Dear Angelus,
Does a Sacrament subsist in the rite of the Sacrament? In mathematical language: sacrament < rite of sacrament.
Yes, the Sacrament is a subset of the entire Rite. But some parts of the larger Rite are necessary to establish proper ministerial intention, as is explained in
Apostolicae Curae.
The validity of the Sacrament depends on matter, form, and intention of the minister.
The Church has defined matter and form pretty clearly over the years. If those change in any way from what has always been done, the Sacrament is at least doubtful and should be avoided (except in case of some extreme necessity).
Determination of the intention of the minister is less obvious. But the quote from
Apostolicae Curae and the quote from Trent explain the essence of the standard. If the Church uses a "received and approved" Rite for the conferral of Sacrament, it is automatically trusted and valid.
But when the surrounding Rite is changed from what has been transmitted by perennial Tradition, doubts are introduced if the changes to the Rite are substantial, as they were in the case of the Anglican Orders and in the post-VII changes.
What is most important to understand is not the academic question of binary validity/invalidity. Rather, you must know that a Catholic is told as a moral imperative to avoid doubtful Sacraments. The standard to follow is to avoid any Sacrament in which there is "positive doubt," meaning you have objective evidence of tampering, as in the case of the changes to Rites of Holy Orders in 1968.
Here is what the
Moral Theology says about receiving doubtful Sacraments:
678. We are obliged always to follow a safe course, that is, not to expose ourselves to the danger of formal sin (see 249, 258); but Tutiorism errs when it teaches that we are also obliged always to follow the safer or safest course, that is, never to expose ourselves even to the danger of material sin. There are cases, however, when we are obliged (because some law requires it) to follow a safer course, that is, not to expose ourselves or others to some great harm. Thus, we must follow the safer side in the following cases:
....
(c) when there is question of the validity or invalidity of a Sacrament, for the virtue of religion requires that the Sacraments be administered with fidelity, and be not exposed to the peril of nullity. Example: It is not lawful to consecrate matter that has probably been substantially adulterated;
....
679. In emergency one may expose a Sacrament to nullity by taking a course that is less safe for the Sacrament, but safer for the subject, relying on the axiom that the Sacraments are for men, and not men for the Sacraments. Example: Titus is called to baptize the dying Caius. No water can be procured except rose water, whose sufficiency is doubtful. Titus not only may, but should, use the doubtful matter, since no other can be had.