As we discussed the definition of the previous sneak peak, we can now write the formal rules of the collapsing-E notation!
Collapsing-E notation
(&E)
Let E a1 %(1) a2 %(2) a3 %(3) ... a[n-1] %(n-1) an
where a1 ~ an, and n are natural numbers (positive integers)
and %(1) ~ %(k-1) are k-1 delimiters from the set x^
and k ∈ {1, 2, 3, 4, 5, 6, ...} be defined as follows: Let
m denote ak-1
n denote ak
@ denote the unchanged remainder of an expression
% denote any portion of a delimiter we chose to omit
*formerly using & since the symbol & is now used as the hyper-delimiter in the curly brackets {} to
diagonalize over uncountable limits of the delimiter structures inside {}.
and %[n] denote the nth member of the fundamental sequence defined for the delimiter %
Formal rules
Rule 1. Base rule.
For k = 1 (with only one argument and no hyperions), we have E[x]n = x^n
Rule 2. Decomposition rule.
For L%(n-1) ≠ #^n (the last cascader is not in the form of #^n):
E[x]@a%b = E[x]@a%[b]a (@ indicates the unchanged remainder of the expression and %[b] is the fundamental sequence of %)
Rule 3. Termination rule.
For L%(n-1) = #^n (the last cascader is in the form of #^n), and the last argument is 1, it can be removed:
E[x]@a%1 = E[x]@a
Rule 4. Expansion rule.
For L%(n-1) = #^n and %k ≠ # (the last cascader is in the form of #^n but not the single hyperion):
E[x]@a%*#b = E[x]@a%a%*#(b-1)
Rule 5. Expansion rule.
Otherwise:
E[x]@a#b = E[x]@(E[x]@a#(b-1))
Decomposition rules
In addition the set of legal delimiters must be defined. Let & be the set of legal delimiters in xE^. The set is defined recursively:
We regard all elements of % in {n} as "transfinite n" (including countable and uncountable delimiters), for % ≥ #.
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){n}(b) for "n ≥ 1 or transfinite n" is an element of %
IV. If a,b are elements of % and c is an element of %+, then (a){n}(b)>(c) for "n ≥ 1 or transfinite n" 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(%) ≠ #^n.
Also, the decompositions of decomposable hyper-delimiters using "&" must be defined. A delimiter containing "&", %, is also decomposable (% is a member of %hdecomp, also known as an alternative of %decomp), if and only if L(%) ≠ #^n in {}.
The decompositions are defined as follows:
Case I. L = a^b, where a, b ∈ %:
A. When b = #:
I.A.1. %(a)^#[1] = %a
I.A.2. %(a)^#[n] = %a*(a)^#[n-1]
B. When b = k*#:
I.B.1. %(a)^(k*#)[1] = %(a)^(k)
I.B.2. %(a)^(k*#)[n] = %(a)^(k)*(a)^(k*#)[n-1]
C. When b ∈ %decomp:
%(a)^(b)[n] = %(a)^(b[n])
Case II. L = a{p}b, where a, b ∈ %, and (p > 1 or 0 < p < # in m+p, and m ≥ #):
(p is copies of multiple carets or a successor ordinal)
A. When b = #:
II.A.1. %(a){p}#[1] = %a
II.A.2. %(a){p}#[n] = %(a){p-1}((a){p}#[n-1])
B. When b = k*#:
II.B.1. %(a){p}(k*#)[1] = %a
II.B.2. %(a){p}(k*#)[n] = %(a){p-1}(k)>((a){p}(k*#)[n-1])
C. When b ∈ %decomp:
%(a){p}(b)[n] = %(a){p}(b[n])
Case III. L = a{p}b>c, where a, b ∈ %, c ∈ %+, and (p > 1 or 0 < p < # in m+p, and m ≥ #):
(p is copies of multiple carets or a successor ordinal)
A. When c = #:
III.A.1. %(a){p}(b)>#[1] = %(a){p}(b)
