Bashicu Matrix System

**Bashicu matrix system** (BMS) is a number notation that is notable for its extremely unrivaled and overpowered strength as well as introducing the concept of difference a completely new family of BM-like hydra and sequence notations. BMS has been proven to be well-founded and terminating by Yto.

## Definition

See https://github.com/kyodaisuu/basmat/blob/master/basic/BM4.bas for the BASIC program used to define it.

#### The following is a informal example expansion of BM4, the definitions of other versions may be hard to find

Example of expansion: (0,0,0,0)(1,1,1,1)(2,2,1,1)(3,3,1,1)(4,2,0,0)(5,1,1,1)(6,2,1,1)(7,3,1,1)(0,0,0,0)[n]

The last column is consists of 0, we may now delete it:

(0,0,0,0)(1,1,1,1)(2,2,1,1)(3,3,1,1)(4,2,0,0)(5,1,1,1)(6,2,1,1)(7,3,1,1)

This marks an "expansion "completed. Each operation, increment the [row] by 1 and move on

We repeat, because we do not have the situation as before (where the last term of each row was a 0) we do as follows: Start with the top term in the final column. Now we find the family of that term. Search for the first term smaller, than smaller than that, then smaller than that. We call this the Bad Root. The Good Root can be the start of the sequence (The concept of the Good Root is not used in BMS). In our example it goes 7, 6, 5, 4, 3, 2, 1, 0:

(**0**,0,0,0)(**1**,1,1,1)(**2**,2,1,1)(**3**,3,1,1)(**4**,2,0,0)(**5**,1,1,1)(**6**,2,1,1)(**7**,3,1,1)

Continue searching down the rows of the matrix. If there is any term that is not a Bad Root of the other term, we cross out the entire column that the term is in from that term and down. We never search a crossed term:

(**0**,**0**,0,0)(**1**,~~1~~,~~1~~,~~1~~)(**2**,~~2~~,~~1~~,~~1~~)(**3**,~~3~~,~~1~~,~~1~~)(**4**,~~2~~,~~0~~,~~0~~)(**5**,**1**,1,1)(**6**,**2**,1,1)(**7**,**3**,1,1)

(**0**,**0**,**0**,0)(**1**,~~1~~,~~1~~,~~1~~)(**2**,~~2~~,~~1~~,~~1~~)(**3**,~~3~~,~~1~~,~~1~~)(**4**,~~2~~,~~0~~,~~0~~)(**5**,**1**,~~1~~,~~1~~)(**6**,**2**,~~1~~,~~1~~)(**7**,**3**,**1**,1)

(**0**,**0**,**0**,**0**)(**1**,~~1~~,~~1~~,~~1~~)(**2**,~~2~~,~~1~~,~~1~~)(**3**,~~3~~,~~1~~,~~1~~)(**4**,~~2~~,~~0~~,~~0~~)(**5**,**1**,~~1~~,~~1~~)(**6**,**2**,~~1~~,~~1~~)(**7**,**3**,**1**,**1**)

The Bad Root is rightmost column of marked terms. Δ is 0 if it is the last row that isn't ending in 0 or if a row ends in 0. If not, it's last Bad Root. So in our case: Δ = 7,3,1,0. We use this to add to the terms with something called an Ascension Matrix. To get the Ascension Matrix, we repeat the family finding algorithm but starting at the second to last term. If a term cant find its Bad Root, mark the column as non-ascending. The first row is never marked and the last row is always marked.

Then copy over the Bad Part by the [n] increment variable and add to the terms by increasing amounts of Δ. Do not ascend terms marked off as non-ascending in the ascension matrix:

(0,0,0,0)(1,1,1,1)(2,2,1,1)(3,3,1,1)(4,2,0,0)(5,1,1,1)(6,2,1,1)(7,3,1,0)(8,4,2,1)... and this keeps going depending on the current state of [n]

Note that in other contexts, the Bad Root can be called the Parent while the term is called the Child.

## Analysis of Growth Rate

Note that the version of BMS used for this section is BM4.

The growth rate of 1-row matrices (aka the primitive sequence system) is

f

ε

0

(
n
)

{\displaystyle f\_{\varepsilon \_{0}}(n)}

 and the growth rate of 2-row matrices (the pair sequence system) is

f

ψ

0

(

Ω

ω

)

(
n
)

{\displaystyle f\_{\psi \_{0}(\Omega \_{\omega })}(n)}

 with respect to Buchholz's function in BM4.

