FANDOM


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.

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