After hyp cos found non-teminating sequence of Bashicu matrix, Bashcu updated his system in his blog post. Please note that it is different from the version that User:KurohaKafka posted on the talk page. KurohaKafka thinks (0,0,...,0)(1,1,...,1) terminates for Bashicu's new system, while KurohaKafka's version may not terminate.
Let us call the original version of Bashicu matrix as Bashicu Matrix version 1 (BM1), and the new version that Bashicu made as version 2 (BM2). The problem in BM2 is that it is too difficult to understand. I would suggest that we use BM1 for up to pair sequence to understand the concept, and use BM2 for higher system.
Now I updated the Bashicu Matrix calculator, and you can calculate both BM1 and BM2. In BM2, the new concept of "C matrix" is introduced. You can check how the calculation proceeds by using the detailed calculation option. You can check the source code of the program to see the calculation process, or you can download the program and run it on your PC.
Example calculations where BM1 and BM2 differs.
I hope this program helps you analyze Bashicu Matrix system.