BHAN, ABHAN, and FGH analysis

In the Googology Discord Server, I analyze ABHAN with FGH and it very strong. However, Boris claim that BHAN is stronger. So I will analyze BHAN, ABHAN, and FGH.

I will use the lists, since I cannot make the tables. I will start from BHAN Extended Arrays.

• FGH = BHAN = ABHAN
• w^w = a(1)b = 0{1}1
• w^w+1 = a(2)b = 1{1}1
• w^w+n = a(n)b = n{1}1
• w^w+w = a(a,b)b = 0,1{1}1
• w^w+w2 = a(a,b,c)b = 0,2{1}1
• w^w*2 = a(a(1)b)b = 0{1}2
• w^w*3 = a(a(a(1)b)b)b = 0{1}3
• w^(w+1+) = a\^1_2 = 0{1}0,1
• w^(w+1)*2 = (a\^1_2)\^1_2 = 0{1}0,2
• w^(w+2) = a\^1_3 = 0{1}0,0,1
• w^(w+n) = a\^1_n = 0{1}0,0,0,...
• w^(w*2) = a\^2_2 = 0{1}0{1}1
• w^(w*2+1) = a\^2_3 = 0{1}0{1}0,1
• w^(w*3) = a\^3_1 = 0{1}0{1}0{1}1
• w^w^2 = a\^1_2\^1_2 = 0{2}1
• w^(w^2*2) = a\^1_2\^1_3 = 0{2}0{2}1
• w^w^3 = a\^1_2\^2_1 = 0{3}1
• w^w^4 = a\^1_2\^1_2\^1_2 = 0{4}1
• w^w^w = a\\^n_m = 0{0,1}1
• w^(w^w+1) = a\\^n_m\^1_2 = 0{0,1}0,1
• w^(w^w*2) = a\\^n_m\\^n_m = 0{0,1}0{0,1}1
• w^w^(w+1) = a\\\^n_m = 0{1,1}1
• w^w^(w+2) = a\\\\^n_m = 0{2,1}1
• w^w^(w+n) = a\_n\^n_m = 0{n,1}1
• w^w^(w2) = a^+b = 0{0,2}1
• w^w^(w2)*2 = a^+(a^+b) = 0{0,2}2
• w^(w^(w2)+1) = a^(+2)b = 0{0,2}0,1
• w^(w^(w2)+2) = a^(+3)b = 0{0,2}0,0,1
• w^(w^(w2)+n) = a^(+n)b = 0{0,2}0,0,0,...
• w^(w^(w2)+w) = a^++b = 0{0,2}0{1}1
• w^(w^(w2)+w)+1 = a^+++b = 1{0,2}0{1}1
• w^(w^(w2)+w)+n = a^{n,+}b = n{0,2}0{1}1
• w^(w^(w2)+w)+w = M1,a = 0,1{0,2}0{1}1
• w^(w^(w2)+w)+w+1 = M2,a = 1,1{0,2}0{1}1
• w^(w^(w2)+w)+w+2 = <1,1,1>,a = 2,1{0,2}0{1}1
• w^(w^(w2)+w)+w2 = M3,a = 0,2{0,2}0{1}1
• w^(w^(w2)+w)+w3 = <2,1,2>,a = 0,2{0,2}0{1}1
• w^(w^(w2)+w)+w^2 = <2,2>,a = 0,0,1{0,2}0{1}1
• w^(w^(w2)+w)+w^3 = <2,2,2>,a = 0,0,0,1{0,2}0{1}1
• w^(w^(w2)+w)+w^w = M4,a = 0{1}1{0,2}0{1}1
• w^(w^(w2)+w)+w^(w2) = <3,2,3>,a = 0{1}0{1}1{0,2}0{1}1
• w^(w^(w2)+w)+w^w^2 = <3,3>,a = 0{2}1{0,2}0{1}1
• w^(w^(w2)+w)+w^w^w = M5,a = 0{0,1}1{0,2}0{1}1
• w^w^w^2 = <4,4>,a = 0{0,0,1}1
• w^w^w^w = M6,a = 0{0{1}1}1
• w^w^w^w^w = M7,a = 0{0{0,1}1}1
• e0 = Ma,b = 0{0\1}1
• e0+1 = M[2]a,b = 1{0\1}1
• e0+2 = M[2][2]a,b = 2{0\1}1
• e0+w = M[2][/a,b/]a,b = 0,1{0\1}1
• e0+w^w = M[2][/a(1)b/]a,b = 0{1}1{0\1}1
• e0*2 = M[2][/Ma,b/]a,b = 0{0\1}2
• e0*w = M[3]a,b = 0{0\1}0,1
• e0*w2 = M[3][/M[3]a,b/]a,b = 0{0\1}0,2
• e0*w^2 = M[4]a,b = 0{0\1}0,0,1
• e0*w^w = M[/a,b/]a,b = 0{0\1}0{1}1
