The power of jurisdiction cannot be intrinsically tied to the physical or sacramental act of consecration. If ordinary jurisdiction came from the ceremony of consecration itself, then every validly consecrated schismatic bishop (such as Eastern Orthodox or Old Catholic bishops) would possess divinely-sanctioned ordinary jurisdiction over their flocks, but the Church teaches that schismatic bishops exercise no jurisdiction even if their orders remain valid. Rather, She teaches that the ceremony of episcopal consecration does NOT automatically grant bishops the authority of governance over the faithful. The essential form of episcopal consecration, those words necessary for a candidate to be validly made a bishop, signifies the power of the order being conferred (the plenitude of the priesthood) and the grace of the Holy Ghost: it evidently does not grant ordinary jurisdiction by its very nature.
“Perfect in Thy priest the fullness of Thy ministry, and, clothed in the raiment of all glory, sanctify him with the dew of heavenly anointing.”—Pontificale Romanum, 1888 (as noted by Pope Pius XII, Sacramentum Ordinis, 1947)
Even though Christ grants men the full power of the priesthood upon being consecrated bishops, both the overwhelming consensus of theologians and the decrees of the popes throughout history make it clear that bishops can be granted ordinary jurisdiction only directly and immediately from a living pope. As one can tell from a reading of the rite of episcopal consecration, the rite bestows an aptitude, but not an active power, to govern the faithful. Like a man finishing law school might have the aptitude to judge but cannot legally try a case until the state appoints him as a judge, a bishop has the sacramental aptitude to rule from his consecration, but he remains completely powerless to exercise ordinary jurisdiction until the Pope assigns him a part of the world.
“...By the laying on of hands, a bishop receives his spiritual character; but by the word of the Roman Pontiff, he receives his distinct territory and the people over whom he may exercise that character.”—Pope Innocent III, Consecratione Pontificis, Sermo 2
That episcopal authority in the world remains dependent upon the Apostolic See as its source was also taught by Pope St. Innocent I, addressing the bishops of Carthage in the fifth century, and by the most eminent theologians down through the years, such as St. Bonaventure (1221–1274).
“[...N]o one receives an ordinary power of governing except through the supreme head, who is the Vicar of Christ. It is from this fountainhead that all local waters of episcopal authority must flow.”—St. Bonaventure, Breviloquium, Part VI, Chapter 12
Likewise, St. Thomas Aquinas (1225–1274) in his Summa (P. II-II, q. 39, a. 3) distinguishes between the power of order and that of jurisdiction, noting that the latter is bound directly to the subjection to the Pope. “[T]he power of jurisdiction,” he says, “does not adhere immovably to the recipient” so therefore “the Pope can take away or give jurisdiction to bishops.” Francis de Vitoria (1483–1546), another Dominican theologian, discussed whether the apostles received jurisdiction directly from Christ or through St. Peter, and in that context he says that the origin of a bishop's authority is immediately from Christ, then clarifying in his Relectiones that the twelve apostles were the first and only persons to receive the power directly from Christ as an extraordinary personal privilege that died with them. While he speculates that one territorial bishop could appoint another of that province without waiting for a papal mandate, he concludes like other theologians that if no canonical mission is granted or recognized by the pope, it would be null and illicit, as the authority of jurisdiction now depends on the see of Peter. St. Robert Bellarmine (1542–1621) came to the same conclusion about the issue.
“[...T]he Apostles themselves received their jurisdiction from Christ, but for the future, bishops were to receive it from the successor of Peter.”—St. Robert Bellarmine, De Romano Pontifice, Book IV, Chapter 24.
Bellarmine teaches that not only the particular assignment, but the power of jurisdiction itself is contingent on a pope. There is a three-fold power in the Bishop: the power of Order, the power of Jurisdiction, and the assignment of a particular place. These last two are not to be exercised without the approval of the Pope, as the magisterium of the Church expresses. The Redemptorist founder St. Alphonsus Maria de Liguori (1696–1787) also taught in his manual of moral and dogmatic theology that “the power of jurisdiction is given immediately by the Roman Pontiff,” and the Jesuit Cardinal Louis Billot (1846–1931) wrote that “it is impossible to conceive of an ordinary power over a portion of the flock except by assignment from him.” This constant teaching is practically settled with the encyclicals of Pope Pius XII. Writing to the true Catholics in China following the Communists' illicit consecrations, Pius XII clearly separated the power of order from jurisdiction.
“The power of holy Orders, which is conferred by the sacred rite of consecration, is distinct from the power of jurisdiction... Jurisdiction is given to bishops only through the Roman Pontiff, as We recalled in Our Encyclical Letter Mystici Corporis.”—Pope Pius XII, Ad Sinarum Gentem, 1954
Up until the year he died, in his encyclical Ad Apostolorum Principis (1958), Pope Pius XII reiterates that bishops who have not been named or confirmed by the Holy See have no power of governing. Any of the sedevacantists now operating without a papal mandate (called episcopi vagantes) have no right to claim ordinary jurisdiction.
Episcopal Consecration confers the potestas ordinis, the power to ordain, as bishops are the usual ministers of the sacrament of Holy Orders, by which men are made priests at the laying on of hands. Only a valid papal appointment can confer the auctoritas jurisdictionis, the moral and legal authority of jurisdiction to command, judge, and teach a group of the faithful. This appointment by the pope to judge a group of the faithful is tied to a tangible part of the world, even if it is a vacant diocese somewhere on the planet. Monsignor G. Van Noort explains that.
“Ordinary jurisdiction is, by definition, bound to an office. But an episcopal office cannot exist abstractly. Christ willed that the Church be ruled by bishops tracking localized flocks. Therefore, when the Pope grants a canonical mission, he simultaneously creates the office by carving out a piece of the earth, and then infuses that office with authority. Consecration gives a man the fitness to rule, but the Pope gives him the territory to rule.”—Gerardus van Noort (1861–1946), Dogmatic Theology, Vol. II: Christ's Church
If a bishop received his jurisdiction immediately from Christ via his consecration as a matter of divine law, the Pope would have no authority to strip a bishop of his jurisdiction based on merely prudential decisions as has happened in Church history: for example, Pope Pius VII forced the resignation of all the bishops in France after agreeing to the Concordat of 1801 with Napoleon I and Pope Benedict XV in 1919 compelled the bishops of Strasbourg and Metz to resign after the Prussian handover of Alsace and Lorraine.
With the teaching of the Church being so clear and settled, it is a sign of extreme bad will that there would be any bishop claiming to be traditional who would arrogate to themselves the power and authority of ordinary jurisdiction. They might have been consecrated under the principle of epikeia to do the things necessary for people to go to heaven, but they do not have any part of the world they rule nor do they have the authority which comes with having ordinary jurisdiction; each Catholic who does not acknowledge the papal claimants since 1958 or 1965 must decide based on their individual prudential decision where they go to Church on Sunday and other things like those, so long as they follow the teachings of the universal Catholic Church, or those in force in their region just before the death of the latest Pope.
Contrary opinions, claiming that bishops are granted jurisdiction by virtue of their consecration “to teach and to govern the entire Church,” were floated at preparatory congregations before the Vatican Council of 1870, but these were never made a teaching of the Church. Among the followers of Vatican Council II (1962-1965), the destructive error has prevailed: Lumen Gentium, one of its principal documents, taught that episcopal consecration “confers the office of teaching and governing,” paving the way for the heretical idea of collegiality, in which not only the Pope, but also all bishops also possess supreme and permanent power over the universal Church by virtue of their consecration.