User blog comment:Hyp cos/Fundamental Sequences in Taranovsky's Notation/@comment-40154718-20190917091450/@comment-35470197-20190917115107

The reason why we can define such an FS for ordinal notations associated to OCFs is because we have an order-preserving map to a sufficently large ordinal induced by the OCFs themselves. Unlike them, there is no known order-preserving map from TON to ordinals. If it admits one, then it automatically ensures the well-foundedness of TON, which is still an open problem.