Kevin Wong offers some astute criticisms:
You wrote: "For one thing, wholes depend on their parts for their existence, and not vice versa. (Unless you thought of parts as abstractions from the whole, which the Persons could not be.) Parts are ontologically prior to the wholes of which they are the parts.This holds even in the cases in which the whole is a necessary being and each part is as well." Chad M. seems to be following William Lane Craig. Craig's partner-in-crime is J. P. Moreland, who argues that with substances, the whole is metaphysically prior to its parts. For example, a heart has its identity only because it is a constituent of the human person. Removed from a human person, it ceases to be a heart.
If a concrete particular such as book counts as an Aristotelian primary substance, and it does, then I should think that the book as a whole is not metaphysically/ontologically prior to its (proper) parts. In cases like this the whole depends for its existence on the prior existence of the parts. First (both temporally and logically) you have the pages, glue, covers, etc., and then (both temporally and logically) you have the book. If, per impossibile, there were a book that always existed, it would still be dependent for its existence on the existence of its proper parts logically, though not temporally. So it is not true in general that "with substances, the whole is metaphysically prior to the parts."
But a book is an artifact whereas Kevin brings up the case of living primary substances such as living animals. The heart of a living animal is a proper part of it. Now does it depend for its existence on the whole animal of which it is a proper part? Is it true, as Kevin says, that the heart is identity-dependent on the animal whose heart it is?
I don't think so. Otherwise, there couldn't be heart transplants. Suppose Tom, whose heart is healthy, dies in a car crash. Tom's heart is transplanted into Jerry whose diseased heart has been removed. Clearly, one and the same heart passes from Tom to Jerry. Therefore, the heart in question is not identity-dependent on being Tom's heart. In principle if not in practice, every part of an animal can be transplanted. So it seems as if the whole is not metaphysically prior to its parts in the case of animals.
Accidents and Parts
Tom's smile cannot 'migrate' from Tom to Jerry, but his heart can (with a little help from the cardiologists). This is the difference between an accident of a substance and a proper part of a substance. If A is an accident of substance S, then not only is A dependent for its existence on a substance, it is dependent for its existence on the very substance S of which it is an accident. This is why an accident cannot pass from one substance to another. The accidents of S cannot exist apart from S, but S can exist without those very accidents (though presumably it must have some accidents or other). So we can say that a substance is metaphysically prior to its accidents. But I don't think it is true that a substance is metaphysically/ontologically prior to its parts. The part-whole relation is different from the accident-substance relation.
So as far as I can see what I originally said is correct.
Further, you wrote, "The divine aseity, however, rules out God's being dependent on anything." Would it not be more accurate to say that divine aseity is the thesis that God's being is not dependent upon anything external and distinct from himself? If that is the case, the dependence of God (proper) upon his members (the Father, Son, and Holy Spirit) would be a dependence upon nothing external to himself (unlike a Platonic rendition of God which postulates that God is dependent upon the properties he instantiates, these properties being external to himself). There is a strong strand in Christian tradition that states that the Son is God of God, that he is begotten of the Father and yet retains full divinity. If his divinity is not in jeopardy because of dependence upon the Father, why should the one God's divinity be in jeopardy because he depends upon the members of the
The reason I said what I said is because it makes no sense to say that God is dependent on God. God can no more be dependent on himself than he can cause himself to exist. I read causa sui privatively, not positively. To say that God is causa sui is to say that he is not caused by another; it is not to say that he causes himself. 'Self-caused' is like 'self-employed': one who is self-employed does not employ himself; he is not employed by another.
For you, however, God can be said to depend on God in the sense that God as a whole depends on his proper parts, the Persons of the Trinity. The problem, however, is that you are assuming the mereological model that I am questioning. You are assuming that the one God is a whole of parts and the each of the Persons (F, S, HS) is a proper part of the whole.
And isn't your second criticism inconsistent with your first? Your first point was that a whole is prior to its parts. But now you are saying that God can depend on God by depending in his proper parts.