Extended Cascading-E Notation

Extended Cascading-E Notation (xE^) is a notation by Sbiis Saibian, and the most recent component of the Extensible-E System. It extends upon Cascading-E notation by applying up-arrow notation to hyperions (#). The limit of its growth rate is comparable to \(f_{\varphi(\omega,0,0)}(n)\).

The Extensible-E System includes xE^ and all extensions to come.

Definition
Let Ea&a2& ... &an be any expression in xE^, where a1 through an are n positive integers, and all & are hyper-products (which may or may not be distinct.) Each individual & may be chosen from the set of legal separators.

Below are the 5 formal rules of xE^. Let \(\&_k\) be the kth hyper-product and \(L(\&_k)\) be the last cascader of the kth hyper-product.

In addition the set of legal delimiters must be defined. Let & be the set of legal delimiters in xE^. The set is defined recursively:
 * Rule 1. Base Rule. With no hyperions, we have \(Ea = 10^a\).
 * Rule 2. Decomposition Rule. If \(L(\&_{n-1}) \neq \#^n\) (the last cascader is not of the form \(\#^{n}\)):
 * \(E@a\text{&}b = E@a\text{&}[b]a\) (@ indicates the unchanged remainder of the expression and &[b] is the fundamental sequence of &)
 * Rule 3. Termination Rule. If the last argument is 1, it can be removed: \(E@a\text{&}1 = E@a\)
 * Rule 4. Expansion Rule. \(L(\&_{n-1}) = \#^n\) and \(\&_k \neq \#\):
 * \(E@a\text{&}*\#b = E@a\text{&}a\text{&}*\#b-1\).
 * Rule 5. Recursion Rule. Otherwise:
 * \(E@a\#b = E@(E@a\#(b-1))\)
 * I. # is an element of &
 * II. If a,b are elements of & then a*b is an element of &
 * III. If a,b are elements of & then (a)^^^...^^^^(b) w/n ^s is an element of &
 * IV. If a,b are elements of & and c is an element of &+, then (a)^^^...^^^(b)>(c) w/n ^s is an element of & for n>1.
 * V. If a is an element of & then a is an element of &+
 * VI. If a,b are elements of &+ then a+b is an element of &+

Lastly the decompositions of decomposable-delimiters must be defined. A delimiter, &, is decomposable (& is a member of &decomp), if and only if \(L(\text{&}) \neq \#^n\).

The decompositions are defined as follows:
 * Case I. L= (α)^(β) where α,β ∈ &
 * A. When β = # :
 * IA1. &(α)^(#)[1] = &α
 * IA2. &(α)^(#)[n] = &α*(α)^(#)[n-1]
 * B. When β = ρ*# :
 * IB1. &(α)^(ρ*#)[1] = &(α)^(ρ)
 * IB2. &(α)^(ρ*#)[n] = &(α)^(ρ)*(α)^(ρ*#)[n-1]
 * C. When β ∈ &decomp:
 * IC1. &(α)^(β)[n] = &(α)^(β[n])


 * Case II. L= (α)^..k..^(β) where α,β ∈ & and k>1
 * A. When β = #:
 * IIA1. &(α)^..k..^(#)[1] = &α
 * IIA2. &(α)^..k..^(#)[n] = &(α)^..k-1..^((α)^..k..^(#)[n-1])
 * B. When β = ρ*#:
 * IIB1. &(α)^..k..^(ρ*#)[1] = &(α)^..k..^(ρ)
 * IIB2. &(α)^..k..^(ρ*#)[n] = &(α)^..k..^(ρ)>(α^..k..^(ρ*#)[n-1])
 * C. When β ∈ &decomp:
 * IIC1. &(α)^..k..^(β)[n] = &(α)^..k..^(β[n])


