User blog comment:QuasarBooster/Obfuscated Bashicu Matrix Sytem in Python/@comment-35470197-20190603223832/@comment-35470197-20190604004214

Such an expectation is derived from table-based analysis, which is not a formal (i.e. mathematically precise) analysis. Roughly speaking, table-based analysis is to write a table which displays the correspondence between matrices and other notations. If we write a table displaying the correspondence between matrices and ordinals, then we can intuitively judge whether it looks consistent or not by comparing fundamental sequences. (It can never be a proof, though.)

Unfortunately, almost all tables on BMS display the correspondence between matrices and UNOCF, which is known to be ill-defined. Although UNOCF does not admit a well-defined system of fundamental sequences, it is believed to work "well" in a traditional thoughts in this community. Through the comparison, people believe that 3-row BMS goes beyond the countable collapse of the least weakly compact cardinal, which is greater than \(\psi(\psi_{I_{\omega}}(0))\) and \(\psi(\psi_{\chi_{\varepsilon_{M+1}(0)}(0)}(0))\) (with respect to any reasonable OCF \(\psi\)). This is the reason. I emphasise that UNOCF is known to be ill-defined, and hence it is impossible to argue on whether the correspondence looks consistent or not.

As far as I know, KurohaKafka's table analysis, which is still being updated, is the only one displaying the correspondence between matrices and actual ordinals. Of course, it does not give a proof, but we can argue whether the correspondence looks consistent or not unlike UNOCF.