Ascending-E Notation

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Ascending-E Notation (E+ for short) is an alternative extension for Extended Hyper-E Notation created by the user SeveralLegend9998.

Definition

The latest update of this notation features a new term:

E[#] Structure: A delimiter that is in form E[B]a$a$a$...$a$a$a where a's can be anything, B is the base (typically a hyperion (#)) and $'s are delimiters.

And a few new symbols:

◆ is the unchanged remainder of the expression.
■ is (the rest of) some delimiter.
△ is some delimiter that is not decomposable. (This one isn't used for some reason.)
▲ is some delimiter that is decomposable.

Evaluate E◆n normally unless ◆ = [■]. In that case: E[■]n = ■^n, (E[■]■₂*#)[n] = (E[■]■₂)^n, and (E[■]▲)[n] = E[■](▲[n])

(E◆#{■}▲)[n] = (E◆#{■}(▲[n]))
((E[■₃]◆#{■}■₄)>(■₂*#))[n] = (E[(E[(E[...(E[(E[(E[■₃]◆#{■}■₄)>■₂]◆#{■}■₄)>■₂]◆#{■}■₄)>■₂...]◆#{■}■₄)>■₂]◆#{■}■₄)>■₂]◆#{■}■₄)>■₂ with n E's
((E[■₂]◆#{■}■₃)>▲)[n] = ((E[■₂]◆#{■}■₃)>(▲[n]))
(E◆#{■}(■₂*#))[n] = (E◆#{■}■₂)>(E◆#{■}■₂)>(E◆#{■}■₂)>...>(E◆#{■}■₂)>(E◆#{■}■₂)>(E◆#{■}■₂) with n (E◆#{■}■₂)'s

The 5 main rules (From xE^) still apply.

Examples

E3(E[#]#{#}#)3
= E3(E[#]#{#}#)[3]3
= E3(E[#]#{#}#[3])3
= E3(E[#]#{#}3)3
= E3(E[#](E[#]#{#}2))3
= E3(E[#](E[#](E[#]#{#}1)))3
= E3(E[#](E[#](E[#]#)))3
= E3(E[#](E[#](E[#]#)))[3]3
= ...
= E3(E[#](E[#](E[#]#[3])))3
= E3(E[#](E[#](E[#]3)))3
= E3(E[#](E[#]###))3
= E3(E[#](E[#]###))[3]3
= E3(E[#](E[#]###)[3])3
= ...
= E3(E[#](E[#]##)*(E[#]##)*(E[#]##))3
= ...

Growth Rate

The ordinal for the intended growth rate of this notation is \(\psi_0(\Omega_\omega)\) in the Fast-growing hierarchy according to the creator's calculations, which are shown below.

