Part 1 is here.
I made a mistake - BEAF-BSC and BEAF-HDI have points of intersection, but they strongly diverge between this.
The FB100Z system of BEAF was wery weak and reached only LVO - but we know it is not so, BEAF is much stronger.
So, I will show you three systems. First system is Ikosarakt innovational system - but it only diverge after LVO, so in the first part of this page it will be absent. Second system is BEAF-BSC, and third is BEAF-HD(I).
The psi-function used in analyses is again Buchholz's function and theta-function was used by Hyp cos in his analyses, but it is unspecified. So, here are some milestones:
Structure | BEAF-BSC | BEAF-HDI |
---|---|---|
{X,X(1)2} | \(\psi_0(\Omega^{\Omega^\omega})\) | \(\psi_0(\Omega^{\Omega^\omega})\) |
{{X,X(1)2},X(1)2}&n={X,3,1,...1,2}&n 2 a.p X+1 | \(\theta(\Omega^\omega,1)\) | \(\theta(\Omega^\omega,1)\) |
{X,X+1(1)2}&n={X,X,...X,X}&n a.p X+1 | \(\psi_0(\Omega^{\Omega^\omega\omega})\) | \(\psi_0(\Omega^{\Omega^\omega\omega})\) |
{X,X+2(1)2}&n | \(\psi_0(\Omega^{\Omega^{\omega+1}\omega})\) | \(\psi_0(\Omega^{\Omega^{\omega+1}\omega})\) |
{X,X^2(1)2}&n | \(\psi_0(\Omega^{\Omega^{\omega^2}})\) | \(\psi_0(\Omega^{\Omega^{\omega^2}})\) |
{X,3,2(1)2}&n | \(\psi_0(\Omega^{\Omega^{\psi(\Omega^{\Omega^\omega)}}})\) | \(\psi_0(\Omega^{\Omega^{\psi(\Omega^{\Omega^\omega)}}})\) |
{X,X,2(1)2}&n | \(\psi_0(\Omega^{\Omega^\Omega})\) | \(\psi_0(\Omega^{\Omega^\Omega})\) |
We're at LVO now, and this is not a legion yet. After this the Ikosarakt's system which he described in comment will diverge: (I had to verify this several times until I got stable results)
Structure | Strongest BEAF | BEAF-BSC | BEAF-HDI |
---|---|---|---|
{X,X,X(1)2}&n | \(\psi_0(\Omega^{\Omega^{\Omega\omega}})\) | \(\psi_0(\Omega^{\Omega^{\Omega\omega}})\) | \(\psi_0(\Omega^{\Omega^\Omega+\omega})\) |
{X,X,1,2(1)2}&n | \(\psi_0(\Omega^{\Omega^{\Omega^2}})\) | \(\psi_0(\Omega^{\Omega^{\Omega^2}})\) | \(\psi_0(\Omega^{\Omega^\Omega+\Omega})\) |
{X,X,1,1,2(1)2}&n | \(\psi_0(\Omega^{\Omega^{\Omega^3}})\) | \(\psi_0(\Omega^{\Omega^{\Omega^3}})\) | \(\psi_0(\Omega^{\Omega^\Omega+\Omega^2})\) |
{X,X(1)3}&n | \(\psi_0(\Omega^{\Omega^{\Omega^\omega}})\) | \(\psi_0(\Omega^{\Omega^{\Omega^\omega}})\) | \(\psi_0(\Omega^{\Omega^\Omega+\Omega^\omega})\) |
{X,X,2(1)3}&n | \(\psi_0(\Omega^{\Omega^{\Omega^\Omega}})\) | \(\psi_0(\Omega^{\Omega^{\Omega^\Omega}})\) | \(\psi_0(\Omega^{\Omega^\Omega2})\) |
{X,X(1)X}&n | \(\psi_0(\varepsilon_{\Omega+1}) = \psi_0(\Omega_2)\) | \(\psi_0(\varepsilon_{\Omega+1}) = \psi_0(\Omega_2)\) | \(\psi_0(\Omega^{\Omega^\Omega\omega})\) |