 * Case III. L= (α)^..k..^(β)>(γ) where α,β ∈& ,γ ∈ &+, and k>1
 * A. When γ = #:
 * IIIA1. &(α)^..k..^(β)>(#)[1] = &(α)^..k..^(β)
 * IIIA2. &(α)^..k..^(β)>(#)[n] = &((α)^..k..^(β)>(#)[n-1])^..k..^(β)
 * B. When γ=ρ+#:
 * IIIB1. &(α)^..k..^(β)>(ρ+#)[1] = &((α)^..k..^(β)>(ρ))^..k..^(β)
 * IIIB2. &(α)^..k..^(β)>(ρ+#)[n] = &((α)^..k..^(β)>(ρ+#)[n-1])^..k..^(β)
 * C. When γ ∈ &decomp:
 * IIIC1. (α)^..k..^(β)>(γ)[n] = (α)^..k..^(β)>(γ[n])
 * D. When γ=ρ+δ where ρ ∈ &+ and δ ∈ &decomp:
 * IIID1. (α)^..k..^(β)>(ρ+δ)[n] = (α)^..k..^(β)>(ρ+(δ[n]))
 * E. When γ=δ*# where δ ∈ &:
 * IIIE1. (α)^..k..^(β)>(δ*#)[1] = (α)^..k..^(β)>(δ)
 * IIIE2. (α)^..k..^(β)>(δ*#)[n] = (α)^..k..^(β)>(δ+δ*#)[n-1]
 * F. When γ =ρ+δ*# where ρ ∈ &+ and δ ∈ &:
 * IIIF1. (α)^..k..^(β)>(ρ+δ*#)[1] = (α)^..k..^(β)>(ρ+δ)
 * IIIF2. (α)^..k..^(β)>(ρ+δ*#)[n] = (α)^..k..^(β)>(ρ+δ+δ*#)[n-1]

Hyper-Extended Cascading-E Notation
Hyper-Extended Cascading-E Notation (#xE^) is an extension of xE^ that goes up to \(f_{\varphi(1,0,0,0)}(n)\), and is currently the last component of ExE that is fully formalized. The rules are as follows:


 * Rule I. When γ = #: &(α){#}#[n] = &(α)^..n..^#
 * Rule II. When γ = ρ+#: &(α){ρ+#}#[n] = &(α){ρ+n}#
 * Rule III. When γ ∈ &decomp: &(α){γ}#[n] = &(α){γ[n]}#
 * Rule IV. When γ = ρ+δ, ρ ∈ &+, δ ∈ &decomp: &(α){ρ+δ}#[n] = &(α){ρ+(δ[n])}#
 * Rule V. When γ = δ*# where δ ∈ &+:
 * V1. &(α){δ*#}#[1] = &(α){δ}#
 * V2. &(α){δ*#}#[n] = &(α){δ+δ*#}#[n-1]
 * Rule VI. When γ = ρ+δ*#, ρ ∈ &+, δ ∈ &:
 * VI1. &(α){ρ+δ*#}#[1] = &(α){ρ+δ}#
 * VI2. &(α){ρ+δ*#}#[n] = &(α){ρ+δ+δ*#}#[n-1]
 * Rule VII. When γ is a successor ordinal, consult the rules for xE^.

Further extensions


After #xE^ has been concluded, new non-fully-formalized extensions are created:


 * In Hyper-Hyper-Extended Cascading-E Notation (##xE^), hyperions can form arrays. In particular, {#,#,1,2} is defined as the limit of #, #{#}#, #{#{#}#}#, .... It continues to {#,#+1,1,2}, the limit of iterations of {X,#,1,2} on #. Then however, he tells that iterating {X,#+1,1,2} cannot give {#,#+2,1,2} in his theory of the climbing method, and goes on to devise an ad-hoc continuation. It starts with &(1), defined as the limit of iterating {X,#+1,1,2} on #, then defines &(2) as the limit of iterating {X,#+1,1,2} on &(1). It continues &(#), &(#^#), &(#{#}#), &(&(#)). Then asterisks are introduced that combine with operators, like #*^^#. Then * is introduced to generalize the idea of asterisk operators.
 * In Solidus-Extended Cascading-E Notation (/xE^), a new symbol is introduced, the solidus (/). It is the next symbol after the asterisk. The next symbol after the solidus is the x, which is then generalized. Then x's can perform exponents, tetrations, pentations, etc. Lastly, multiple levels of / are introduced, such as /2.
 * Sbiis Saibian also came up with some googolisms that currently do not have full definitions, such as General Gogulus = E100{#,#,#,#}100, for other future extensions to his notation. Those extensions will apply array notation to hyperions, in contrast with /xE^ and the second half of ##xE^, which are ad-hoc extensions to the notation. Lastly, there are two names for numbers that don't even have expressions yet.