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Thanks to @Koteitan in the beginning for helping me actually make this entire thing valid, because I'm apparently bad at explaining things.

So, this will be Sudden Sequence System defined not by Bashicu's BASIC code smh like how am I supposed to run that but my own python code in my past blog post. Or whatever code I've been using that I only remembered to post in discord but forgot to put in wikia, which I will put here after. I will use Extended Weak Buchholz psi defined here User blog:P進大好きbot/Extended Weak Buchholz's OCF in Programming Language Qp#Automatic Conversion into English idk I think it's the original definition, and besides since the standard terms in extended weak buchholz is a subset of standard terms in extended normal buchholz it should work if it's ordinalnotationified

To convert any (hopefully standard) SSS expression to EWB (or back), the process operates on 3 very simple rules which how they work will be shown below:

  1. empty is 0
  2. ascension is subscripting
  3. concatenation is nesting

1 is straight forward so I'll start with 2.

Ascension is subscript: \( O([0,a_1,\dots,a_i]) = \psi_{O([a_1-1,\dots,a_i-1])}(0)\)

Here \(a_i\) isn't 0 for any \(i\); there may be any amount of terms in the sequence including 0 or 1 terms. Use this rule if some expression contains only a single 0 in the beginning.

Concatenation is nesting:

if \( O([0,a_1,\dots,a_i]) = \psi_{\alpha}(0) \) and \(O([0,b_1,\dots,b_i]) = \beta \), then \(O([0,a_1,\dots,a_i]+[0,b_1,\dots,b_i]) = \psi_\alpha(\beta)) \)

Here \( a_i \) isn't 0 but \(b_i\) can be anything, including 0. List addition is concatenation. Use this rule if some expression contains more than one 0.

And we are done.

Except, if you actually do that, you will find that for every standard SSS expresion that contains a non-0, the result of O(said expression) is always uncountable. So 0,1 = Ω and stuff. This really isn't an issue, since if you're using the ordinal notation EWB then the range of O has obviously a countable order-type. To fix that simply put O(expression) in a ψ0 to countableify it. Weirdly if an SSS expression is 0 concatenated with another standard expression then putting it through O gives you exactly that. However originally Bashicu made the standard form of SSS things that expand from (0,1,2,3,...,n) for some n and these are all uncountable, except behaviour of SSS is the exact same when you put an extra 0 in the beginning.

Here's an analysis table. I chose what to put here, since it's just examples anyway.

