## FANDOM

10,835 Pages

Here are 2 ways BEAF reaches BHO.

We can define some more structures. Here I use square-bracketed array to represent L-array, and the base number is always L. This is a "type-2" array. The limit of original BEAF is a legion in type-2 array ----we can name it "type-2 legion". A type-2 legion is represented by L2.The [&] operator now become &2. Now let's continue the FGH comparisons.

BEAF FGH Ordinal
$$\{[L,X/2]\}_{n,n}$$ $$\theta(\Omega^{\Omega}2)$$
$$\{[L,2X/2]\}_{n,n}$$ $$\theta(\Omega^{\Omega}2,1)$$
$$\{[L,X^2/2]\}_{n,n}$$ $$\theta(\Omega^{\Omega}2,\omega)$$
$$\{[L,X,2/2]\}_{n,n}$$ $$\theta(\Omega^{\Omega}2+1)$$
$$\{[L,X,3/2]\}_{n,n}$$ $$\theta(\Omega^{\Omega}2+2)$$
$$\{[L,X,1,2/2]\}_{n,n}$$ $$\theta(\Omega^{\Omega}2+\Omega)$$
$$\{[L,X,1,1,2/2]\}_{n,n}$$ $$\theta(\Omega^{\Omega}2+\Omega^2)$$
$$\{[L,X(1)2/2]\}_{n,n}$$ $$\theta(\Omega^{\Omega}2+\Omega^\omega)$$
$$\{[L,X(1)1,2/2]\}_{n,n}$$ $$\theta(\Omega^{\Omega}2+\Omega^{\omega+1})$$
$$\{[L,X(1)(1)2/2]\}_{n,n}$$ $$\theta(\Omega^{\Omega}2+\Omega^{\omega2})$$
$$\{[L,X(2)2/2]\}_{n,n}$$ $$\theta(\Omega^{\Omega}2+\Omega^{\omega^2})$$
$$\{[L,X(0,1)1,2/2]\}_{n,n}$$ $$\theta(\Omega^{\Omega}2+\Omega^{\omega^\omega})$$
$$\{[X\uparrow\uparrow X\&_2L/2]\}_{n,n}$$ $$\theta(\Omega^{\Omega}2+\Omega^{\varepsilon_0})$$
$$\{[L\&_2L/2]\}_{n,n}$$ $$\theta(\Omega^{\Omega}2+\Omega^{\theta(\Omega^\Omega)})$$
$$\{[L\&_2L\&_2L/2]\}_{n,n}$$ $$\theta(\Omega^{\Omega}2+\Omega^{\theta(\Omega^\Omega+\Omega^{\theta(\Omega^\Omega)})})$$
$$\{[L,X/3]\}_{n,n}$$ $$\theta(\Omega^{\Omega}2+\Omega^{\theta(\Omega^{\Omega}2)})$$
$$\{[L,X,2/3]\}_{n,n}$$ $$\theta(\Omega^{\Omega}2+\Omega^{\theta(\Omega^{\Omega}2)}+1)$$
$$\{[L,X,1,2/3]\}_{n,n}$$ $$\theta(\Omega^{\Omega}2+\Omega^{\theta(\Omega^{\Omega}2)}+\Omega)$$
$$\{[L,X(1)2/3]\}_{n,n}$$ $$\theta(\Omega^{\Omega}2+\Omega^{\theta(\Omega^{\Omega}2)}+\Omega^\omega)$$
$$\{[L,X(0,1)2/3]\}_{n,n}$$ $$\theta(\Omega^{\Omega}2+\Omega^{\theta(\Omega^{\Omega}2)}+\Omega^{\omega^\omega})$$
$$\{[L\&_2L/3]\}_{n,n}$$ $$\theta(\Omega^{\Omega}2+\Omega^{\theta(\Omega^{\Omega}2)}+\Omega^{\theta(\Omega^\Omega)})$$
$$\{[L,X/4]\}_{n,n}$$ $$\theta(\Omega^{\Omega}2+\Omega^{\theta(\Omega^{\Omega}2)}2)$$
$$\{[L,X/5]\}_{n,n}$$ $$\theta(\Omega^{\Omega}2+\Omega^{\theta(\Omega^{\Omega}2)}3)$$