A complete analysis of the growth rate of matrices with 3 (trio sequence system/TSS) or more rows is on going and has been proven to be excruciatingly difficult due to TSS's near unrivaled strength. People are analyzing the system to some extent.[1][2] Unfortunately, many of the analyses are done with ill-defined and semi-well-defined notations, and aren't the most concrete as they lack proof and evidence of correctness, and some have been proven to be incorrect.

All explicitly written analysis past (0,0,0)(1,1,1) are only lists of correspondences that may lack a reasoning of the surjectivity. All explicitly written analysis past the matrix are only lists of correspondences that lack a reasoning of the surjectivity, although the following correspondences are strongly believed to be concrete:

ψ

Ω

(

ψ

I

(
0
)
)

{\displaystyle \psi \_{\Omega }(\psi \_{I}(0))}

 (0,0,0)(1,1,1)(2,1,1)(3,1,0)(4,2,0)

ψ

Ω

(

ε

I
1

)

{\displaystyle \psi \_{\Omega }(\varepsilon \_{I+1})}

in Hypcos's OCF,

P
T
O
(
K
P
i
)

{\displaystyle PTO(KPi)}
 (0,0,0)(1,1,1)(2,1,1)(3,1,1)

ψ

Ω

(

I

ω

)

{\displaystyle \psi \_{\Omega }(I\_{\omega })}

 (0,0,0)(1,1,1)(2,2,0)

α

{\displaystyle \alpha }

 that is

Π

n

−
r
e
f
l

{\displaystyle \Pi \_{n}-refl}

 for all n,

P
T
O
(
K
P

Π

n
<
ω

−
r
e
f
)

{\displaystyle PTO(KP+\Pi \_{n<\omega }-ref)}

, {1{1',,2}2} in SAN

Bashicu coined the following ordinals corresponding to several standard Bashicu matrices with respect to BM4.

α

{\displaystyle \alpha }

 that is

Π

n

−
r
e
f
l

{\displaystyle \Pi \_{n}-refl}

 (0,0,0)(1,1,1)(2,2,1)(3,3,0)

α

{\displaystyle \alpha }

 that is

d
o
u
b
l
y

(
1
)
−
s
t
b

{\displaystyle doubly\ (+1)-stb}
 (0,0,0)(1,1,1)(2,2,2)

α

{\displaystyle \alpha }

 that is

ω
−
p
l
y
−
s
t
b

{\displaystyle \omega -ply-stb}

The ordinal in FGH approximating the growth of an n-row matrix (0,0....,0)(1,1...,1) which corresponds to the limit of BMS is not yet known but it is strongly believed to correspond with Y(1,3) in Y-sequence and it is well-ordered given BMS's proof of termination.

## Sources

1. ↑ 1.0 1.1 https://arxiv.org/abs/2307.04606
2. ↑ A blog post introducing primitive sequence system
3. ↑ [:ja:ユーザーブログ:BashicuHyudora/バシク行列の解析 Analysis of trio sequence system by Bashicu] with his own ordinal notations
4. ↑ [:ja:ユーザーブログ:BashicuHyudora/トリオ数列の解析 Bashicu's calculation in more detail]
5. ↑ [:ja:ユーザーブログ:KurohaKafka/バシク行列の解析 by Kuroha Analysis by KurohaKafka]
6. ↑ Analysis by Bubby3
7. ↑ Analysis by Googleaarex
8. ↑ https://docs.google.com/spreadsheets/d/1fN6hFcOil-AqM2ZMeekawa0dvGh-XX\_zpPEwMv3HPoY/edit#gid=0
9. ↑ https://docs.google.com/spreadsheets/d/1oZuXL7Ob0miy3YXrzA9Og63DirhDPEV55cYsY9rjar0/edit#gid=0
10. ↑ https://docs.google.com/spreadsheets/d/1Iw4-8HOmru6SpWVJSGv-vjAd0bjQzHY1IcLKw7CzuQo/edit#gid=1838341102
11. ↑ https://docs.google.com/spreadsheets/d/1aBjyJdl6WK21drtu7eDBkfgwCyHpp4ciXq2FxPqvT8U/edit#gid=0
12. ↑ Bashicu, バシク行列の解析, Japanese Googology Wiki user blog, retrieved at 02/07/2020.