III.A.2. %(a){p}(b)>#[n] = %((a){p}(b)>#[n-1]){p}(b)
B. When c = k+#:
III.B.1. %(a){p}(b)>(k+#)[1] = %((a){p}(b)>(k)){p}(b)
III.B.2. %(a){p}(b)>#[n] = %((a){p}(b)>(k+#)[n-1]){p}(b)
C. When c ∈ %decomp:
%(a){p}(b)>(c)[n] = %(a){p}(b)>(c[n])
D. When c = k+d where k ∈ &+ and d ∈ %decomp:
%(a){p}(b)>(k+d)[n] = %(a){p}(b)>(k+d[n])
E. When c = d*# where d ∈ %:
III.E.1. %(a){p}(b)>(d*#)[1] = %(a){p}(b)>(d)
III.E.2. %(a){p}(b)>(d*#)[n] = %(a){p}(b)>(d+d*#)[n-1]
F. When c = k+d*# where k ∈ %+ and d ∈ %:
III.F.1. %(a){p}(b)>(k+d*#)[1] = %(a){p}(b)>(k+d)
III.F.2. %(a){p}(b)>(k+d*#)[n] = %(a){p}(b)>(k+d+d*#)[n-1]
Case IV. L = a{p}b, where L(p) ≠ m:
(L denotes the last sum of delimiters in {}, and m denotes the natural number)
A. When p = #:
&(a){#}#[n] = &(a)^^^^...^^^^# with n ^'s
B. When p = k+#:
&(a){k+#}#[n] = &(a){k+n}#
C. When p ∈ %decomp:
&(a){p}#[n] = &(a){p[n]}#
D. When p = k+c, k ∈ %+, c ∈ %decomp:
&(a){k+c}#[n] = &(a){k+c[n]}#
E. When p = c*# where c ∈ %+:
IV.E.1. &(a){c*#}#[1] = &(a){c}#
IV.E.2. &(a){c*#}#[n] = &(a){c+c*#}#[n-1]
F. When p = k+c*#, k ∈ %+, c ∈ %:
IV.E.1. &(a){k+c*#}#[1] = &(a){c}#
IV.E.2. &(a){k+c*#}#[n] = &(a){k+c+c*#}#[n-1]
Natural language equivalent of the formal rules:
The formal rules of the notation are very similar to those of Extended Cascading-E and Hyper-Extended Cascading-E notation extensions, except they are just the rewritten versions of the xE^ and #xE^ rules.
Case I. If L is an exponent operator (^), where a, b belong to previous member of %:
When the expression ends with copies of #: Copy the previous delimiter using hyper-product and one hyperion mark less after #. If the latter of the delimiter is not copies of #, or the last hyper-product is not copies of #, consult the rules for decomposing intermediate delimiter structures.
Case II. If L is in the form of a{p}b, where a, b ∈ %, and (p is natural number greater or equal to 2, or p is the successor ordinal):
Let the hyper operator iterate recursively via the up-arrow notation rules. If there are more than one hyperion after carets, or the last hyper-product is in the form of copies of hyperions, decompose the delimiter structure via the caret-tops, and with the identical delimiter structures. If the latter of the delimiter is not copies of #, consult the rules for decomposing intermediate delimiter structures.
Case III. If L in in the form of a{p}b>c, where a, b ∈ %, c ∈ %+, and (p is natural number greater or equal to 2, or p is the successor ordinal):
For each delimiter structures based on the hyper-operators from the least tetrational operator (^^#), with caret-tops, let it decompose into copies of hyperoperators from left to right. If there are plus signs followed by a single hyperion mark, let it decompose in the similar fashion, by removing the plus sign and a hyperion mark. If there are two or more consecutive hyperion marks, or the last hyper-product is in the form of copies of hyperions, use the hyperion-addition rule. If the latter of the delimiter is not copies of #, consult the rules for decomposing intermediate delimiter structures.
