User blog:Rpakr/Bashicu Matrix Version 3.3

In BM4, (0,0,0)(1,1,1)(2,1,0)(1,1,1) expands to (0,0,0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,2,0)(2,2,0)(3,3,1)(4,3,0)(3,3,0)(4,4,1)(5,4,0)... not (0,0,0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,1,0)(2,2,0)(3,3,1)(4,1,0)(3,3,0)(4,4,1)(5,1,0)...,and this is known to make the analysis of BM4 very difficult past \(\psi(\Omega_\omega\times\Omega+\Omega_\omega)\) in UNOCF. In BM4, if a standard matrix A does not contain (1,0) and \(A=\psi(C)\) in UNOCF, A(1,1) is known to correspond to \(\psi(C+\Omega)\) in UNOCF. However, even if a standard matrix A does not contain (1,0,0) and (1,1,0) and \(A=\psi(C)\), A(1,1,1) does not necessarily correspond to \(\psi(C+\Omega_\omega)\). Because this makes the analysis a lot harder, many people have tried to make a version in which this holds. BM3, BM3.1 and BM3.2 was able to make (0,0,0)(1,1,1)(2,1,0)(1,1,1) expand to (0,0,0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,1,0)(2,2,0)(3,3,1)(4,1,0)(3,3,0)(4,4,1)(5,1,0)..., but they are all known to have strange behavior. BM3 does not terminate at (0,0,0)(1,1,1)(2,1,1)(1,1,1), and in BM3.1 and BM3.2 the above does not hold at (0,0,0)(1,1,1)(2,2,0)(1,1,1). Since August 2018, some people in discord thought of a hypothetical (not defined) version of BMS in which the above holds and called it "Idealized BMS". In March 2019, me and ecl1psed in discord collaborated and made a definition that seems to work like idealized BMS, and the version is currently called "BM3.3". I made programs that can expand BM3.3. This program expands the input matrix showing detail output. This program only shows the result of the expansion. In this blog post I will explain the definition of BM3.3.

The definition of BM3.3 is very similar to BM4, and the only difference is the ascension matrices. Let t be the column number of th lowermost nonzero element in the rightmost column. In columns of the bad part such that row t is a descendant of the bad root (including the bad root itself), the ascension matrix is 1 for all t rows in both versions. The ascension matrices are differnt in columns such that row t is not a descendant of the bad root. In BM4, the ascension matrix is 1 if the parent's ascension matrix is 1 and the parent is the bad root or to the right of the bad root for column. (This is not the actual definition but it is equivalent.) On the other hand, in BM3.3 the ascension matrix is 1 if the parent's ascension matrix is 1 and the parent is to the right of the bad root. For example, consider the matrix (0,0,0)(1,1,1)(2,1,0)(1,1,1). t=3, and the bad root is (0,0,0). For (2,1,0), the 3rd row is 0 and is not a descendant of the bad root, so the second case applies. The first row, 2, has a parent which has ascension matrix 1 and to the right of the bad root, so it has ascension matrix 1, but in the second row, 1, its parent has ascension matrix 1 but is not to the bad root, so it has ascension matrix 0.

According to analysis, BM3.3 is weaker than BM4 before (0,0,0)(1,1,1)(2,2,2), but they might catch up. In fact, ecl1psed conjectures that BM3.3 catch up with BM4 at (0,0,0)(1,1,1)(2,2,2). Some analysis results of mine of BM3.3 are as follows. Note that these results are not proven formally. \(\psi\) is UNOCF.

(0,0,0)(1,1,1)(2,1,0)(1,1,1)=\(\psi(\Omega_\omega\times\Omega+\Omega_\omega)\) (0,0,0)(1,1,1)(2,1,0)(3,2,1)(4,2,0)=\(\psi(\Omega_\omega\times\Omega_2)\) (0,0,0)(1,1,1)(2,1,1)=\(\psi(\Omega_\omega^2)\) (0,0,0)(1,1,1)(2,1,1)(3,1,1)=\(\psi(\Omega_\omega^{\Omega_\omega})\) (0,0,0)(1,1,1)(2,2,0)=\(\psi(\Omega_{\omega+1})\) (0,0,0)(1,1,1)(2,2,0)(3,3,1)=\(\psi(\Omega_{\omega\times2})\) (0,0,0)(1,1,1)(2,2,1)=\(\psi(\Omega_{\omega^2})\) (0,0,0)(1,1,1)(2,2,1)(2,2,1)=\(\psi(\Omega_{\omega^3})\) (0,0,0)(1,1,1)(2,2,1)(3,0,0)=\(\psi(\Omega_{\omega^\omega})\) (0,0,0)(1,1,1)(2,2,1)(3,1,0)=\(\psi(\Omega_\Omega)\) (0,0,0)(1,1,1)(2,2,1)(3,1,1)=\(\psi(I)\)