SSS, Extended Buchholz and Extended Weak Buchholz
SSS Countable EWB Countable Buchholz Default notation
empty 0 0 0
0 \(\psi_0(0)\) \(\psi_0(0)\) 1
0,0 \(\psi_0(\psi_0(0)) \) \( \psi_0(0)+\psi_0(0) \) 2
0,1 \(\psi_0(\psi_1(0))\) \( \psi_0(\psi_0(0)) \) \(\omega\)
0,1,0,0,1 \(\psi_0(\psi_1(\psi_0(\psi_1(0))))\) \( \psi_0(\psi_0(0))+\psi_0(\psi_0(0)) \) \(\omega\cdot 2\)
0,1,0,1 \(\psi_0(\psi_1(\psi_1(0)))\) \( \psi_0(\psi_0(0)+\psi_0(0)) \) \(\omega^2\)
0,1,1 \(\psi_0(\psi_2(0))\) \( \psi_0(\psi_0(\psi_0(0))) \) \(\omega^\omega\)
0,1,1,2 \(\psi_0(\psi_\omega(0))\) \(\psi_0(\psi_1(0))\) \(\varepsilon_0\)
0,1,1,2,0,1,1,2 \(\psi_0(\psi_\omega(\psi_\omega(0)))\) \(\psi_0(\psi_1(0)+\psi_1(0))\) \(\varepsilon_1\)
0,1,1,2,1 \(\psi_0(\psi_{\omega+1}(0))\) \(\psi_0(\psi_1(\psi_0(0)))\) \(\varepsilon_\omega\)
0,1,1,2,1,1,2 \(\psi_0(\psi_{\omega\cdot 2}(0))\) \(\psi_0(\psi_1(\psi_0(\psi_1(0))))\) \(\varepsilon_{\varepsilon_0}\)
0,1,1,2,1,2 \(\psi_0(\psi_{\omega^2}(0))\) \(\psi_0(\psi_1(\psi_1(0)))\) \(\zeta_0\)
0,1,1,2,1,2,1,2 \(\psi_0(\psi_{\omega^3}(0))\) \(\psi_0(\psi_1(\psi_1(0)+\psi_1(0)))\) \(\eta_0\)
0,1,1,2,2 \(\psi_0(\psi_{\omega^\omega}(0))\) \(\psi_0(\psi_1(\psi_1(\psi_0(0))))\) \(\phi(\omega,0)\)
0,1,1,2,2,3 \(\psi_0(\psi_{\varepsilon_0}(0))\) \(\psi_0(\psi_1(\psi_1(\psi_0(\psi_1(0)))))\) \(\phi(\varepsilon_0,0)\)
0,1,2 \(\psi_0(\psi_\Omega(0))\) \(\psi_0(\psi_1(\psi_1(\psi_1(0))))\) \(\Gamma_0\)
0,1,2,0,1,0,1,1,2,3 \(\psi_0(\psi_\Omega(\psi_1(\psi_{\Gamma_0}(0))))\) \(\psi_0(\psi_1(\psi_1(\psi_1(0)))+ \psi_0(\psi_1(\psi_1(\psi_1(0)))))\) \(\Gamma_0^2\)
0,1,2,0,1,0,1,1,2,3,0,1,1,2 \(\psi_0(\psi_\Omega(\psi_1(\psi_{\Gamma_0}(\psi_\omega(0)))))\) \(\psi_0(\psi_1(\psi_1(\psi_1(0)))+\psi_1(0))\) \(\varepsilon_{\Gamma_0+1}\)
0,1,2,0,1,0,1,1,2,3,0,1,1,2,3 \(\psi_0(\psi_\Omega(\psi_1(\psi_{\Gamma_0}(\psi_{\Gamma_0}(0)))))\) \(\psi_0(\psi_1(\psi_1(\psi_1(0)))+ \psi_1(\psi_1(\psi_0(\psi_1(\psi_1(\psi_1(0)))))))\) \(\phi(\Gamma_0,1)\)
0,1,2,0,1,0,1,1,2,3,1 \(\psi_0(\psi_\Omega( \psi_1(\psi_{\Gamma_0+1}(0))))\) \(\psi_0(\psi_1(\psi_1(\psi_1(0)))+ \psi_1(\psi_1(\psi_0(\psi_1(\psi_1(\psi_1(0)))))+1))\) \(\phi(\Gamma_0,\omega)\)
0,1,2,0,1,0,1,1,2,3,1,2 \(\psi_0(\psi_\Omega( \psi_1(\psi_{\psi_0(\psi_\Omega(\psi_1(0)))}(0))))\) \(\psi_0(\psi_1(\psi_1(\psi_1(0)))+ \psi_1(\psi_1(\psi_0(\psi_1( \psi_1(\psi_1(0)))))+\psi_1(0)))\) \(\phi(\Gamma_0+1,0)\)
0,1,2,0,1,0,1,2 \(\psi_0(\psi_\Omega(\psi_1(\psi_{\Omega}(0))))\) \(\psi_0(\psi_1(\psi_1(\psi_1(0)))+ \psi_1(\psi_1(\psi_1(0))))\) \(\Gamma_1\)