Case IV. If p in a{p}b is not a natural number or the successor ordinal
Consult the rule for decomposing delimiter structures inside curly braces, in a similar fashion to the rules on caret-tops.
With the new rules introduced in the Collapsing-E notation, the rules are as follows:
Rule I. When c = &:
I1. %a{&}#[1] = %a
I2. %a{&}#[2] = %a{a}#
I3. %a{&}#[n] = %a{a{&}#[n-1]}# for n > 2
Rule II. When c = d+&
II1. %a{d+&}#[1] = %a{d}#
II2. %a{d+&}#[n] = %a{d+a{d+&}#[n-1]}#
Rule III. When c = d*& where d ∈ %+
III1. %a{d*&}#[1] = %a{d}#
III2. %a{d*&}#[n] = %a{d*a{d*&}#[n-1]}#
Rule IV. When c = k+d*&, k ∈ %+, d ∈ %:
IV1. %a{k+d*&}#[1] = %a{k+d}#
IV2. %a{k+d*&}#[n] = %a{k+d*a{k+d*&}#[n-1]}#
Rule V. When c = d^& where d ∈ %+:
V1. %a{d^&}#[1] = %a{d}#
V2. %a{d^&}#[n] = %a{d^a{d^&}#[n-1]}#
Rule VI. When c = k+d^&, k ∈ %+, d ∈ %:
VI1. %a{k+d^&}#[1] = %a{k+d}#
VI2. %a{k+d^&}#[n] = %a{k+d^a{k+d^&}#[n-1]}#
Rule VII.
When d^& where d ∈ %+, and the exponentiation rules of &, based off the Cascading-E notation rules, are applicable, consult the rules for case I in the decomposition rule:
VII1. %a{k*d}#[n] = %a{k*d[n]}#
VII2. %a{k^d}#[n] = %a{k^d[n]}#
VII3. %a{&^^#}#[n] = %a{&^^n}# = %a{&^&^&^...^&^&^&}# with n &'s
In a natural language equivalent:
I. For expressions with the delimiter structures containing just "&" delimiter inside curly braces {}, Repeat the left-hand side delimiter structure by nesting it inside curly braces recursively.
II. For expressions with the last hyper-operator-sum containing "&", let the left-hand side delimiter structure substitute recursively in place of &.
III. For ampersand-product and ampersand-exponent, consult the similar rules as the ampersand-sum decomposition rule.
Remember, we can continue by extrapolating a pattern observed with the existing separators within a new "hyper-product" paradigm, involving hyper-product between countable delimiters (#) and uncountable hyper-delimiters (&). Observe that every succeeding countable delimiter is multiplying the previous delimiter by the uncountable one, meaning that it can be written without "*" if the "&" delimiter is immediately followed by the "#" delimiter. For example:
&*# = &#
&*## = &##
&*#^# = &#^#
&*#^^# = &#^^#
&*#{&}# = &#{&}#
&&*# = &&#
&&*#{&&}# = &&#{&&}#
&&&*# = &&&#
&&&&*# = &&&&#
etc.
On the other hand:
&^&*# ≠ &^&#
&^&*#{&}# ≠ &^&#{&}#
&^&^&*#{&^&^&}# ≠ &^(&^&*#{&^&^&}#) ≠ &^&^&#{&^&^&}#
And finally, these are formal definitions of the Collapsing-E notation! The definition of the notation extension is very lengthy, but these are really nice, requiring 5 basic rules and 25 additional decomposition rules!
In that case, this notation is really neat for constructing larger numbers, as we now have a fully functional system that eventually reach:
the "Bachmann-Howard ordinal" (BHO)!!!
Proceed to &E numbers
Now that we have defined the Collapsing-E notation (&E), before inventing some new googolisms, I am not done yet. Before proceeding to begin, let's peek at the fourth sneak peak for something extraordinary ...