{X,X(1)1,2}&n | \(\psi_0(\varepsilon_{\Omega2})\) | \(\psi_0(\varepsilon_{\Omega2})\) | \(\psi_0(\Omega^{\Omega^{\Omega+1}})\) |
{X,X(1)1,3}&n | \(\psi_0(\varepsilon_{\Omega3})\) | \(\psi_0(\varepsilon_{\Omega3})\) | \(\psi_0(\Omega^{\Omega^{\Omega+1}2})\) |
{X,X(1)X,X}&n | \(\psi_0(\varepsilon_{\Omega\omega})\) | \(\psi_0(\varepsilon_{\Omega\omega})\) | \(\psi_0(\Omega^{\Omega^{\Omega+1}\omega})\) |
{X,X(1)1,1,2}&n | \(\psi_0(\varepsilon_{\Omega^2})\) | \(\psi_0(\varepsilon_{\Omega^2})\) | \(\psi_0(\Omega^{\Omega^{\Omega+2}})\) |
{X,X(1)1,1,1,2}&n | \(\psi_0(\varepsilon_{\Omega^3})\) | \(\psi_0(\varepsilon_{\Omega^3})\) | \(\psi_0(\Omega^{\Omega^{\Omega+3}})\) |
{X,X(1)(1)2}&n | \(\psi_0(\varepsilon_{\Omega^\omega})\) | \(\psi_0(\varepsilon_{\Omega^\omega})\) | \(\psi_0(\Omega^{\Omega^{\Omega+\omega}})\) |
{X,X,2(1)(1)2}&n | \(\psi_0(\varepsilon_{\Omega^\Omega})\) | \(\psi_0(\varepsilon_{\Omega^\Omega})\) | \(\psi_0(\Omega^{\Omega^{\Omega2}})\) |
{X,X(2)2}&n | \(\psi_0(\varepsilon_{\varepsilon_{\Omega+1}})\) | \(\psi_0(\varepsilon_{\varepsilon_{\Omega+1}})\) | \(\psi_0(\Omega^{\Omega^{\Omega\omega}})\) |
{X,X,2(2)2}&n | \(\psi_0(\varepsilon_{\varepsilon_{\Omega2}})\) | \(\psi_0(\varepsilon_{\varepsilon_{\Omega2}})\) | \(\psi_0(\Omega^{\Omega^{\Omega^2}})\) |
{X,X(0,1)2}&n | \(\psi_0(\zeta_{\Omega+1})\) | \(\psi_0(\zeta_{\Omega+1})\) | \(\psi_0(\Omega^{\Omega^{\Omega^\omega}})\) |
{X,X((1)1)2}&n | \(\psi_0(\eta_{\Omega+1})\) | \(\psi_0(\eta_{\Omega+1})\) | \(\psi_0(\Omega^{\Omega^{\Omega^{\Omega^\omega}}})\) |
X^^X&X&n | \(\psi_0(\varphi(\omega,\Omega+1))\) | \(\psi_0(\varphi(\omega,\Omega+1))\) | \(\psi_0(\varepsilon_{\Omega+1}) = \psi_0(\Omega_2)\) |
{X,X,2(X^^X)2}&X&n | \(\psi_0(\Gamma_{\Omega+1})\) | \(\psi_0(\Gamma_{\Omega+1})\) | ? |
.My conjecture is that I will get to \(\psi_0(\Omega_3)\) earlier than X&X&X&n (or, {X,X(1)2}&X&n) and I will have \(\psi_0(\Omega_\omega)\) as the limit of sub-legion arrays, that is, I will verify Hyp cos' assumption. In part 1 I said that we will reach the limit of Bashicu's pair sequence system. It is supposed to be the limit of sub-legion arrays. Why is it so? 'Cause we have \(\psi_0(\Omega)\) as X^^X&n, \(\psi_0(\Omega_2)\) as {X,X(1)X}&n, and we almost reached \(\psi_0(\Omega_3)\) when we have expressions like A&X&n, where A is some expression using X. I strongly suppose that if we add some more &, then we will increment index at \(\Omega_m\) and finally come to a limit of sub-legion arrays, and that is \(\psi_0(\Omega_\omega)\), as Hyp cos predicted.