User blog comment:Edwin Shade/A Complete Analysis of Taranovsky's Notation/@comment-30118230-20180129200050/@comment-27513631-20180129205114

There's little evidence that (say) sDAN actually works, except that we haven't found a non-terminating sequence yet.

In general, the only way to be sure that a notation works is to express the recursion used in terms of some well-founded ordering (normally an ordinal), which is scarcely ever done. Strong heuristic arguments are more common, but those are difficult for theories which outgrow the simpler OCFs (eg sub WCC).

Depressingly much of googology is wishful thinking, if you're one who cares for rigour.