If V1 states that should the pope teach new doctrines, the Holy Ghost's protection from error will not be present, then that's plain enough for everyone to understand, so that's what I take it to mean.

It doesn't actually say that, because it's never happened before, and my opinion is that God would not allow the devil to get this close to the line.  V1 is just explaining the proper use of infallibility and what it's supposed to provide the Church.
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In the sense of V1, a new doctrine is in fact heresy because that in the same sentence, V1 says why the divine protection *is* promised, namely, so "that, by his assistance, they might religiously guard and faithfully expound the revelation or deposit of faith transmitted by the apostles".

V1 is explaining the PURPOSE of infallibility; it is not giving examples of how it cannot be used.
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A "new doctrine" is a contradiction in terms, because "doctrine", by definition, is already infallible.  If you want to say "new ideas" or "new preachings" are heresy, then we agree.

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If V1 states that should the pope teach new doctrines, the Holy Ghost's protection from error will not be present, then that's plain enough for everyone to understand, so that's what I take it to mean.

It doesn't actually say that, because it's never happened before, and my opinion is that God would not allow the devil to get this close to the line.  V1 is just explaining the proper use of infallibility and what it's supposed to provide the Church.
V1 is explaining the PURPOSE of infallibility; it is not giving examples of how it cannot be used.

Clearly V1 is defining both when and why - First, when there is no divine protection, and Second, why there is divine protection. Later in the decree, V1 defines when there is divine protection - which is when he defines a doctrine ex cathedra.

And yes, per Fr. Hesse, it has happened before, I posted this earlier in the thread for LT, it's pretty short, just listen from 19:39 for a few minutes:

The whole talk is well worth hearing, it's almost as if Fr. Hesse was reading this thread, but Fr. Hesse has a few examples for you, start at the 19:39 mark and listen till 28:00.

https://youtu.be/1FAXaqsVOBg?t=1176

V1 is explaining the PURPOSE of infallibility; it is not giving examples of how it cannot be used.
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A "new doctrine" is a contradiction in terms, because "doctrine", by definition, is already infallible.  If you want to say "new ideas" or "new preachings" are heresy, then we agree.

Stating when the divine protection is absent, admits, at least implicitly, that situation can happen - and in fact did happen with V2.

Also "new doctrine" is not a contradiction in terms, we know this because V1 used that exact term.  

Vatican I is the best refutation of neo-Gallican R&R absurdities (The Council was literally a supreme triumph of "ultramontanism" over Gallicanism" that there is. Just 3 random brief excerpts. I encourage everyone interested in understanding the Church's Magisterium's mind on the subject to read the whole Council: https://www.ewtn.com/catholicism/library/first-vatican-council-1505

I. 2. Wherefore we teach and declare that, by divine ordinance, the Roman Church possesses a pre-eminence of ordinary power over every other Church, and that this jurisdictional power of the Roman Pontiff is both episcopal and immediate. Both clergy and faithful, of whatever rite and dignity, both singly and collectively, are bound to submit to this power by the duty of hierarchical subordination and true obedience, and this not only in matters concerning faith and morals, but also in those which regard the discipline and government of the Church throughout the world. 3. In this way, by unity with the Roman Pontiff in communion and in profession of the same faith , the Church of Christ becomes one flock under one Supreme Shepherd. 4. This is the teaching of the Catholic truth, and no one can depart from it without endangering his faith and salvation. [50]."

II. The sentence of the Apostolic See (than which there is no higher authority) is not subject to revision by anyone, nor may anyone lawfully pass judgment thereupon [54]. And so they stray from the genuine path of truth who maintain that it is lawful to appeal from the judgments of the Roman pontiffs to an ecuмenical council as if this were an authority superior to the Roman Pontiff.

3. "in accordance with the divine promise of our Lord and Savior to the prince of his disciples: I have prayed for you that your faith may not fail; and when you have turned again, strengthen your brethren [60].7. This gift of truth and never-failing faith was therefore divinely conferred on Peter and his successors in this See so that they might discharge their exalted office for the salvation of all, and so that the whole flock of Christ might be kept away by them from the poisonous food of error and be nourished with the sustenance of heavenly doctrine. Thus the tendency to schism is removed and the whole Church is preserved in unity, and, resting on its foundation, can stand firm against the gates of hell."

Xavier, you rant and rave against Sede-ism and now you're ranting against "R&R".  This makes no sense, except then I remembered that you're a fake-trad who accepts V2 and the new mass wholeheartedly.  There's really no reason for you to be on this site, which was founded on the very principles you, and your friends in new-rome, reject.

Maybe your conscience pricks you, Pax Vobis, for being out of communion with the Catholic Church, and for acting as if there is no way you can commit the sin of schism today, that you react so maliciously. If you want to be out of communion with the Pope and the Bishops, I weep for your soul. I know before God I'm a Traditional Catholic, and I have no need to prove anything to you. I want to avoid heresy, schism, vice, sin, become a saint, serve the Church, and save my soul. That's it. 

Not long ago, this forum said: "A forum for SSPX and other Traditional Catholics to discuss matters pertaining to the Catholic Faith". That was when I first signed up.

