User blog:B1mb0w/Alpha Function and f epsilon nought

Alpha Function and \(f_{\epsilon_0}(m)\)
This blog will compare the growth rate of the Alpha Function to the fast growing hierarchy function \(f_{\epsilon_0}(n)\). Starting with:

\(D(n+1,n-2) >> f_n(n) >> f_{\omega}(n)\) more information at this link

\(D(1,m,0) >> f_{\omega}^m(f_{\omega+1}(m))\) more information at this link

\(D(2,m,0) >> f_{\omega+1}^m(f_{\omega+2}(m))\) more information at this link

and

\(D(l,m,0) >> f_{\phi}^m(f_{\phi+1}(m))\) when \(D(l-1,m,0) >> f_{\phi-1}^m(f_{\phi}(m))\)  more information at this link

The following is speculated based on the progression above. Proofs will be added soon.

\(f_{\omega.2}(m)\)
\(D(m,m,0) >> f_{\omega}^m(f_{\omega+m}(m)) = f_{\omega+m}(m) = f_{\omega.2}(m)\)

\(f_{\omega^2}(m)\)
\(D(m.2,m,0) >> f_{\omega.2 + m}(m) = f_{\omega.2 + \omega}(m) = f_{\omega.3}(m)\)

\(D(m.3,m,0) >> f_{\omega.3 + m}(m) = f_{\omega.3 + \omega}(m) = f_{\omega.4}(m)\)

\(D(m.(m-1),m,0) = D(m^2-m,m,0) >> f_{\omega.(m-1) + m}(m) = f_{\omega.(m-1) + \omega}(m)\)

\(= f_{\omega.m}(m) = f_{\omega^2}(m)\) Proofs will be added soon

\(f_{\omega^{\omega}}(m)\)
\(D(m^3-m,m,0) >> f_{\omega^3}(m)\)

\(D(m^4-m,m,0) >> f_{\omega^4}(m)\)

\(D(m^m-m,m,0) >> f_{\omega^m}(m) = f_{\omega^{\omega}}(m)\) Proofs will be added soon

\(f_{\omega^{\omega^{\omega}}}(m)\)
\(D(m^{m^2}-m,m,0) >> f_{\omega^{\omega^2}}(m)\)

\(D(m^{m^3}-m,m,0) >> f_{\omega^{\omega^3}}(m)\)

\(D(m^{m^m}-m,m,0) >> f_{\omega^{\omega^m}}(m) = f_{\omega^{\omega^{\omega}}}(m)\) Proofs will be added soon

\(f_{\epsilon_0}(m)\)
\(D(m\uparrow\uparrow m-m,m,0) >> f_{\epsilon_0}(m)\) Proofs will be added soon

and

\(D(4,n-2) >> f_3(n) >> n\uparrow\uparrow n\)

then

\(D(D(4,n-2),n,0) >> f_{\epsilon_0}(n)\)

or

\(D(D(4,n-1),0,0) >> f_{\epsilon_0}(n)\) Proofs will be added soon