Apologies for the necroposting! This topic interests me greatly and I'd love to chime in. I hope I make a good first impression
:)
Here it is (in a nutshell):
1. Composed beings are made up of parts.
2. A composed being exists contingently upon its parts in their specific arrangement.
3. A part of a composed being is either composed or uncomposed.
4. A part that is a composed being does not, in turn, exist in-itself but, rather, exists contingently upon its parts and their specific arrangement.
5. An infinite series of composed beings for any given composed being (viz., a composed being of which its parts are also, in turn, composed and so on ad infinitum) would not have the power to exist on their own.
6. Therefore, an infinite series of composed beings is impossible.
7. Therefore, a series of composed beings must have, ultimately, uncomposed parts as its first cause. (6 & 3)
8. An uncomposed being (such as an uncomposed part) is purely simple, since it lacks any parts.
9. Two beings can only exist separately if they are distinguishable in their parts.
10. Two purely simple beings do not have any different parts (since they have none).
11. Therefore, only one purely simple being can exist.
12. The purely simple being would have to be purely actual—devoid of any passive potency—because passive potency requires a being to have parts which can be affected by an other.
13. No composed being could be purely actual, because a composed being always has parts which, as parts, must have passive potency.
14. Therefore, there can only be one purely actual being which is also purely simple. (11 & 12 & 13)
15. The purely actual being is changeless (immutable), because it lacks any passive potency which could be actualized.
16. The purely actual being is eternal, because it is changeless and beyond time (as time’s subsistence of existence).
17. The effect must be some way in the cause.
18. The physical parts of a composed being cannot exist in something which is purely simple and actual; for, then, it would not be without parts.
19. Therefore, the forms of the composed beings must exist in the purely simple and actual being.
20. Intelligence is having the ability to apprehend the form of things (and not its copies!).
21. The purely simple and actual being apprehends the forms of things. (19)
22. Therefore, the purely simple and actual being must be an intelligence.
23. To know the forms of every composed being is what it means to be omniscient.
24. Therefore, the purely simple and actual being is omniscient.
25. To cause the existence of a thing in correspondence to its form from knowledge (intelligence) requires a will.
26. Therefore, the purely simple and actual being must have a will.
27. To be good is to lack any privation of what the thing is.
28. The purely simple and actual being cannot have any privations, since it is fully actual.
29. Therefore, the purely simple and actual being is all-good.
30. To will the good of another independently of one’s own good is love.
31. The purely simple and actual being wills the good of all composed beings by willing their existence.
32. Therefore, the purely simple and actual being is all-loving.
33. Power is the ability to actualize potentials.
34. The purely simple and actual being is the ultimate cause of all actualization of potentials.
35. Therefore, the purely simple and actual being is omnipotent.
36. The existence of all composed things subsists through this purely simple and actual being.
37. Therefore, the purely simple and actual being is omnipresent.
38. A being which is absolutely simple, absolutely actual, eternal, immutable, all-loving, omniscient, omnipresent, omnipotent, all-good, one, unique, and necessary just is God.
39. Therefore, the purely simple and actual being is God.
40. The world we live in is made up of composed beings.
41. The composed beings must subsist through an absolutely simple and actual being.
42. Therefore, God exists.
So, this argument somewhat resembles an argument given by Ed Feser in his book
Five Proofs for the Existence of God which he names the "Neo-Platonic Proof." If I'm right in assuming that your OP is inspired by that argument, then I'm sorry to say, but you haven't given an accurate Thomistic presentation of that argument.
The thing I want to note (and this is something that @NotAristotle and @JuanZu alluded to) is that this argument doesn't conclude in a Being whose existence is separate from the world it creates, rather it is a Being who is a "part" of everything in that world. This argument is an argument from
material causality, whereas Thomistic arguments involve
efficient causality. Thomas Aquinas would vehemently reject the idea that God is a part of anything, indeed he argues against the idea that God "can enter into composition with other things" in the Summa Theologiae (part 1, question 3, article 8).
If your argument succeeds, then it would turn out that everything around us has God as a metaphysical constituent of it. Chairs, apples, trees, planets, quarks, people... all are "made of" God. And such a world would be a far cry from the classical theism of Feser and Aquinas, instead it would be more akin to some kind of pantheism.
Now, to be fair, Feser in the book does generously help himself to instances of composition when trying to illustrate the idea of causality ordered
per se. This makes it a little confusing when he speaks of God as first cause, and I wouldn't blame someone for thinking that Feser is making the case that God is an absolutely simple uncomposed part of each thing as it exists. But that's not quite what the argument states. It's rather that the arrangement of parts as they make up the whole need an external sustaining cause to exist as they do. So basically, every composite needs a composer.
That's what the argument from composition is about. It's not about every composite being composed of uncomposed parts.
Nonetheless, I do think your argument is effective. Not as a dialectical defense of Thomism or classical theism, but rather as a
reductio of the whole system. I actually agree with the argument that there cannot be an infinite series of composition and that every composed object is composed of uncomposed, purely actual parts! I'm not willing to (or at least I don't want to) grant that the purely actual part of the object must be unique in being purely actual. It seems to me that purely actual beings can differ in attributes, such as the having of the very relation they do to the whole they constitute. But in that case, we don't need an external efficient sustaining cause to explain the existence of the composite, rather we just have these "atomic" purely actual parts explaining its existence. So what we finally get in the end is a form of existential inertia where wholes persist in virtue of their parts, not classical theism.