Don't the Popes have the power to bind and loose? Did Paul IV not use this power, and would this power not extend to what he said about the legitimacy of a heretic Pope?

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I think a lot of people who dismiss the Universal Peaceful Acceptance idea don't grasp the problem it is intended to solve. The problem is, we must accept with certainty what the pope teaches to the whole Church. In the case of an extraordinary ex cathedra definition, we must accept such a definition with the certitude of faith.

But such a definition, and such teaching, can only come from a valid pope. An ex cathedra definition from someone who is not a valid pope is not part of faith, and might even be heretical. Catholics, therefore, need a way to know that any given papal claimant is a valid pope, and that he was validly elected.

No conclusion can be stronger than its premises. If we say that such and such a dogma is part of our faith because it is defined by a pope, while having doubt about whether that pope is a valid pope or not, results in our faith in that (supposedly) defined dogma being in doubt as well.

So what criterion will tell you that someone is a valid pope? The assertion of the college of cardinals? Maybe. But didn't they all assert that the first pope in the Great Western Schism was pope, and then the French cardinals "changed their minds" later on? So, if you use that criterion, then your acceptance of someone as pope would have to be contingent on waiting to see if any cardinals will change their minds at some point.

That's how Cardinal Billot argues that the universal, peaceful acceptance of the Church of someone as pope must be an infallible sign that the man is pope. This is a clear sign that everyone can see, and is pretty objective, so it meets what we are looking for as an objective, external proof that someone is pope. And it fits the principle that the entire Church cannot adhere to a false rule of faith, and the pope is the rule of faith for the whole Church.

Cardinal Billot says that it's not just his opinion about Universal Peaceful Acceptance, but that of the moral unanimity of theologians, and he says it is certain.

People reject it because they misunderstand what is being said. Some people think this means the whole Church ratifies the election and makes it valid. This is false. The acceptance of a pope is a sign that his election was valid. It does not make it valid, nor does the whole Church elect the pope. Other people object that this idea is false because the whole Church could theoretically adhere to a false pope. But according to Billot this is false.

It can be explained by an analogy. If electricity is passing through a light bulb, the filament will emit light. The light is an infallible sign that electricity is passing through the bulb, since it is impossible for light to come from the filament without electricity. Are we saying the light coming from the bulb causes electricity to flow through the bulb? No. This is backwards. In the same way, if someone is validly elected (electricity passing through him, something we  can't see), then the whole Church will adhere to him (something eminently visible, as the light from a bulb).

Now, this is not something that universally takes place. It has happened that a valid pope has been elected and not everyone has adhered to him. The situation with Antipope Anacletus II comes to mind, where the Church was split between the true pope and an antipope. That is possible, according to the UPA thesis. It's possible that not the entire Church will adhere to a true pope.

What is not possible, according to the Universal Peaceful Acceptance teaching, is that either:

1. The whole Church reject a valid pope, or that
2. The whole Church adhere to a false pope, believing him to be a true pope.

For people who disagree with this idea, I'm really curious what objective, publicly verifiable criterion they think proves whether someone is a valid pope or not?