Yes, unless it contradicts something of greater weight.

Yes

Lad-

So, if something appears in the AAS, from an undisputedly legitimate Pope, are we required to assent to it?

Hence it follows that the authority of the encyclicals is not at all the same as that of the solemn definition, the one properly so-called. The definition demands an assent without reservation and makes a formal act of faith obligatory. The case of the encyclical’s authority is not the same.

This authority (of the papal encyclicals) is undoubtedly great. It is, in a sense, sovereign. It is the teaching of the supreme pastor and teacher of the Church. Hence the faithful have a strict obligation to receive this teaching with an infinite respect. A man must not be content simply not to contradict it openly and in a more or less scandalous fashion. An internal mental assent is demanded. It should be received as the teaching sovereignly authorized within the Church.

Ultimately, however, this assent is not the same as the one demanded in the formal act of faith. Strictly speaking, it is possible that this teaching (proposed in the encyclical letter) is subject to error. There are a thousand reasons to believe that it is not. It has probably never been (erroneous), and it is normally certain that it will never be. But, absolutely speaking, it could be, because God does not guarantee it as He guarantees the teaching formulated by way of definition’.

Lercher teaches that the internal assent due to these pronouncements cannot be called certain according to the strictest philosophical meaning of the term. The assent given to such propositions is interpretative condicionatus, including the tacit condition that the teaching is accepted as true “unless the Church should at some time peremptorially define otherwise or unless the decision should be discovered to be erroneous.”   Lyons and Phillips use the same approach in describing the assent Catholics are in conscience bound to give to the Church’s non-infallible teachings. Fr. Yves de la Brière speaks of the “submission and hierarchical obedience” due to these pronouncements.

Msgr. Manzoni lists the encyclicals among the docuмents in which non-infallible teaching is to be found. He holds that the definition of which the Vatican Council speaks in proposing the doctrine of papal infallibility is to be found only in the exercise of the solemn, as distinguished from the ordinary magisterium. In explaining the binding force of these non-infallible pronouncements, he, like Bishop Francis Egger, and Fathers Mangenot, MacGuinness, and Dieckmann, employs an explanation formulated by Cardinal Franzelin in his Tractatus de divina traditione et scriptura.

Franzelin holds that the Roman Pontiff can command all Catholics to assent to a given proposition (either directly or by condemning the contradictory statement), for either one of two different reasons. First the Holy Father can intend to define this proposition infallibly as true or as de fide. Again he can will merely to look after the security of Catholic doctrine. The magisterium of the Church has been equipped with help from God by reason of which the first sort of teaching gives infallible truth, while the second affords infallible security. Employing the plentitude of its power, the teaching Church operates as the auctoritas infallibilitatis. Working, not to define, but merely to take those steps it deems necessary to safeguard the faith, it is the auctoritas providentiae doctrinalis. To this auctoritas providentiae doctrinalis and to the teachings it sets forth, the faithful owe the obedience of respectful silence and of an internal mental assent according to which the proposition thus presented is accepted, not as infallibly true, but as safe, as guaranteed by that authority which is divinely commissioned to care for the Christian faith.
...
It might be definitely understood, however, that the Catholic’s duty to accept the teachings conveyed in the encyclicals even when the Holy Father does not propose such teachings as a part of his infallible magisterium is not based merely upon the dicta of the theologians. The authority which imposes this obligation is that of the Roman Pontiff himself. To the Holy Father’s responsibility of caring for the sheep of Christ’s fold, there corresponds, on the part of the Church’s membership, the basic obligation of following his directions, in doctrinal as well as disciplinary matters. In this field, God has given the Holy Father a kind of infallibility distinct from the charism of doctrinal infallibility in the strict sense. He has so constructed and ordered the Church that those who follow the directives given to the entire kingdom of God on earth will never be brought into the position of ruining themselves spiritually through this obedience. Our Lord dwells within His Church in such a way that those who obey disciplinary and doctrinal directives of this society can never find themselves displeasing God through their adherence to the teachings and the commands given to the universal Church militant. Hence there can be no valid reason to discountenance even the non-infallible teaching authority of Christ’s vicar on earth.
...
It is, of course, possible that the Church might come to modify its stand on some detail of teaching presented as non-infallible matter in a papal encyclical. The nature of the auctoritas providentiae doctrinalis within the Church is such, however, that this fallibility extends to questions of relatively minute detail or of particular application. The body of doctrine on the rights and duties of labor, on the Church and State, or on any other subject treated extensively in a series of papal letters directed to and normative for the entire Church militant could not be radically or completely erroneous. The infallible security Christ wills that His disciples should enjoy within His Church is utterly incompatible with such a possibility.

Freedom is a right of every person: each individual enjoys the freedom of belief, thought, expression and action. The pluralism and the diversity of religions, color, sex, race and language are willed by God in His wisdom, through which He created human beings. This divine wisdom is the source from which the right to freedom of belief and the freedom to be different derives.

