Why do you say that Paul IV doesn't exercise his office as supreme teacher, when actually he explicitly invokes his apostolic authority? Do you want to deceive your readers? Do you think they didn't read the quotes and fall for your denial?
There are various reasons why a pope invokes his apostolic authority, doctrine being only a subset. There are also governmental and canon law reasons.
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The pope doesn't need to speak ex cathedra for a general Council to be infallible:
Absolutely, positively, 100% ridiculous.
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If the pope is infallible beyond the definitions of Vatican 1, please let us all know.
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And let the record confirm that +Vigano confirms that V2 was not infallible.