$$\{[L,X/X]\}_{n,n}$$ $$\theta(\Omega^{\Omega}2+\Omega^{\theta(\Omega^{\Omega}2)}\omega)$$
$$\{[L,X/1,2]\}_{n,n}$$ $$\theta(\Omega^{\Omega}2+\Omega^{\theta(\Omega^{\Omega}2)+1})$$
$$\{[L,X/1,1,2]\}_{n,n}$$ $$\theta(\Omega^{\Omega}2+\Omega^{\theta(\Omega^{\Omega}2)+2})$$
$$\{[L,X/1(1)2]\}_{n,n}$$ $$\theta(\Omega^{\Omega}2+\Omega^{\theta(\Omega^{\Omega}2)+\omega})$$
$$\{[L,X/1/2]\}_{n,n}$$ $$\theta(\Omega^{\Omega}2+\Omega^{\theta(\Omega^{\Omega}2)2})$$
$$\{[L,X/1/1/2]\}_{n,n}$$ $$\theta(\Omega^{\Omega}2+\Omega^{\theta(\Omega^{\Omega}2)3})$$
$$\{[L,X(/1)2]\}_{n,n}$$ $$\theta(\Omega^{\Omega}2+\Omega^{\theta(\Omega^{\Omega}2)\omega})$$
$$\{[L,X(/0,1)2]\}_{n,n}$$ $$\theta(\Omega^{\Omega}2+\Omega^{\theta(\Omega^{\Omega}2)\omega^\omega})$$
$$\{X\uparrow\uparrow X\&_2\&_2L\}_{n,n}$$ $$\theta(\Omega^{\Omega}2+\Omega^{\theta(\Omega^{\Omega}2)\varepsilon_0})$$
$$\{L\&_2\&_2L\}_{n,n}$$ $$\theta(\Omega^{\Omega}2+\Omega^{\theta(\Omega^{\Omega}2)\theta(\Omega^\Omega)})$$
$$\{[L,X/2]\&_2\&_2L\}_{n,n}$$ $$\theta(\Omega^{\Omega}2+\Omega^{\theta(\Omega^{\Omega}2)^2})$$
$$\{[L,2X/2]\&_2\&_2L\}_{n,n}$$ $$\theta(\Omega^{\Omega}2+\Omega^{\theta(\Omega^{\Omega}2,1)})$$
$$\{L\&_2\&_2L\&_2\&_2L\}_{n,n}$$ $$\theta(\Omega^{\Omega}2+\Omega^{\theta(\Omega^{\Omega}2+\Omega^{\theta(\Omega^{\Omega}2)\theta(\Omega^\Omega)})})$$
$$\{[L,X//2]\}_{n,n}$$ $$\theta(\Omega^{\Omega}3)$$
$$\{[L,X,2//2]\}_{n,n}$$ $$\theta(\Omega^{\Omega}3+1)$$
$$\{[L,X/2//2]\}_{n,n}$$ $$\theta(\Omega^{\Omega}3+\Omega^{\theta(\Omega^{\Omega}2)})$$
$$\{[L,X//3]\}_{n,n}$$ $$\theta(\Omega^{\Omega}3+\Omega^{\theta(\Omega^{\Omega}3)})$$
$$\{[L,X//1//2]\}_{n,n}$$ $$\theta(\Omega^{\Omega}3+\Omega^{\theta(\Omega^{\Omega}3)2})$$
$$\{L\&_2\&_2\&_2L\}_{n,n}$$ $$\theta(\Omega^{\Omega}3+\Omega^{\theta(\Omega^{\Omega}3)\theta(\Omega^\Omega)})$$
$$\{[L,X///2]\}_{n,n}$$ $$\theta(\Omega^{\Omega}4)$$
$$\{[L,X(1)/2]\}_{n,n}$$ $$\theta(\Omega^{\Omega}\omega)$$
$$\{[L,X(2)/2]\}_{n,n}$$ $$\theta(\Omega^{\Omega}\omega^\omega)$$
$$\{[L,X(3)/2]\}_{n,n}$$ $$\theta(\Omega^{\Omega}\omega^{\omega^2})$$
$$\{[L,X(0,1)/2]\}_{n,n}$$ $$\theta(\Omega^{\Omega}\omega^{\omega^\omega})$$

L2's can make type-1 legiattic arrays. Type-1 arrays use bp as initial rule.

