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I would say that the bottom line is if independent reliable sources describe Cox as Catholic, then we can go with that. The difference is that Cox belongs to a schismatic organization that is several levels removed from the Catholic Church. Williamson, in his association with the SSPX, was directly ordained by a Catholic Church bishop and remained Catholic, even while not in full communion. Note that the Catholic Church affirmed that attendance at an SSPX chapel would fulfill the Sunday obligation, i.e. in a ""Catholic rite"". I do not know if the Catholic Church has commented on the canonical status of the Irish Orthodox Catholic and Apostolic Church, or the validity of the Thuc line of apostolic succession. My suggestion for a succinct description of Cox would be ""Independent Catholic bishop"", as that is how his church is described.   "