0,1,2,0,1,1 \(\psi_0(\psi_\Omega(\psi_2(0)))\) \(\psi_0(\psi_1(\psi_1(\psi_1(0))+1))\) \(\Gamma_\omega\)
0,1,2,0,1,1,0,1,1,2,3 \(\psi_0(\psi_\Omega( \psi_2(\psi_{\Gamma_0}(0))))\) \(\psi_0(\psi_1(\psi_1(\psi_1(0))+ \psi_0(\psi_1(\psi_1(\psi_1(0))))))\) \(\Gamma_{\Gamma_0}\)
0,1,2,0,1,1,0,1,2 \(\psi_0(\psi_\Omega( \psi_2(\psi_\Omega(0))))\) \(\psi_0(\psi_1(\psi_1(\psi_1(0))+\psi_1(0)))\) \(\phi(1,1,0)\)
0,1,2,0,1,1,1 \(\psi_0(\psi_\Omega(\psi_3(0)))\) \(\psi_0(\psi_1(\psi_1(\psi_1(0))+\psi_1(\psi_0(0))))\) \(\phi(1,\omega,0)\)
0,1,2,0,1,1,1,1 \(\psi_0(\psi_\Omega(\psi_4(0)))\) \(\psi_0(\psi_1(\psi_1(\psi_1(\psi_0(0)))))\) \(\phi(1,0,\dots,0)\)
0,1,2,0,1,1,2 \(\psi_0(\psi_\Omega(\psi_\omega(0)))\) \(\psi_0(\psi_2(0))\) \(\psi(\Omega_2)\)
0,1,2,0,1,1,2,1,2 \(\psi_0(\psi_\Omega(\psi_{\omega^2}(0)))\) \(\psi_0(\psi_2(\psi_2(0)))\) \(\psi(\Omega_2^2)\)
0,1,2,0,1,1,2,2 \(\psi_0(\psi_\Omega(\psi_{\omega^\omega}(0)))\) \(\psi_0(\psi_2(\psi_2(1)))\) \(\psi(\Omega_2^\omega)\)
0,1,2,0,1,1,2,3 \(\psi_0(\psi_\Omega(\psi_{\Gamma_0}(0)))\) \(\psi_0(\psi_2(\psi_2(\psi_0( \psi_1(\psi_1(\psi_1(0)))))))\) \(\psi(\Omega_2 ^{\Gamma_0})\)
0,1,2,0,1,2 \(\psi_0(\psi_\Omega(\psi_\Omega(0)))\) \(\psi_0(\psi_2(\psi_2(\psi_1(0)))))\) \(\psi(\Omega_2^{\Omega})\)
0,1,2,1 \(\psi_0(\psi_{\Omega+1}(0))\) \(\psi_0(\psi_2(\psi_2(\psi_1(0)))+1))\) \(\psi(\Omega_2^{\Omega} \cdot \omega)\)
0,1,2,1,2 \(\psi_0(\psi_{\Omega\cdot 2}(0))\) \(\psi_0(\psi_2(\psi_2(\psi_1(0))+\psi_1(0)))\) \(\psi(\Omega_2^ {\Omega}\cdot \Omega)\)
0,1,2,1,2,2 \(\psi_0(\psi_{\psi_1(\psi_2(0))}(0))\) \(\psi_0(\psi_2(\psi_2(\psi_1(0))+\psi_2(0)))\) \(\psi(\Omega_2^ {\Omega+1})\)
0,1,2,1,2,3 \(\psi_0(\psi_{\psi_1(\psi_\Omega(0))}(0))\) \(\psi_0(\psi_2(\psi_2(\psi_1(0))+\psi_2(\psi_1(0)))))\) \(\psi(\Omega_2^ {\Omega\cdot 2})\)
0,1,2,2 \(\psi_0(\psi_{\psi_2(0)}(0))\) \(\psi_0(\psi_2(\psi_2(\psi_2(0))))\) \(\psi(\Omega_2 ^{\Omega_2})\)
0,1,2,2,0,1,1,2 \(\psi_0(\psi_{\psi_2(0)}(\psi_\omega(0)))\) \(\psi_0(\psi_3(0))\) \(\psi(\Omega_3)\)
0,1,2,2,3 \(\psi_0(\psi_{\psi_\omega(0)}(0))\) \(\psi_0(\psi_\omega(0))\) \(\psi(\Omega_\omega)\)
0,1,2,3 \(\psi_0(\psi_{\psi_\Omega(0)}(0))\) \(\psi_0(\psi_\Omega(0))\) \(\psi(\Omega_\Omega)\)
0,2 \(\psi_0(\Omega_{\Omega_{ \Omega_{\Omega_{\ddots}}}})\) \(\psi_0(\psi_{\psi_{\psi_{\psi_{\ddots}(0)}(0)}(0)}(0))\) \(\psi(\psi_I(0))\)
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