BEAF FGH Ordinal
$$\{\{L_2,X\uparrow\uparrow X\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega}\varepsilon_0)$$
$$\{\{L_2,L\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega}\theta(\Omega^\Omega))$$
$$\{\{L_2,L_2\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega}\theta(\Omega^{\Omega}2))$$
$$\{\{L_2,3,2\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega}\theta(\Omega^{\Omega}\theta(\Omega^\Omega2)))$$
$$\{\{L_2,X,2\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega+1})$$
$$\{\{L_2,2X,2\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega+2})$$
$$\{\{L_2,L_2,2\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega+\theta(\Omega^\Omega2)})$$
$$\{\{L_2,X,3\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega2})$$
$$\{\{L_2,X,L_2\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega\theta(\Omega^\Omega2)})$$
$$\{\{L_2,X,1,2\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^2})$$
$$\{\{L_2,X,2,2\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^2+\Omega})$$
$$\{\{L_2,X,1,3\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{2}2})$$
$$\{\{L_2,X,1,1,2\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{3}})$$
$$\{\{L_2,X(1)2\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\omega}})$$
$$\{\{L_2,X(1)1,2\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\omega+1}})$$
$$\{\{L_2,X(2)2\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\omega^2}})$$
$$\{X\uparrow\uparrow X@L\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\varepsilon_0}})$$
$$\{L@L\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\theta(\Omega^\Omega)}})$$
$$\{[L,X/2]@L\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\theta(\Omega^\Omega2)}})$$
$$\{L@L@L\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\theta(\Omega^{\Omega^{\theta(\Omega^\Omega)}})}})$$

L2's can make type-2 legiattic arrays. For example, [L,2] is type-1 lugion, and [L,3] is type-1 lagion. Generally speaking, {[L,A+1]}n,n=n @& n @& n ... n @& n , where @& is "[L,A]-attic array of".

BEAF FGH Ordinal
$$\{[L_2,2]_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega})$$
$$\{\{[L_2,2],L\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega}+\Omega^\Omega\theta(\Omega^\Omega))$$
$$\{\{[L_2,2],L_2\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega}+\Omega^\Omega\theta(\Omega^\Omega2))$$
$$\{\{[L_2,2],[L_2,2]\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega}+\Omega^\Omega\theta(\Omega^{\Omega^\Omega}))$$
$$\{\{[L_2,2],X,2\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega}+\Omega^{\Omega+1})$$
$$\{\{[L_2,2],X,1,2\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega}+\Omega^{\Omega2})$$
$$\{\{[L_2,2],X(1)2\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega}+\Omega^{\Omega^\omega})$$

$$\{\{[L_2,2],X/2\}_{L,X}\}_{n,n}=$${L%L}n,n                        $$\theta(\Omega^{\Omega^\Omega}+\Omega^{\Omega^{\theta(\Omega^\Omega)}})$$

$$\{\{[L_2,2],X\backslash2\}_{L,X}\}_{n,n}=$${[L,2]%L}n,n              $$\theta(\Omega^{\Omega^\Omega}+\Omega^{\Omega^{\theta(\Omega^\Omega,1)}})$$

{[L_2,2]L,X%L}n,n                       $$\theta(\Omega^{\Omega^\Omega}+\Omega^{\Omega^{\theta(\Omega^{\Omega^\Omega})}})$$

{L%L%L}n,n                                  $$\theta(\Omega^{\Omega^\Omega}+\Omega^{\Omega^{\theta(\Omega^{\Omega^\Omega}+\Omega^{\Omega^{\theta(\Omega^\Omega)}})}})$$