If the Pope should happen to fall into heresy, he is no longer a member of the Church, much less its head. It is understood that the Pope cannot be guilty of heresy when he speaks infallibly ex cathedra. The supposition is only possible should the Pope teach heretical doctrine in a private capacity.
(Rev. Matthew Ramstein, A Manual of Canon Law [Hoboken, NJ: Terminal Printing & Publishing Co., 1948], p. 193; underlining added.)
The Pope is the Teacher and Shepherd of the whole Church, thus, the whole Church is so bound to hear and follow him that if he would err, the whole Church would err. Now our adversaries respond that the Church ought to hear him so long as he teaches correctly, for God must be heard more than men. On the other hand, who will judge whether the Pope has taught rightly or not? For it is not for the sheep to judge whether the shepherd wanders off, not even and especially in those matters which are truly doubtful. Nor do Christian sheep have any greater judge or teacher to whom they might have recourse. As we showed above, from the whole Church one can appeal to the Pope yet, from him no one is able to appeal; therefore necessarily the whole Church will err if the Pontiff would err.
(St. Robert Bellarmine, On the Roman Pontiff, Book IV, Chapter 3; , p. 160; underlining added.)

• The Open Letter accusing Francis of Heresy: A Sedevacantist Analysis

It must first be kept in mind that every quest of the human spirit for truth and goodness, and in the last analysis for God, is inspired by the Holy Spirit. The various religions arose precisely from this primordial human openness to God. At their origins we often find founders who, with the help of God’s Spirit, achieved a deeper religious experience. Handed on to others, this experience took form in the doctrines, rites and precepts of the various religions.
(Antipope John Paul II, General Audience, Vatican.va, Sep. 9, 1998)

…[F]aith, which is the basis and foundation of all religion, must consist in a certain interior sense, originating in a need of the divine [according to the Modernists]. This need of the divine, which is experienced only in special and favorable circuмstances, cannot of itself appertain to the domain of consciousness, but is first latent beneath consciousness, or, to borrow a term from modern philosophy, in the subconsciousness, where also its root lies hidden and undetected.
It may perhaps be asked how it is that this need of the divine which man experiences within himself resolves itself into religion? To this question the Modernist reply would be as follows: Science and history are confined within two boundaries, the one external, namely, the visible world, the other internal, which is consciousness. When one or other of these limits has been reached, there can be no further progress, for beyond is the unknowable. In presence of this unknowable, whether it is outside man and beyond the visible world of nature, or lies hidden within the subconsciousness, the need of the divine in a soul which is prone to religion excites — according to the principles of Fideism, without any previous advertence of the mind — a certain special sense, and this sense possesses, implied within itself both as its own object and as its intrinsic cause, the divine reality itself, and in a way unites man with God. It is this sense to which Modernists give the name of faith, and this is what they hold to be the beginning of religion.
…How far this position is removed from that of Catholic teaching! We have already seen how its fallacies have been condemned by the Vatican Council. Later on, we shall see how these errors, combined with those which we have already mentioned, open wide the way to Atheism. Here it is well to note at once that, given this doctrine of experience united with that of symbolism, every religion, even that of paganism, must be held to be true. What is to prevent such experiences from being found in any religion? In fact, that they are so is maintained by not a few. On what grounds can Modernists deny the truth of an experience affirmed by a follower of Islam? Will they claim a monopoly of true experiences for Catholics alone? Indeed, Modernists do not deny, but actually maintain, some confusedly, others frankly, that all religions are true. That they cannot feel otherwise is obvious. For on what ground, according to their theories, could falsity be predicated of any religion whatsoever? Certainly it would be either on account of the falsity of the religious sense or on account of the falsity of the formula pronounced by the mind. Now the religious sense, although it maybe more perfect or less perfect, is always one and the same; and the intellectual formula, in order to be true, has but to respond to the religious sense and to the believer, whatever be the intellectual capacity of the latter. In the conflict between different religions, the most that Modernists can maintain is that the Catholic has more truth because it is more vivid, and that it deserves with more reason the name of Christian because it corresponds more fully with the origins of Christianity. No one will find it unreasonable that these consequences flow from the premises. …
There is yet another element in this part of their teaching which is absolutely contrary to Catholic truth. For what is laid down as to experience is also applied with destructive effect to tradition, which has always been maintained by the Catholic Church. Tradition, as understood by the Modernists, is a communication with others of an original experience, through preaching by means of the intellectual formula. To this formula, in addition to its representative value they attribute a species of suggestive efficacy which acts firstly in the believer by stimulating the religious sense, should it happen to have grown sluggish, and by renewing the experience once acquired, and secondly, in those who do not yet believe by awakening in them for the first time the religious sense and producing the experience. In this way is religious experience spread abroad among the nations; and not merely among contemporaries by preaching, but among future generations both by books and by oral transmission from one to another. Sometimes this communication of religious experience takes root and thrives, at other times it withers at once and dies. For the Modernists, to live is a proof of truth, since for them life and truth are one and the same thing. Thus we are once more led to infer that all existing religions are equally true, for otherwise they would not survive.
(Pope Pius X, Encyclical Pascendi Dominici Gregis, nn. 7,14-15; underlining added.)

Theologians have taught that the pope is infallible in universal disciplinary laws and that this proposition is theologically certain.  However, I don't ever recall seeing the syllogism spelled out that reaches this conclusion.  Can anyone show me where I can find it or present it here?  The terms used in the syllogism should be defined.

https://isidore.co/calibre#book_id=7428&library_id=CalibreLibrary&panel=book_details

Theologians distinguished between these authorities (what's at issue is the Pope, and not just any authority) as PRIVATE PERSONS vs. acting in their official capacity.

To assert that Montini was acting as a private person in promulgating the NOM is the height of absurdity.

Ex Cathedra does not mean "infallibly".  In addition to teaching from the Chair (i.e. in his official capacity as Pope), the other notes have to be applied.  This is an unfortunate misapplication of a "shorthand" term that leads to a misunderstanding of the term itself.  See again the definition of the term in Pastor Aeternus.