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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.

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