• e0^2 = M[/Ma,b/]a,b = 0{0\1}0{0\1}1
• e0^w = M[1,2]a,b = 0{1\1}1
• e0^w*w = M[1,2][1,2]a,b = 0{1\1}0,1
• e0^w*w^w = M[1,2][1,/a,b/]a,b = 0{1\1}0{1}1
• e0^w2 = M[1,2][1,/M[1,2]a,b/]a,b = 0{1\1}0{1\1}1
• e0^w^2 = M[1,3]a,b = 0{2\1}1
• e0^w^w = M[1,/a,b/]a,b = 0{0,1\1}1
• e0^e0 = M[1,/Ma,b/]a,b = 0{0{0\1}1\1}1
• e1 = M[2,2]a,b = 0{0\2}1
• e1*w = M[2,2][1,2]a,b = 0{0\2}0,1
• e1^w = M[2,2][2,2]a,b = 0{1\2}1
• e2 = M[2,3]a,b = 0{0\3}1
• e(w) = M[2,/a,b/]a,b = 0{0\0,1}1
• e(e0) = M[2,/Ma,b/]a,b = 0{0\0{0\1}1}1
• z0 = M[3,2]a,b = 0{0\0\1}1
• e(z0+1) = M[3,2][3,2]a,b = 0{0\1\1}1
• z1 = M[3,3]a,b = 0{0\0\2}1
• z(z0) = M[3,/M[3,2]a,b/] = 0{0\0\0{0\0\1}1}1
• n0 = M[4,2]a,b = 0{0\0\0\1}1
• phi(4,0) = M[5,2]a,b = 0{0\0\0\0\1}1
• phi(w,0) = M[/a,b/,2]a,b = 0{0\{1}1}
• phi(e0,0) = M[/Ma,b/,2]a,b = 0{0\{0{0\1}1}1}1
• gamma0 = M[1,1,2]a,b = 0{0\{0\1}1}1
• gamma1 = M[1,1,2][1,1,2]a,b = 0{0\{0\1}2}1
• gamma(gamma0) = M[1,1,2][1,1,/M[1,1,2]a,b/]a,b = 0{0\{0\1}0{0\{0\1}1}1}1
• phi(1,1,0) = M[1,1,3]a,b = 0{0\{0\1}0\1}1
• phi(1,1,phi(1,1,0)) = M[1,1,/M[1,1,3]a,b/]a,b = 0{0\{0\1}0\0{0\{0\1}0\1}1}1
• phi(1,2,0) = M[1,2,2]a,b = 0{0\{0\1}0\0\1}1
• phi(1,gamma0,0) = M[1,/M[1,1,2]a,b/,2]a,b = 0{0\{0\1}0\{0{0\{0\1}1}1}1}1
• phi(2,0,0) = M[2,1,2]a,b = 0{0\{0\1}0\{0\1}1}1
• phi(2,1,0) = M[2,2,2]a,b = 0{0\{0\1}0\{0\1}0\1}1
• phi(3,0,0) = M[3,1,2]a,b = 0{0\{0\1}0\{0\1}0\{0\1}1}1
• phi(gamma0,0,0) = M[/M[1,1,2]a,b/,1,2]a,b = 0{0\{0{0\{0\1}1}1\1}1}1
• phi(1,0,0,0) = M[1,1,1,2]a,b = 0{0\{0\2}1}1
• phi(1,1,0,0) = M[1,2,1,2]a,b = 0{0\{0\2}0\{0\1}1}1
• phi(2,0,0,0) = M[2,1,1,2]a,b = 0{0\{0\2}0\{0\2}1}1
• phi(phi(1,0,0,0),0,0,0) = M[/M[1,1,1,2]a,b/,1,1,2]a,b = 0{0\{0{0\{0\2}1}1\2}1}1
• phi(1,0,0,0,0) = M[1,1,1,1,2]a,b = 0{0\{0\3}1}1
• phi(phi(1,0,0,0,0),0,0,0,0) = M[/M[1,1,1,1,2]a,b/,1,1,1,2]a,b = 0{0\{0{0\{0\3}1}1\3}1}1
• phi(1,0,0,0,0,0) = M[1,1,1,1,1,2]a,b = 0{0\{0\4}1}1
• psi(W^W^w) = M[a(1)b]a,b = 0{0\{0\0,1}1}1
• psi(W^(W^w+1)) = M[a(1)b+1]a,b = 0{0\{0\0,1}0\1}1
• psi(W^(W^w+psi(W^W^w))) = M[a(1)/M[a(1)b]a,b/]a,b = 0{0\{0\0,1}0\{0{0\{0\0,1}1}1}1}1
• psi(W^(W^w+W)) = M[a(2)b]a,b = 0{0\{0\0,1}0\{0\1}1}1
• psi(W^(W^w+W2)) = M[a(3)b]a,b = 0{0\{0\0,1}0\{0\1}0\{0\1}1}1
• psi(W^(W^w+W^2)) = M[a(1,2)b]a,b = 0{0\{0\0,1}0\{0\2}1}1
• psi(W^(W^w*2)) = M[a(a(1)b)b]a,b = 0{0\{0\0,1}0\{0\0,1}1}1
• WIP