BEAF FGH Ordinal
$$\{[L_2,3]_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega}2)$$
$$\{[L_2,4]_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega}3)$$
$$\{[L_2,X]_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega}\omega)$$
$$\{[L_2,L_2]_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega}\theta(\Omega^\Omega2))$$
$$\{[L_2,X,2]_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega+1})$$
$$\{[L_2,2X,2]_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega+1}2)$$
$$\{[L_2,X,3]_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega+2})$$
$$\{[L_2,X,1,2]_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega+\Omega})$$
$$\{[L_2,X,1,3]_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega+\Omega2})$$
$$\{[L_2,X,1,1,2]_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega+\Omega^2})$$
$$\{[L_2,X(1)2]_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega+\Omega^\omega})$$
$$\{[L_2,X(0,1)2]_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega+\Omega^{\omega^\omega}})$$
$$\{X\uparrow\uparrow X@_2L\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega+\Omega^{\varepsilon_0}})$$
$$\{L@_2L\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega+\Omega^{\theta(\Omega^\Omega)}})$$
$$\{L@_2L@_2L\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega+\Omega^{\theta(\Omega^{\Omega^\Omega+\Omega^{\theta(\Omega^\Omega)}})}})$$
$$\{[L,X\backslash2]\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega2})$$
$$\{[L,X\backslash\backslash2]\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega2}+\Omega^\Omega)$$
$$\{[L,X(0,1)\backslash2]\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega2}+\Omega^\Omega\omega^{\omega^\omega})$$
$$\{\{L_22,X,2\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega2}+\Omega^{\Omega+1})$$
$$\{\{L_22,X,1,2\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega2}+\Omega^{\Omega2})$$
$$\{\{L_22,X((1)1)2\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega2}+\Omega^{\Omega^{\omega^{\omega^\omega}}})$$
$$\{[L_22,2]_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega2}+\Omega^{\Omega^\Omega})$$
$$\{[L_22,3]_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega2}+\Omega^{\Omega^\Omega}2)$$
$$\{[L_22,X]_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega2}+\Omega^{\Omega^\Omega}\omega)$$
$$\{[L_22,L_22]_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega2}+\Omega^{\Omega^\Omega}\theta(\Omega^\Omega2))$$
$$\{[L_22,X,2]_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega2}+\Omega^{\Omega^\Omega+1})$$
$$\{[L_22,X,1,2]_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega2}+\Omega^{\Omega^\Omega+\Omega})$$

{X^^X%2L}n,n               $$\theta(\Omega^{\Omega^\Omega2}+\Omega^{\Omega^\Omega+\Omega^{\varepsilon_0}})$$

{L%2L}n,n                                         $$\theta(\Omega^{\Omega^\Omega2}+\Omega^{\Omega^\Omega+\Omega^{\theta(\Omega^\Omega)}})$$

{[L,X|2]}n,n                                                       $$\theta(\Omega^{\Omega^\Omega2}2)$$

$$\{[L,X-2]\}_{n,n}$$                                             $$\theta(\Omega^{\Omega^\Omega2}3)$$

$$\{[L_2X]_{L,X}\}_{n,n}$$                                    $$\theta(\Omega^{\Omega^\Omega2}\omega)$$

$$\{[L_2L_2]_{L,X}\}_{n,n}$$                                 $$\theta(\Omega^{\Omega^\Omega2}\theta(\Omega^\Omega2))$$

Next come "type-3 arrays". Now I use {2  } to represent type-2 arrays, and use {3  } to represent type-3 arrays. Generally speaking, {m+1  } represents a type-(m+1) array, and its base number is always Lm, which represents a legion in type-m array. The initial rule of type-(m+1) array is:

{m{m+1Lm,A+1}}b,p=b @& b @& b ... b @& b - p times , where @& is "{m+1Lm,A}-attic type-m-array of".

"Type-m array of" is represented by symbol &m, and the same as &&'s, &&&'s, ..., @'s, %'s and #'s.

