Really, really hating to come on this...
We observe indirectly the fact of a deeper reality with the principle of entanglement
No we don't, we see quantum mechanics at work.
We know there is some sort of hidden mechanism that can cause action at a distance.
No, there is not a hidden mechanism - this was dismissed long ago - there is quantum mechanics. QM is the mechanism.
And there is no "action at a distance" in the usual sense. There is certainly no superluminal transfer of anything...
First, to claim that physical things always must have matter is unproven and probably false. it appears that physical things exist in an undefined state that gives rise to specific form, we call matter, but that prior to that specific form, to say the physical thing has matter is conjecture. The physical thing has the potential for matter in various forms.
This makes no sense.
Secondly, the principle of entanglement shows that there is something about physical reality that is hidden and from our perspective acts superluminally
No it does not.
Thirdly, since science is dealing already with this arena, it seems wrong to claim "material" must contain matter
I didn't see anyone claim that material must contain matter. I saw Percy repeatedly state that matter and energy constitute material. But all of this is a very naive appreciation of the concept of material from the POV of theoretical physicists.
The fact is we already play in the arena of the "deeper reality", looking at moduli spaces, the "reality" of the Lagrangian, superspace methods, etc. We don't look for the explanations of entanglement... it's already explained by QM. And this deeper reality does not contain anything close to what I call spirit or spirituality... unless you call general wackyness and disassociation with everyday concepts of reality, spirituality. I don't.
I would be more than delighted to see evidence of God in theoretical physics... the fact is it is not there, or at least no more there than it is in observations of the heavens, views across the Alps, the sight of my 2yr old asleep.
Additionally, if anything observed or that contains energy is material, then if God or angels exist, they are material and within the scope of science
This does not follow at all. Does God contain "energy"? Energy is purely a local function of this universe, and simply an expression of deep symmetry. Observations of God or other supernatural apparitions could easily be internal mental observations.