Proof One

Suppose God is only part of the Universe.

Let us call the irreducible information content of the non-God part of the Universe $I_N$, the irreducible information content of the God part of the Universe $I_G$, and the irreducible information content of the entire Universe $I_U$ (where of course any of these quantities could be infinite).

It is obvious that:

$$I_U \equiv I_G + I_N$$ (1)

as the irreducible information content of the whole must match the total irreducible information content of the disjoint parts.

Also,

$$I_N \not\equiv I_G$$ (2)

as the non-God and God parts are disjoint and cannot be mutually reducible.

Clearly,

$$I_G \equiv I_U$$ (3)

as that's the meaning of omniscience.

Therefore:

$$I_N \equiv \emptyset$$ (4)

The only way the irreducible information content of the non-God part of the irreducible information content of the entire Universe can be encoded in the irreducible information content of the God part is if the irreducible information content of the non-God part is null - no information at all.

No existing system has null information content (this is why we needed the principle that a physical system is its own minimal encoding, that a system ``knows'' its own state in an incompressible way), so we conclude that if God exists at all, the non-God part of the Universe is (quite literally) no-thing, nothing. It does not exist¹⁰.

For readers already familiar with (information or physical) entropy, the proof is largely unnecessary. All that is required is to note that entropy is an extensive quantity, one that scales with system size, one that is the log of the missing information required to precisely ``know'' the state of the system. Any disjunctive partitioning of an irreducible system therefore introduces entropy of the whole relative to any nonzero part.

There are numerous ways to illustrate this entropy. Some of these ways are quite literally ``ancient wisdom'' of a sort; others are quite modern. Before we proceed to a second proof of this theorem that proceeds from very different grounds, let us look at a few of them as they may be more compelling to a lay person who doesn't have a good idea of what ``entropy'' actually is.