BEAF FGH Ordinal
$$\{\{_2\{_3L_2,L_2\}\}_{L,X}\}_{n,n}$$

$$\theta(\Omega^{\Omega^\Omega2}\theta(\Omega^\Omega2))$$

$$\{\{_2\{_3L_2,X,2\}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega2+1})$$
$$\{\{_2\{_3L_2,X,3\}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega2+2})$$
$$\{\{_2\{_3L_2,X,1,2\}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega2+\Omega})$$
$$\{\{_2\{_3L_2,X,1,1,2\}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega2+\Omega^2})$$
$$\{\{_2\{_3L_2,X(1)2\}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega2+\Omega^\omega})$$
$$\{\{_2\{_3L_2,X(2)2\}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega2+\Omega^{\omega^2}})$$
$$\{\{_2L_2\&_3L_2\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega2+\Omega^{\theta(\Omega^\Omega2)}})$$
$$\{\{_2L_2\&_3L_2\&_3L_2\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega2+\Omega^{\theta(\Omega^{\Omega^\Omega2+\Omega^{\theta(\Omega^\Omega2)}})}})$$
$$\{\{_2\{_3L_2,X/2\}\}_{L,X}\}_{n,n}$$ it means an $$L_3$$ $$\theta(\Omega^{\Omega^\Omega3})$$
$$\{\{_2\{_3L_2,2X/2\}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega3}2)$$
$$\{\{_2\{_3L_2,X,1,2/2\}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega3+\Omega})$$
$$\{\{_2\{_3L_2,X(2)2/2\}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega3+\Omega^{\omega^2}})$$
$$\{\{_2\{_3L_2\&_3L_2/2\}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega3+\Omega^{\theta(\Omega^\Omega2)}})$$
$$\{\{_2\{_3L_2\&_3L_2\&_3L_2/2\}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega3+\Omega^{\theta(\Omega^{\Omega^\Omega2+\Omega^{\theta(\Omega^\Omega2)}})}})$$
$$\{\{_2\{_3L_2,X/3\}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega3+\Omega^{\theta(\Omega^{\Omega^\Omega3})}})$$
$$\{\{_2\{_3L_2,X/4\}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega3+\Omega^{\theta(\Omega^{\Omega^\Omega3})}2})$$
$$\{\{_2\{_3L_2,X/1,2\}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega3+\Omega^{\theta(\Omega^{\Omega^\Omega3})+1}})$$
$$\{\{_2\{_3L_2,X/1,1,2\}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega3+\Omega^{\theta(\Omega^{\Omega^\Omega3})+2}})$$
$$\{\{_2\{_3L_2,X/1/2\}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega3+\Omega^{\theta(\Omega^{\Omega^\Omega3})2}})$$
$$\{\{_2\{_3L_2,X(/1)2\}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega3+\Omega^{\theta(\Omega^{\Omega^\Omega3})\omega}})$$
$$\{\{_2L_2\&_3\&_3L_2\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega3+\Omega^{\theta(\Omega^{\Omega^\Omega3})\theta(\Omega^\Omega2)}})$$
$$\{\{_2L_2\&_3\&_3L_2\&_3\&_3L_2\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega3+\Omega^{\theta(\Omega^{\Omega^\Omega3+\Omega^{\theta(\Omega^{\Omega^\Omega3})\theta(\Omega^\Omega2)}})}})$$
$$\{\{_2\{_3L_2,X//2\}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega4})$$
$$\{\{_2\{_3L_2,X///2\}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega5})$$
$$\{\{_2\{_3L_2,X(1)/2\}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega\omega})$$
$$\{\{_2\{_3L_2,X(0,1)/2\}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega\omega^{\omega^\omega}})$$
$$\{\{_2\{L_3,L_2\}_{L_2,X}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega\theta(\Omega^\Omega2)})$$
$$\{\{_2\{L_3,L_3\}_{L_2,X}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^\Omega\theta(\Omega^{\Omega^\Omega3})})$$
$$\{\{_2\{L_3,X,2\}_{L_2,X}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\Omega+1}})$$
$$\{\{_2\{L_3,2X,2\}_{L_2,X}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\Omega+2}})$$
$$\{\{_2\{L_3,X,3\}_{L_2,X}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\Omega2}})$$
$$\{\{_2\{L_3,X,1,2\}_{L_2,X}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\Omega^2}})$$
$$\{\{_2\{L_3,X(1)2\}_{L_2,X}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\Omega^\omega}})$$
$$\{\{_2L_2@L_2\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\Omega^{\theta(\Omega^\Omega2)}}})$$
$$\{\{_2L_2@L_2@L_2\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\Omega^{\theta(\Omega^{\Omega^{\Omega^{\theta(\Omega^\Omega2)}}})}}})$$
$$\{\{_2\{_2L_3,2\}_{L_2,X}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\Omega^\Omega}})$$
$$\{\{_2\{\{_2L_3,2\},2\}_{L_2,X}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\Omega^\Omega}+\Omega^\Omega})$$
$$\{\{_2\{\{_2L_3,2\},X\}_{L_2,X}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\Omega^\Omega}+\Omega^{\Omega^\omega}})$$
$$\{\{_2L_2\%L_2\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\Omega^\Omega}+\Omega^{\Omega^{\theta(\Omega^\Omega2)}}})$$
$$\{\{_2\{_2L_3,3\}_{L_2,X}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\Omega^\Omega}2})$$
$$\{\{_2\{_2L_3,4\}_{L_2,X}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\Omega^\Omega}3})$$
$$\{\{_2\{_2L_3,X\}_{L_2,X}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\Omega^\Omega}\omega})$$
$$\{\{_2\{_2L_3,L_3\}_{L_2,X}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\Omega^\Omega}\theta(\Omega^{\Omega^\Omega3})})$$
$$\{\{_2\{_2L_3,X,2\}_{L_2,X}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\Omega^\Omega+1}})$$
$$\{\{_2\{_2L_3,X,1,2\}_{L_2,X}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\Omega^\Omega+\Omega}})$$
$$\{\{_2\{_2L_3,X(1)2\}_{L_2,X}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\Omega^\Omega+\Omega^\omega}})$$
$$\{\{_2L_2@_2L_2\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\Omega^\Omega+\Omega^{\theta(\Omega^\Omega2)}}})$$
$$\{\{_2\{_3L_3,2\}_{L_2,X}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\Omega^\Omega2}})$$
$$\{\{_2\{_2\{_3L_3,2\},2\}_{L_2,X}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\Omega^\Omega2}+\Omega^{\Omega^\Omega}})$$
$$\{\{_2\{_2\{_3L_3,2\},X,1,2\}_{L_2,X}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\Omega^\Omega2}+\Omega^{\Omega^\Omega+\Omega}})$$
$$\{\{_2L_2\%_2L_2\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\Omega^\Omega2}+\Omega^{\Omega^\Omega+\Omega^{\theta(\Omega^\Omega2)}}})$$
$$\{\{_2\{_3L_3,3\}_{L_2,X}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\Omega^\Omega2}2})$$
$$\{\{_2\{_3L_3,X\}_{L_2,X}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\Omega^\Omega2}\omega})$$
$$\{\{_2\{_3L_3,X,2\}_{L_2,X}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\Omega^\Omega2+1}})$$
$$\{\{_2\{_3L_3,X,1,2\}_{L_2,X}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\Omega^\Omega2+\Omega}})$$
$$\{\{_2L_2@_3L_2\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\Omega^\Omega2+\Omega^{\theta(\Omega^\Omega2)}}})$$
$$\{\{_2\{_3L_2,X\backslash2\}\}_{L,X}\}_{n,n}$$ it means an type-3 lugion $$\theta(\Omega^{\Omega^{\Omega^\Omega3}})$$
$$\{\{_2L_2\%_3L_2\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\Omega^\Omega3}+\Omega^{\Omega^\Omega2+\Omega^{\theta(\Omega^\Omega2)}}})$$

