As you well know, this is not a scientific question. One can ask for evidence (for or against) a scientific theory, but never proof. Are you parodying creationists and their constant bellowing for "proof," i.e. certainty?
Can you show me how the theory of gravity can be used to calculate which slit a bucky ball will fall through in the slit experment?
According to Newton, it will fall through the one it's above. According to Einstein it will fall through the one that intersects the geodesic it will follow. Is this a trick question?
"Fully Deterministic" could be a pretty bold claim--is it in this case?,
Yes, both are fully deterministic. Surely you're familiar with the claim that a complete knowledge of the positions and velocities of all particles completely determines both past and future in Newton's physics.
... can you tell me how the orbital velocities of the electrons were measured?
A web search found this. Maybe you could follow up on your own. It's just something I read about some time ago.
F=mA is just symbols in a row without some philosophy attached to it.
Maybe now we're getting to the issue. I'd say that they are just symbols in a row without some interpretation attached to them. In mathematics they call the interpretation a model of a theory although usually it simply relates the symbols to something in another mathematical theory. You seem to agree with this meaning given your next sentence. Are you calling an interpretation or model a philosophy?
I'm curious what elegant criteria you have discovered that tells you that some things that seem to be highly abstract statements about the way the universe is organized are philosophy, and some are not?
It's not so elegant. If it is a statement subject to falsification then it is not a philosophy. On the other hand, a statement about the form a scientific theory should take is philosophical.
A theory is not a material entity; it is not a tangible force; it is not an part of a formal mathematical proof
In fact, a scientific theory is a mathematical theory plus a physical interpretation of the theory's undefined terms. It is applied by making deductions from the mathematical theory and relating them to the world through the interpretation.
Do you think the parable of the cave is not philosophy?
I'm only passingly familiar with it, but I would say that, as I understand it, it is at least not a scientific claim. For it to be so it would have to say something about what is not possible to observe. So if it must be either scientific or philosophical or both, I'd have to conclude that it is philosophical.
I guess. It goes through both slits simultaneously--a determinist might conceivably find this a tad annoying.
What is the mathematical theory at the heart of evolutionary biology?