ordinal (LVO, ψ(Ω^Ω^Ω)) in the FGH.

And this time, we define the collapsing form to reach the Bachmann-Howard ordinal (BHO)!

• X{#}X has ordinal level ψ(Ω^Ω^Ω)

• X^^X|(X{#}X+1) has ordinal level ψ(Ω^Ω^Ω + 1)

• X^^X^2|(X{#}X+1) has ordinal level ψ(Ω^Ω^Ω + Ω)

• X^^^X|(X{#}X+1) has ordinal level ψ(Ω^Ω^Ω + Ω^Ω)

• X^^^^X|(X{#}X+1) has ordinal level ψ(Ω^Ω^Ω + Ω^Ω^2)

• X{X}X|(X{#}X+1) has ordinal level ψ(Ω^Ω^Ω + Ω^Ω^ω)

• X{X{#}X}X|2 has ordinal level ψ(Ω^Ω^Ω + Ω^Ω^ψ(Ω^Ω^Ω))

• X{#}X|2 has ordinal level ψ(Ω^Ω^Ω·2)

• X{#}X|X has ordinal level ψ(Ω^Ω^Ω·ω)

• X{#}X|X{#}X has ordinal level ψ(Ω^Ω^Ω·ψ(Ω^Ω^Ω))

• X{#}X>X has ordinal level ψ(Ω^(Ω^Ω + 1))

• X{#}X>X^2 has ordinal level ψ(Ω^(Ω^Ω + 2))

• X{#}X>X^X has ordinal level ψ(Ω^(Ω^Ω + ω))

• X{#}X>X{#}X has ordinal level ψ(Ω^(Ω^Ω + ψ(Ω^Ω^Ω)))

• X{#}X>>X has ordinal level ψ(Ω^(Ω^Ω + Ω))

• X{#}X>>>X has ordinal level ψ(Ω^(Ω^Ω + Ω^2))

• X{#}X[X]>X has ordinal level ψ(Ω^(Ω^Ω + Ω^ω))

• X{#}X[#]>X has ordinal level ψ(Ω^(Ω^Ω + Ω^ψ(Ω^Ω^Ω)))

• X{#}X^2 has ordinal level ψ(Ω^(Ω^Ω·2))

• X{#}X^X has ordinal level ψ(Ω^(Ω^Ω·ω))

• X{#}X{#}X has ordinal level ψ(Ω^(Ω^Ω·ψ(Ω^Ω^Ω)))

• X{#+1}X has ordinal level ψ(Ω^Ω^(Ω + 1))

• X{#+2}X has ordinal level ψ(Ω^Ω^(Ω + 2))

• X{#+X}X has ordinal level ψ(Ω^Ω^(Ω + ω))

• X{#+X{#}X}X has ordinal level ψ(Ω^Ω^(Ω + ψ(Ω^Ω^Ω)))

• X{#·2}X has ordinal level ψ(Ω^Ω^(Ω·2))

• X{#·2+1}X has ordinal level ψ(Ω^Ω^(Ω·2 + 1))

• X{#·3}X has ordinal level ψ(Ω^Ω^(Ω·3))

• X{#·4}X has ordinal level ψ(Ω^Ω^(Ω·4))

• X{#^2}X has ordinal level ψ(Ω^Ω^Ω^2)

• X{#^2+1}X has ordinal level ψ(Ω^Ω^(Ω^2 + 1))

• X{#^2+#}X has ordinal level ψ(Ω^Ω^(Ω^2 + Ω))

• X{#^2·2}X has ordinal level ψ(Ω^Ω^(Ω^2·2))

• X{#^2·X}X has ordinal level ψ(Ω^Ω^(Ω^2·ω))

• X{#^2·X{#^2}X}X has ordinal level ψ(Ω^Ω^(Ω^2·ψ(Ω^Ω^Ω^2)))

• X{#^3}X has ordinal level ψ(Ω^Ω^Ω^3)

• X{#^4}X has ordinal level ψ(Ω^Ω^Ω^4)

• X{#^X}X has ordinal level ψ(Ω^Ω^Ω^ω)

• X{#^X{#}X}X has ordinal level ψ(Ω^Ω^Ω^ψ(Ω^Ω^Ω))

• X{#^#}X has ordinal level ψ(Ω^Ω^Ω^Ω)

• X{#^#^2}X has ordinal level ψ(Ω^Ω^Ω^Ω^2)

• X{#^#^X}X has ordinal level ψ(Ω^Ω^Ω^Ω^ω)

• X{#^#^#}X has ordinal level ψ(Ω^Ω^Ω^Ω^Ω)

• X{#^#^#^#}X has ordinal level ψ(Ω^Ω^Ω^Ω^Ω^Ω)

• X{#^#^#^#^#}X has ordinal level ψ(Ω^Ω^Ω^Ω^Ω^Ω^Ω)

• ...

And the limit of this system is the Bachmann-Howard ordinal level (X{#^#^#^#^#^#^...}X).

Can you reach the limit of Bashicu matrix system (BMS) or even Y sequence?