E[#]# ≈ \(\omega^\omega\)
E[#]## ≈ \(\omega^{\omega^2}\)
E[#](E[#]#) ≈ \(\omega^{\omega^\omega}\)
E[#]#{#}# ≈ \(\varepsilon_0\)
E[E[#]#{#}#]#{#}# ≈ \(\varepsilon_1\)
(E[#]#{#}#)># ≈ \(\varepsilon_\omega\)
(E[#]#{#}#)>(E[#]#{#}#) ≈ \(\varepsilon_{\varepsilon_0}\)
(E[#]#{#}##) ≈ \(\zeta_0\)
(E[#]#{#}###) ≈ \(\eta_0\)
(E[#]#{#}(E[#]#)) ≈ \(\varphi(\omega,0)\)
(E[#]#{#}#{#}#) ≈ \(\varphi(1,0,0)\)
(E[#]#{#}#{#}##) ≈ \(\varphi(1,1,0)\)
(E[#]#{#}#{#}#{#}#) ≈ \(\varphi(2,0,0)\)
(E[#]#{##}#) ≈ \(\varphi(\omega,0,0)\)
(E[#]#{##}#{#}#) ≈ \(\varphi(1,0,0,0)\)
(E[#]#{##}#{#}##) ≈ \(\varphi(1,0,1,0)\)
(E[#]#{##}#{#}#{#}#) ≈ \(\varphi(1,1,0,0)\)
(E[#]#{##}#{##}#) ≈ \(\varphi(1,\omega,0,0)\)
(E[#]#{##}#{##}#{#}#) ≈ \(\varphi(2,0,0,0)\)
(E[#]#{###}#) ≈ \(\varphi(\omega,0,0,0)\)
(E[#]#{###}#{#}#) ≈ \(\varphi(1,0,0,0,0)\)
(E[#]#{###}#{###}#{#}#) ≈ \(\varphi(2,0,0,0,0)\)
(E[#]#{####}#{#}#) ≈ \(\varphi(1,0,0,0,0,0)\)
E[#]#{E[#]#}# ≈ \(\psi_0(\Omega^{\Omega^\omega})\)
E[#]#{E[#]#}#{#}# ≈ \(\psi_0(\Omega^{\Omega^\Omega})\)
E[#]#{E[#]#}#{#}## ≈ \(\psi_0(\Omega^{\Omega^\Omega+1})\)
E[#]#{E[#]#}#{#}#{#}# ≈ \(\psi_0(\Omega^{\Omega^\Omega+\Omega})\)
E[#]#{E[#]#}#{##}# ≈ \(\psi_0(\Omega^{\Omega^\Omega+\Omega\omega})\)
E[#]#{E[#]#}#{##}#{#}# ≈ \(\psi_0(\Omega^{\Omega^\Omega+\Omega^2})\)
E[#]#{E[#]#}#{###}#{#}# ≈ \(\psi_0(\Omega^{\Omega^\Omega+\Omega^3})\)
E[#]#{E[#]#}#{E[#]#}#{#}# ≈ \(\psi_0(\Omega^{\Omega^\Omega2})\)
E[#]#{(E[#]#)*#}# ≈ \(\psi_0(\Omega^{\Omega^\Omega\omega})\)
E[#]#{(E[#]#)*#}#{#}# ≈ \(\psi_0(\Omega^{\Omega^{\Omega+1}})\)
E[#]#{(E[#]#)*##}#{#}# ≈ \(\psi_0(\Omega^{\Omega^{\Omega+2}})\)
E[#]#{(E[#]#)*(E[#]#)}#{#}# ≈ \(\psi_0(\Omega^{\Omega^{\Omega2}})\)
E[#]#{E[#]##}#{#}# ≈ \(\psi_0(\Omega^{\Omega^{\Omega^2}})\)
E[#]#{E[#]#{#}2}#{#}# ≈ \(\psi_0(\Omega^{\Omega^{\Omega^\Omega}})\)
E[#]#{E[#]#{#}3}#{#}# ≈ \(\psi_0(\Omega^{\Omega^{\Omega^{\Omega^\Omega}}})\)
E[#]#{E[#]#{#}#}# ≈ \(\psi_0(\Omega_2)\)
E[#]#{E[E[#]#{#}#]#{#}#}# ≈ \(\psi_0(\Omega_22)\)
E[#]#{(E[#]#{#}#)>#}# ≈ \(\psi_0(\Omega_2\omega)\)
E[#]#{E[#]#{#}##}# ≈ \(\psi_0(\Omega_2^2)\)
E[#]#{E[#]#{#}(E[#]#)}# ≈ \(\psi_0(\Omega_2^\omega)\)
E[#]#{E[#]#{#}#{#}#}# ≈ \(\psi_0(\Omega_2^{\Omega_2})\)
E[#]#{E[#]#{##}#}# ≈ \(\psi_0(\Omega_2^{\Omega_2\omega})\)
E[#]#{E[#]#{E[#]#}#}# ≈ \(\psi_0(\Omega_2^{\Omega_2^\omega})\)
E[#]#{E[#]#{E[#]#{#}#}#}# ≈ \(\psi_0(\Omega_3)\)
E[#]#{E[#]#{E[#]#{E[#]#{#}#}#}#}# ≈ \(\psi_0(\Omega_4)\)