2\}\}_{L,X}\}_{n,n}\) it means an type-3 lagion

(Oh! How to type a lagion mark here?)

$$\theta(\Omega^{\Omega^{\Omega^\Omega3}2})$$
$$\{\{_2\{_3L_2,X-2\}\}_{L,X}\}_{n,n}$$ type-3 ligion $$\theta(\Omega^{\Omega^{\Omega^\Omega3}3})$$
$$\{\{_2\{_3\{_4L_3,X\}\}_{L_2,X}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\Omega^\Omega3}\omega})$$
$$\{\{_2\{_3\{_4L_3,X,2\}\}_{L_2,X}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\Omega^\Omega3+1}})$$
$$\{\{_2\{_3\{_4L_3,X,1,2\}\}_{L_2,X}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\Omega^\Omega3+\Omega}})$$
$$\{\{_2\{_3\{_4L_3,X(2)2\}\}_{L_2,X}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\Omega^\Omega3+\Omega^{\omega^\omega}}})$$
$$\{\{_2\{_3L_3\&_4L_3\}_{L_2,X}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\Omega^\Omega3+\Omega^{\theta(\Omega^{\Omega^\Omega3})}}})$$
$$\{\{_2\{_3\{_4L_3,X/2\}\}_{L_2,X}\}_{L,X}\}_{n,n}$$ $$\theta(\Omega^{\Omega^{\Omega^\Omega4}})$$

A type-2 legion has limit ordinal $$\theta(\Omega^\Omega2)$$

A type-3 legion has limit ordinal $$\theta(\Omega^{\Omega^\Omega3})$$

A type-4 legion has limit ordinal $$\theta(\Omega^{\Omega^{\Omega^\Omega4}})$$

A type-5 legion has limit ordinal $$\theta(\Omega^{\Omega^{\Omega^{\Omega^\Omega5}}})$$

......

Then this extension of BEAF has limit ordinal $$\theta(\varepsilon_{\Omega+1})$$.

Next idea is from Ikosarakt1. He puts X's in every kind of array.

{n,n/2}, {X,X/2}&n, {X,X/2}&X&n, {X,X/2}&X&X&n, ... their growth rates are all LVO.

