Bashicu matrix system

Bashicu matrix system is a large number notation, invented by BashicuHyudora in 2014, that has not currently been proven to be well-founded. It is an extension of primitive sequence system and pair sequence system. 1-row matrices reach \(\varepsilon_0\), and 2-row \(\psi_0(\Omega_\omega\) with respect to Buchholz's function, in the fast-growing hierarchy. We do not currently know how strong 3-row matrices are (an analysis that said that \(\Big({}^{{}^0_0}_{{}^0_0}\;{}^{{}^1_1}_{{}^1_1}\Big)\) corresponded to the collapse of the least gap ordinal was shown to be incorrect.) It is widely believed that it is well-founded, but there are no proofs. The matrices can also be used to represent ordinals using fundamental sequences.

Definition
The definition given here is the one of BM4, the latest version. A matrix \(S\) is any array of non-negative integers, like \(\big({}^0_0\;{}^1_1\;{}^2_1\big)\). A matrix \(\Big({}^{{}^{a_{00}}_{a_{01}}}_{{}^{a_{02}}_{\vdots}}\;{}^{{}^{a_{10}}_{a_{11}}}_{{}^{a_{12}}_{\vdots}}\;{}^{{}^{a_{20}}_{a_{21}}}_{{}^{a_{22}}_{\vdots}}\;{}^{{}^{\dots}_{\dots}}_{{}^{\dots}_{\ddots}}\;\Big)\) is abbreviated as (a00,a01,a02,...)(a10,a11,a12,...)(a20,a21,a22,...)... in many places, such as in the definition below.

We introduce some terminology: And here are the rules: I also let FSn(S) be G + B1 + B1 + ... + Bn-1. Then I define a function Trans(M) for a matrix M, defined as follows: A table of matrices and values for Trans will be added soon.
 * Let S, G, B, M, A denote any matrices.
 * Let Si denote the (i+1)th column of S.
 * Let Si, j denote the (i+1)th element of the (j+1)th column of \(S\).
 * Let S + T for two matrices S and T be S concatenated to T.
 * Let |S| be the length of S, that is, the number of columns in it.
 * 1) [n] = n
 * 2) (S + (0))[n] = S[n2]
 * 3) If neither rule 1 or 2 applies, apply the rules below, in the order they are written.
 * 4) Let the kth-row parent of the mth column be denoted pk(m). This is defined as the highest x such that Sk,x < Sk,m and that ak-1,m(x) = 1.
 * 5) Let ak,m(t) be defined as returning 0 if m = 0, 1 if there exists a c > 0 such that pkc(m) = t, and 0 otherwise.
 * 6) Let z be the highest t such that Sundefined > 0.
 * 7) Let r, the bad root, be pt(|S|-1).
 * 8) Let B, the bad part, be Sr + Sr+1 + Sr+2 + ... + Sundefined.
 * 9) Let G, the good part, be S0 + S1 + S2 + ... + Sr-1.
 * 10) Let &Delta;b be Sb, - Sb,r if b < z, else 0.
 * 11) Let Ai, j be ai,r+x(r).
 * 12) Let Bki, j be Bi, j + Ai, j &times; &Delta;i &times; k.
 * 13) Let Bki be (Bk0,i,Bk1,i,Bk2,i,...).
 * 14) Let Bk be Bk0 + Bk1 + Bk2 + ....
 * 15) Finally, let S[n] = (G + B0 + B1 + ... + Bn)[n2].
 * Trans = 0
 * Trans(S + (0)) = Trans(S) + 1
 * Otherwise, Trans(M) = sup{Trans(FSi(M)) | i a natrual number}