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.

• 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:
$$E@a\#b = E@(E@a\#(b-1))$$

In addition the set of legal delimiters must be defined. Let & be the set of legal delimiters in xE^. The set is defined recursively:

• 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)^(#) = &α
• IA2. &(α)^(#)[n] = &α*(α)^(#)[n-1]
• B. When β = ρ*# :
• IB1. &(α)^(ρ*#) = &(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..^(#) = &α
• IIA2. &(α)^..k..^(#)[n] = &(α)^..k-1..^((α)^..k..^(#)[n-1])
• B. When β = ρ*#:
• IIB1. &(α)^..k..^(ρ*#) = &(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)>(#) = &(α)^..k..^(β)
• IIIA2. &(α)^..k..^(β)>(#)[n] = &((α)^..k..^(β)>(#)[n-1])^..k..^(β)
• B. When γ=ρ+#:
• IIIB1. &(α)^..k..^(β)>(ρ+#) = &((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*#) = (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*#) = (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 (#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*#}# = &(α){δ}#
• V2. &(α){δ*#}#[n] = &(α){δ+δ*#}#[n-1]
• Rule VI. When γ = ρ+δ*#, ρ ∈ &+, δ ∈ &:
• VI1. &(α){ρ+δ*#}# = &(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^.