Do Catholics eat the substance of the Father during "communion"?

Just in case it helps, while transubstantiation does mean that the bread and wine of the eucharist become the actual body and blood of Christ, Aquinas, I know, argued (working from Aristotle) that while the substance of the bread and wine really does change, that needn't mean that one is actually consuming blood and flesh (as though the bread and wine change into a rare steak or something), since those qualities are "incidental" to the real substance.

The "of one substance" language isn't directly related to the Eucharist, as far as I'm aware. It comes from the Council of Nicea (325 A.D) and its ruling on the relationship between Christ and God the father, who it said to be "of one substance" (gk. h*m*ousios). The term is more related to Christology than to the eucharist.

"Consubstantiation" is normally connected to Martin Luther's take on the eucharist, who believed that the "real presence" of the body and blood of Christ exists, in a way, right alongside the bread and wine. So the bread and wine don't change substance, but the real presence of Christ is there in the sacrament.

Sorry, adding this as an addition answer since it was too long for a comment:

Eques: Sorry if I wasn't entirely clear. I am thinking of the Thomistic differentiation between substance and accident. For Aquinas, substance is the true nature of a thing--what it really is divorced from any accidental characteristic.

For example, we might see accidental variation in dogs when we see them in different colors, shapes, sizes, times and places, but it would be absurd to say define "dog" in terms of any of those accidental characteristics (if we see a brown dog on a street corner at noon, we don't, for that reason, conclude that anything that shows up wearing brown on a street corner at noon is a dog).

Aquinas makes that distinction between the substance of the body and blood of Christ in the sacrament and the accidental characteristics like the flavor, texture, color, etc. of flesh and blood. The substance of the eucharist is the body and blood of Christ, and that doesn't change. So, while the substance of the bread and wine really do change into the substance of the body and blood of Christ, the accidental properties don't change--hence we really are consuming the flesh and blood of Christ even though we're experiencing the accidental properties of bread and wine (which is why those are the things we taste). This is an important distinction because if we, by this reasoning, were only looking at a transformation from flesh and blood into bread and wine, the real presence of Christ wouldn't be there in the sacrament; only accidental properties will have changed, like turning a chocolate lab yellow. It's also sort of important to our own sensory experience, since we know we're tasting bread and wine. (It's also part of the difference between the sacrament and cannibalism!) Hope that's more clear.

You have to take into account at the same time two dogmas, namely those on:

1. Incarnation (if considered on fieri) or Hypostatic Union (if considered in esse), and

2. Transubstantiation.

According to the first, the Incarnate Word has two substances or natures, the divine and the human, united in one divine Person. BTW, the first definition of the 5th Ecumenical Council of Constantinople 553 explicitely identifies ousía (substantia) with physis (natura) and hypostasis (subsistentia) with prosopon (persona) [1].

According to the second, the substance of the bread is converted into the substance of the body of Christ. Regarding this:

a. Since the body of any living animal (including humans) is not dry but permeated by blood, the substance of the bread is actually converted into the substance of the irrigated body of Christ, i.e. his body, as abstracted in a dry state, and his blood.

b. Since the body of any living human being is infused by a spiritual soul, the consecrated host, being the body of Jesus Christ, is infused by his spiritual soul. But the substance of the bread has not been converted into the substance of the soul. So, one does not eat directly the soul but the body infused by the soul, just as someone who hugs you does not hug your soul but your body infused by your soul.

To note, the previous two points would also be the case if the bread were transubstantiated into the body of any ordinary human person, something which God could perfectly do but which of course there would be no point in doing. I point this out to emphasize that transubstantiation has nothing to do with Jesus' body having any special property after his resurrection, as Lutherans claim about their "sacramental union". This should already be clear from the fact that the first transubstantiation occurred at the Last Supper while Jesus' body was in a passible state.

c. The soul [2] of Jesus Christ is united to his divine Person or Hypostasis, Who, as Any of the three divine Persons, is the one and only divine substance or nature.

So, just as St. Mary hugging baby Jesus was not hugging the substance of the Father but the Son of God in his assumed human nature, (speaking even more strictly, she was hugging his body, which was infused by his soul just as any living human body, with Jesus' soul being in turn hypostatically united to his divine Person,) a person receiving Communion does not eat the divine substance but the body of Jesus, which is infused by his soul, which is hypostatically united to his divine Person, Who, as Any of the three divine Persons, is the one and only divine substance or nature.

References

[1] Pavouris, Raphael (2001), The condemnation of the Christology of the three chapters in its historical and doctrinal context: the assessment and judgement of Emperor Justinian and the Fifth Ecumenical Council (553). PhD thesis, University of Glasgow. http://theses.gla.ac.uk/1503/

[2] St. Thomas Aquinas, Summa Theologica, III, q.6, a.1, "Whether the Son of God assumed flesh through the medium of the soul?" https://dhspriory.org/thomas/summa/TP/TP006.html#TPQ6A3THEP1

Transubstantiation

Transubstantiation is a Latin word consisting of trans (a transition or change) and substantia (substance, or what [makes a thing what] a thing is). Properly, then, it means 'change of what it is.'

This was naturally adopted by a Christians (the East use the Greek equivalent, however, metaousia), who believed that the words of consecration, "This is my body" were literally true of the bread, but not that this demanded the makeup of bread to disappear from it physically—it was a change by God of what the thing now is.

St. Cyril of Jerusalem's words sum this belief up quite well (Catechetical Lectures, 22):

For you have just heard him say distinctly, That our Lord Jesus Christ in the night in which He was betrayed, took bread, and when He had given thanks He broke it, and gave to His disciples, saying, Take, eat, this is My Body: and having taken the cup and given thanks, He said, Take, drink, this is My Blood. Since then He Himself declared and said of the Bread, This is My Body, who shall dare to doubt any longer? And since He has Himself affirmed and said, This is My Blood, who shall ever hesitate, saying, that it is not His blood? ... Consider therefore the Bread and the Wine not as bare elements, for they are, according to the Lord's declaration, the Body and Blood of Christ; for even though sense suggests this to you, yet let faith establish you. Judge not the matter from the taste, but from faith be fully assured without misgiving, that the Body and Blood of Christ have been vouchsafed to you.

As such, Catholics don't believe that Jesus' body is of one substance with the Father (but that the Son is). The Son is distinct from the body He assumed, given what this sentence just said. Nor do they even believe that Jesus is of one substance with His body (the Son and the body being conflated in such a case). They believe that the substance of the bread has become the substance of the body of Jesus, which is united to His soul and divinity (given that He is a real man with a soul, and the incarnate God).

In other words, they believe it is true when Jesus says, "This is my body" (not 'this bread' is my body; nor a game of charades).

(All accusations of 'cannabalism' will fall on deaf ears, too, when it comes to Catholics, since we know and have always known that there is no change in the accidents—the bread is the same to us before and after and to our body, physically. What has changed alone is the spiritual benefit when recieved, since "God has called" (Acts 10:15; Luke 22:19) it the Bread of Life, which is "[Jesus] flesh" (John 6:51). And so it is in all but appearance.)