Previous: 4.5.1 Linear Threes Notation (L3N)
Threes notation reaches a new level with the multi-comma delimiter, allowing deeper recursive expansion-
Base rules:
Basic Multi-commas:
T[a,,2] expands to T[3,3,3,...,3,3,3] (With a-1 occurences of 3.)
Equivalent representations:
T[a,,2] = {3,a(1)2} BEAF = {3,a[2]2} BAN.
Basic Application of L3N rules
Example: T[3,2,,2] = {3,3,2(1)2} = {3,{3,2,2(1)2}(1)2} = {3,{3,3(1)2}(1)2} = {3,{3,3,3}(1)2} = {3,3^^^3(1)2} = {3,tritri(1)2}.
Another Example: T[4,2,,2] = {3,4,2(1)2} = {3,{3,3,2(1)2}(1)2}
A small little explanation...
As a little convention here (to simplify the ruleset definitions) let any a,b,c,...,x,y,z array with more than 0 entries and can also have multi-commas be #.
Extended Threes Notation Deleter Case
T[#(,n)1] = T[#] where (,n) denotes any number of commas.
Multi-Level Recursive Reduction
T[a,,n] = T[3,3,...,3,3,,n-1] with a-1 3s involved.
Example decompositions:
T[3,,2] = T[3,3]
T[4,,2] = T[3,3,3]
T[3,,3] = T[3,3,,2]
T[3,,4] = T[3,3,,3]
Divergence from BEAF/BAN
Extended Threes introduces recursive behaviors beyond conventional BEAF/BAN notation
Deep Recursion Cases:
Decomposition Case: T[n,,1,2] = T[3,,T[n-1,,1,2]]
Special Case: T[1#] = 3 also applies so...
Self-Referential Growth
Example:
T[3,,1,2] = T[3,,T[2,,1,2]] = T[3,,T[3,,3]]
T[3,3,,1,2] = T[T[3,2,,1,2],2,,1,2]
Degenerative case: T[1#] = 3
Deleting case: T[#(,n)1] = T[#]
Reduction case: T[a,,1,c] = T[3,,T[a-1,,1,c],c-1]
4. Chaotic case: T[a,,b,c] = T[3,3,3,...,3,3,3,,b-1,c] with a-1 3s
5. Decomposition case: in T[L(,n)#] where L is a linear array (e.g. 4,3,9 for example) and # is any linear or multi comma array.
Rules:
Degenerative case: T[1#] = 3
Deleting case: T[#1] = T[#]
Chaotic Reduction case: T[p,,0,1,...,1,a,L] = T[3,,p,3,...,3,a-1,L]
Chaotic case: T[p,,a,b,c,...,x,y,z] expands to T[3,3,3,...,3,3,3,,a-1,b,c,...,x,y,z] w/ p-1 3's.
Decomposition case: T[L,,#] expands to T[expand(L),,#] (use L3N rules)
This is enough to formalize the notation to w^w2 level, but not enough to reach the end of planar arrays...
Base Case:
Trilinear notation follows the recursive multi-comma delimiter system.
Zero-Based Expansion Rule: T[a,,0,,1,,...,,1,,2] = T[a,,a,,a,,...,,a,,1] (Then apply the Deleter case to reduce one entry.)
Correction for Invalid Arrays: T[a,,1,,1,,...,,1,,2] = T[a,2,,0,,1,,...,,1,,2] (This is theoretically invalid, but included for clarity.)
Decomposition Case:
The deletion rule persists: T[#(,n)1] = T[#] (Where (,n) denotes any number of commas.)
Recursive Strength:
Extends decomposition from bilinear cases into nested transformations: T[a,,n,,m] = T[3,3,...,3,,n-1,,m-1] (With a−1 occurrences of 3.)
Self-Referential Growth
Examples:
Trilinear recursion: T[3,,1,2,,2] = T[3,,T[2,,1,2,,2]] = T[3,,T[3,,3,,2]]
Apeirotly nested trilinear recursion: T[3,3,,1,2,,2] = T[T[3,2,,1,2,,2],2,,1,2,,2]
Linearly nested trilinear recursion: T[3,,2,2,,2]
Use more structures similarly
Reduction Cases
Degenerative case: T[1#] = 3
Deleting case: T[#(,n)1] = T[#]
Chaotic Reduction: T[a,,1,c,,2] = T[3,,T[a-1,,1,c,,2],c-1,,2]
Extended decomposition: T[L,,#,,2] = T[expand(L),,#,,2] (Using L3N rules.)
Growth Rate Comparison:
Previous notation reached w^(w2)level.
3L3N extends to w^(w3) and a system to go beyond.
Deep Recursion Patterns:
Recursive self-replication enables structures equivalent to tetrated transfinite exponentiation.
Notation supports higher-plane arrays without being restricted by previous decomposition methods.
Trilinear Threes Notation (3L3N) solidifies hierarchical recursion while ensuring deep nesting patterns continue beyond bilinear constructs. This notation bridges between extended transfinite recursion and higher-order decomposition methods applicable in large-scale ordinal growth modeling.
All of this is still part of ex3N!!
As we wrap up, let's refine and categorize some currently unformalized expressions, ensuring a precise recursive structure within Trilinear Threes Notation.
Core Structural Expansions
Recursive Nesting and Pattern Alignment
We can have T[3,,3...3,,3] w/ p-1 a's as T[p,,,2]
Basic Transformations and Equivalency Mapping
T[3,3,,,2] = T[3,,,3] = T[2,,,,2]
T[3,3,3,,,2] = T[4,,,3]
Pattern-Based Accelerated Enumeration
Given the increasingly complex growth, let's optimize by leveraging transfinite shorthand representations to generalize:
Recursive Multi-Level Expansion
T[3,,,3,3], T[3,,,3,,3]
T[3,,,3,3,3,3], T[3,,,3,,,3]
T[3,,,3,,3,,3,3], T[3,,,3,,,3,,,3]
Chaotic and Degenerative Transformations
T[3,,,,3] = T[3,,,3,,,3] = T[3,,,3,,3,3,3] = [too chaotic]
With the refined notation, let's systematize ordinal classification based on recursive depth and transfinite exponentiation. This provides a framework for mapping large transfinite constructs, pushing beyond previous hierarchical growth limits.