SVism and R&R being the only two traditional possibilities is your false dichotomy, not mine. I don't have to answer that, and I've already explained before that there is a third alternative RPWR: Recognize, Pray, Work and Restore. It's neither sede-ism nor R&R, but +ABL said this, "We wish to remain attached to Rome and to the Successor of Peter, while refusing his Liberalism through fidelity to his predecessors. We are not afraid to speak to him, respectfully but firmly, as did St. Paul with St. Peter. And so, far from refusing to pray for the Pope, we redouble our prayers and supplications that the Holy Ghost will grant him light and strength in his affirmations and defense of the Faith. Thus, I have never refused to go to Rome at his request or that of his representatives. The Truth must be affirmed at Rome above all other places. It is of God, and He will assure its ultimate triumph." Recognize, Pray, Work, Restore, they all appear, and those are the 4 pillars of Authentic Traditional Restoration. https://www.sspxasia.com/Docuмents/Archbishop-Lefebvre/Apologia/Vol_two/Chapter_40.htm It's because you don't do this that you opposed Bp. Fellay. Whereas everyone who thinks like a Catholic sees that Bp. Fellay is perfectly just in obtaining Ordinary Jurisdiction for the Three Society Bishops. 

unlike the SSPX who remain irregular

This is your unproved opinion, Ladislaus; the Pope has called them Catholics, and that alone is sufficient to refute your opinion. But further, as Bp. Fellay has explained, and I've proven to you before, the Pope has granted them ordinary jurisdiction, extending their faculties beyond the Year of Mercy. Now, it is impossible to have ordinary jurisdiction, and be irregular. I don't have to go into your other non-sequiturs in your post, because you neither understand what schism is (hint, belonging to an objectively Catholic society isn't it), nor, plainly, have you experienced for a long time, the blessing of what it is and what it means, to enjoy the benefits of Catholic Communion with the Universal Church: "We are very grateful to our Holy Father Pope Benedict XVI for issuing, last July, the Motu Proprio Summorum Pontificuм which called us to come into undisputed and peaceful Communion with him.

Now we have that undisputed communion! It is a pearl of great price; a treasure hidden in the field; a sweetness that cannot be imagined by those who have not tasted it or who have not known it, now for many years. Its value cannot be fully expressed in earthly language and therefore we hope that all traditional priests who have not yet done so, will answer Pope Benedict's call to enjoy the grace of peaceful and undisputed communion with him. Believe us, the price to pay is nothing; even all the angry voices that have shouted against us and calumniated us are as nothing when weighed in the scales against undisputed communion with the Vicar of Christ; others have died for it; what are raucous voices?" https://rorate-caeli.blogspot.com/2008/07/transalpine-redemptorists-regularized.html

If I thought the SSPX was non-Catholic, I would NEVER join them. Some of you are so partisan that you cannot see that the Fraternity of St. Peter has done so much good for the Church that anyone who truly loves the Church will appreciate. And the SSPX has also done even more good that very many "on the other side" also appreciate: 

All I can tell some of you is that, at least Summorum Pontificuм, views like yours are increasingly rare. One of the moderators at OnePeterFive, who is an FSSP Parishioner, is very pro-SSPX. And very many Traditional Catholics are like that, whether you know it or not, whether you believe it or not. And they are right, and Bishop Fellay who said, "I rejoice whenever a Traditional Mass is celebrated" is exactly right in that Catholic mindset, that has only the good of the Church at heart. 

I reject the absurd idea that the SSPX is not Catholic, or that belonging to a Catholic society is schismatic, or any other such "sede-doubtist" subjective nonsense. I believe Bishop Fellay is a perfect Traditional Catholic. So why precisely should I refuse communion with him or refuse to belong to the Fraternity he belongs to?

I also reject the general attitude that to belong to the SSPX I must hate the FSSP, or vice versa. If I chose the FSSP, I'm not obliged to hate or refuse communion with the SSPX, since both Fraternities are Catholic. That in fact is a schismatic attitude; schism is a sin against Catholic Charity. You can commit schism in two ways, per St. Thomas. Refusing subjection to the Vicar of Christ. Which some of you do, instead of at least taking the safer position of not refusing subjection by being in communion. Or, by refusing communion with the members of the Church subject to the Pope, which some of you also do. Tread carefully. No one can cause you to be in schism, just like any other sin, if you don't want to. No one can also cause you not to be in schism if you want to remain there. It's a free will choice. 

One is either schismatic or not.  Quo Primum and canon law allow Trads to do what we do, nay, they command it.  Thus, schism has no part in the discussion.
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“Full communion” and it’s various “degrees” are made-up, novel terms that the V2 heretics had to invent to explain (to novus ordo Catholics such as yourself) how Catholics loyal to the Faith (ie Trads) could ignore Modernistic heresies.  V2 officials aren’t going to come out and say “Well, Trads are within their rights of law, morality and history to continue with the Latin mass, because the new mass is illegal and not obligatory.”  But this is the truth.  

This is your unproved opinion, Ladislaus; the Pope has called them Catholics, ...

NOTE, I did not say that the SSPX are not considered to be Catholics, just that they are canonically irregular.  When your theology is no different than that of the FSSP, you have no justification whatsoever for going with an irregular group over the regular FSSP.  Consequently, you are a schismatic.  If I believed as you did, then I would not go anywhere near the SSPX until they were fully regularized.