Sorry to play gotcha with you but your theorem proves what I have always said about your work. I am not saying that your theorem is incorrect as your post above seems to express.
What I am saying is that if we accept your theorem as correct, as I do, then what it says is that the unknowable data must follow the same rules as the known data.
Let me give you and the rest of the readers an explicit example:
Suppose the known data only involved the collection of photons. Then the rules that apply to the known data are Maxwell's equations.
So according to your theorem, the unknown data must be "constrained", to use your terminology, by Maxwell's equations. Then the equations derived should be Maxwell's equations.
For the moment lets neglect the fact that Maxwell's equations apply to fields and not photons.
Back to the general case:
Now in the derivation following F=0 you set the rules for your known data to have certain symmetries. Therefore the equations that you derive are consistent with those rules, which in this case are symmetries.
The derived equations cannot apply to any data that are inconsistent with those rules as stated by your own theorem.
For sure the derived equations do not apply to any or all sets of rules.
Now as before you will just say that I do not understand, or deflect the argument by some other means.
But I am not concerned with what you think. The argument based on yout theorem is so straight forward that any reader on this forum will understand it's implications regarding your work regardless of what kind of a smokescreen you put up. Your own theorem proves your work is limited by the set of rules you assume.
Now back to the photons. Your theory is able to derive Maxwell's equations but is not able to derive the behaviour of a photon. In other words your theory applies to fields but not to particles. It says nothing about how fields collapse into particles or about any of the other standard interpretations of quantum mechanics. It is quite far from being a theory of everything.
This is great fun,