Even Benedict XVI acknowledges that he can err in matters of doctrine. I see nothing in the relevant constitutions which say a pope is always and everywhere protected from teaching error.
In fact, there is a suggestion that the definition of papal infallibility was influenced by the mess around Honorius I. If so, perhaps the influence was shown in the precise description of the circumstances under which a pope is protected from error.
My reading suggests that Honorius's evident approval of Monotheilitism was not promulgated in an encyclical.
It is clear, then, that the council did not think that it stultified itself by asserting that Honorius was a heretic (in the above sense) and in the same breath accepting the letter of Agatho as being what it claimed to be, an authoritative exposition of the infallible faith of the Roman See. The fault of Honorius lay precisely in the fact that he had not authoritatively published that unchanging faith of his Church, in modern language, that he had not issued a definition ex cathedra.It is embarrassing to read something proposed as an ineluctable and baffling dilemma and then to find that the case was oversimplified to the point of mendacity.
And maybe this is the place to point out that the matter of Honorius I and papal infallibility has nothing to do with the ludicrous OP of this thread or with any of the ninety-eleven unrelated attacks on Catholicism. Not only is any old stick good enough to beat a catholic with, but any old pretext is fit to introduce a vast and scattered array of charges.
This is why I only engage in "constructives" in response to assaults. There is no evidence that many of our adversaries really are interested in the Truth anything like as much as they are interested in some fleeting rhetorical victory.
The fact is Honorius was convicted of heresy and excommunicated.
I don't put much stock in the "retroactive" definitions you are so happy to use.