User blog:B1mb0w/Growth Rate of the S Function

Growth Rate of the S Function
This blog will provide a detailed calculation and references for the growth rate of The S Function that I have developed.

Introduction
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Growth Rate up to \(S(n,T(1),1)\)
The number of restricted S Function sequences that can be constructed has a growth rate faster than \(f_{LVO}(n)\). Here are the growth rates for sub-sets of restricted S Function sequences:

\(S(n,S(T(0),3,2),1) >> f_{\varphi(1,0)}(n) = f_{\epsilon_0}(n)\)

\(S(n,S(T(0),5,1),1) >> f_{\varphi(2,0)}(n) = f_{\zeta_0}(n)\)

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\(S(n,S(T(0),3,2),1) >> f_{\psi(0)}(n) = f_{\epsilon_0}(n)\)

\(S(n,S(T(0),5,1),1) >> f_{\psi(\Omega)}(n) = f_{\zeta_0}(n)\)

\(S(n,T(1),1) >> f_{\varphi(\omega,0)}(n)\)

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Growth Rate up to \(S(n,T(m),1)\)
\(S(n,T(2),1) >> f_{\varphi(1,0,0)}(n) = f_{\Gamma_0}(n)\)

\(S(n,T(2),1) >> f_{\psi(\Omega^{\Omega})}(n) = f_{\Gamma_0}(n)\)

\(S(n,T(3),1) >> f_{\psi(\Omega^{\Omega^2})}(n) = f_{\varphi(1,0,0,0)}(n)\)

\(S(n,T(m),1) >> f_{\varphi(1,0_{[m]})}(n)\)

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Growth Rate up to \(S(n,T(T(0)),1)\)
\(S(n,T(T(0)),1) = S(n,T(n),1) >> f_{SVO}(n)\)

\(S(n,T(T(0)),1) = S(n,T(n),1) >> f_{\psi(\Omega^{\Omega^{\omega}})}(n) = f_{svo}(n)\)

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Growth Rate up to \(S(n,T^m(0),1)\)
The growth rate should be faster than \(f_{LVO}(n)\) and a comparable rate to some Ordinal Collapsing Functions. My initial estimates are based on information contained in this comparison table. Here are my results:

\(S(n,T(T(1)),1) >> f_{\psi(\Omega\uparrow\uparrow 3)}(n) = f_{LVO}(n)\) To be confirmed

\(S(n,T(T(T(0))),1) >> f_{\psi(\Omega\uparrow\uparrow\omega)}(n) =\) Bachmann-Howard ordinal To be confirmed

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Detailed estimate of S(n,T(T(1)),1) Growth Rates
These examples give a better explanation of my claim that S(n,T(T(1)),1) has a growth rate faster than \(f_{LVO}(n)\)

\(S(n,T(T(0)),1) = S(n,T(n),1) >> f_{svo}(n)\)

\(S(3,T(T(0)),1) = S(3,T(3),1) >> f_{svo}(3)\)

\(S(3,T(T(0)),2) = S(S(3,T(T(0)),1),T(3),1) >> f_{svo}^2(3)\)

\(S(3,T(T(0)),T(0)) = S(3,T(T(0)),3) = S(3,S(T(T(0)),0,1),1) >> f_{svo + 1}(3)\)

\(S(3,S(T(T(0)),0,T(0)),1) >> f_{svo + 3}(3) = f_{svo + \omega}(3)\)

\(S(3,S(T(T(0)),1,1),1) = S(3,S(T(T(0)),0,T(T(0))),1) >> f_{svo.2}(3)\)

\(S(3,S(T(T(0)),1,T(0)),1) >> f_{svo.2^{svo}}(3)\)

\(S(3,S(T(T(0)),2,1),1) = S(3,S(T(T(0)),1,T(T(0))),1) >>\) TBA

\(S(3,S(T(T(0)),2,T(0)),1) >>\) TBA

\(S(3,S(T(T(0)),T(0),1),1) = S(3,S(T(T(0)),2,T(T(0))),1) >> f_{svo\uparrow\uparrow svo}(3)\) To be confirmed

\(S(3,S(T(T(0)),S(T(0),0,1),1),1) >>\) TBA

\(S(3,S(T(T(0)),S(T(0),1,1),1),1) >>\) TBA

\(S(3,S(T(T(0)),T(1),1),1) = S(3,S(T(T(0)),S(T(0),T(0),1),1),1) >>\) TBA

\(S(3,S(T(T(0)),T(2),1),1) >>\) TBA

\(S(3,T(S(T(0),0,1)),1) = S(3,S(T(T(0)),T(T(0)),1),1) >>\) TBA

\(S(3,T(S(T(0),1,1)),1) = S(3,T(S(T(0),0,T(0))),1) >>\) TBA

\(S(3,T(S(T(0),2,1)),1) = S(3,T(S(T(0),1,T(0))),1) >>\) TBA

\(S(3,T(T(1)),1) = S(3,T(S(T(0),T(0),1)),1) >> f_{LVO}(3)\)

\(S(n,T(T(1)),1) >> f_{LVO}(n)\)

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