But,

{X,X/2}+1&n={n,n/n}, and has growth rate $$\theta(\Omega^\Omega)\omega$$

{X,X/2}*2&n={n,n/1/2}, and has growth rate $$\theta(\Omega^\Omega)^2$$

{X,X/2}^2&n={n,n/2}&&n, and has growth rate $$\theta(\Omega^\Omega)^{\theta(\Omega^\Omega)}$$

{X,X/2}^{X,X/2}&n={n,n/2}&&n&&n, and has growth rate $$\theta(\Omega^\Omega)^{\theta(\Omega^\Omega)^{\theta(\Omega^\Omega)}}$$

{X,X/2}^^X&n={n,n//2}, and has growth rate $$\varepsilon_{\theta(\Omega^\Omega)+1}$$

{X,X/2}^^{X,X/2}&n={L,L}n,n, and has growth rate $$\varepsilon_{\theta(\Omega^\Omega)2}$$

{{X,X/2},X,3}&n={L,X,2}n,n, and has growth rate $$\zeta_{\theta(\Omega^\Omega)+1}$$

{{X,X/2},X,1,2}&n={L,X,1,2}n,n, and has growth rate $$\Gamma_{\theta(\Omega^\Omega)+1}$$

{{X,X/2},X(1)2}&n={L,X(1)2}n,n, and has growth rate $$\theta(\Omega^\omega,\theta(\Omega^\Omega)+1)$$

{X,2X/2}&n={n,n\2}, and has growth rate $$\theta(\Omega^\Omega,1)$$

{X,X,2/2}&n={[L,X,2]}n,n, and has growth rate $$\theta(\Omega^\Omega+1)$$

{X,X,1,2/2}&n={[L,X,1,2]}n,n, and has growth rate $$\theta(\Omega^\Omega+\Omega)$$

{X,X/3}&n={L&2L}n,n, and has growth rate $$\theta(\Omega^\Omega+\Omega^{\theta(\Omega^\Omega)})$$

{X,X//2}&n reaches the limit of BEAF on "n", and has growth rate $$\theta(\Omega^\Omega2)$$

If we add one more &X, it will make a big difference.

{X,X/2}+1&X&n={X,X/X}&n, and has growth rate $$\theta(\Omega^\Omega+\Omega^{\theta(\Omega^\Omega)}\omega)$$

{X,X/2}*2&X&n={X,X/1/2}&n, and has growth rate $$\theta(\Omega^\Omega+\Omega^{\theta(\Omega^\Omega)2})$$

{X,X/2}^2&X&n={X,X/2}&&X&n, and has growth rate $$\theta(\Omega^\Omega+\Omega^{\theta(\Omega^\Omega)^2})$$

{X,X/2}^^X&X&n={X,X//2}&n, and has growth rate $$\theta(\Omega^\Omega2)$$

{X,2X/2}&X&n={X,X\2}&n, and has growth rate $$\theta(\Omega^{\Omega^\Omega})$$

{X,X,2/2}&X&n={[L,X,2]}X,X&n, and has growth rate $$\theta(\Omega^{\Omega^\Omega+1})$$

{X,X/3}&X&n={L&2L}X,X&n, and has growth rate $$\theta(\Omega^{\Omega^\Omega+\Omega^{\theta(\Omega^\Omega)}})$$

{X,X//2}&X&n reaches the limit of {X-array}&n, and has growth rate $$\theta(\Omega^{\Omega^\Omega2})$$

We also have:

{X,X/2}+1&X&X&n={X,X/X}&X&n

{X,X/2}*2&X&X&n={X,X/1/2}&X&n

{X,X/2}^2&X&X&n={X,X/2}&&X&X&n

{X,X/2}^^X&X&X&n={X,X//2}&X&n

{X,2X/2}&X&X&n={X,X\2}&X&n

{X,X,2/2}&X&X&n={[L,X,2]}X,X&X&n

{X,X/3}&X&X&n={L&2L}X,X&X&n

{X,X//2}&X&X&n reaches the limit of {X-array}&X&n, at growth rate $$\theta(\Omega^{\Omega^{\Omega^\Omega2}})$$

......

Normal {n-array} reach $$\theta(\Omega^\Omega2)$$

{X-array}&n reach $$\theta(\Omega^{\Omega^\Omega2})$$

{X-array}&X&n reach $$\theta(\Omega^{\Omega^{\Omega^\Omega2}})$$

{X-array}&X&X&n reach $$\theta(\Omega^{\Omega^{\Omega^{\Omega^\Omega2}}})$$

etc.

So this way can also reach BHO.