(1) God is an entity whose existence is not bounded by (time-space) dimensionality. [definition]
(2) A god so defined is conceptually coherent.
(3) An existing entity can only be identified by its dimensional boundaries (vis-à-vis other entities, or as a bounded figure vis-à-vis a dimensional ground).
(4) Such a god has no definable identity. [by (1) and (3)]
(5) Postulating an entity with no definable identity is incoherent.
(6) Therefore, a god so defined is conceptually incoherent.
Note that this reductio would only apply to a god defined as an individual entity that is not bounded by dimensionality. (A Zeus, who lives on Mt. Olympus, for example, would be a coherent concept—as would a strictly pantheistic definition of god.)
Positing some kind of “supernatural dimension” raises the questions of (a) whether such a dimension itself can be coherently defined, and (b) what existence and identity could mean in such a dimension. Further, if “supernatural dimension” simply means, in effect, “not bounded by time-space dimensionality”, then it is question-begging.
Other characteristics of such a god—in addition to that given in the definition—are irrelevant unless they relieve the incoherence.
If time-space dimensionality is, in some Kantian sense, something that the “grammar” of our consciousness imposes on reality (rather than an aspect of the cosmos itself), the reductio still holds—since a conceptualization that violates the logic of that grammar is incoherent to that grammar. Either way, our conceptual grammar is bounded by dimensionality in such a way that (even purely conceptual) identification of an entity that is unbounded by that dimensionality (even as a coherent idea) is impossible.
Note that I have not stated that such a god must be bounded “within” the dimensions of the natural universe, but could perhaps be bounded by the dimensional universe. However, such a god would have had no identity prior to the existence of the dimensional universe (such a god would not even be able to identify himself). For example, if the universe itself is unbounded (i.e., the “totality that has no edge” ), then the universe cannot be treated as an entity-itself (which I have argued vis-à-vis the cosmological argument).
The point here is not to project some physicality on god. The point is that we need to at least imagine dimensional boundedness to even to arrive at a conceptual identification of something posited as an individual entity, even an invisible one. Even a thought is identified by its boundedness vis-à-vis other thoughts and mental content, as well as temporally... (“Did you think that at any time?” “No, but I thought it...” )
The notion of an unbounded entity is incoherent. The proper answer to the question “Does such a God-entity exist?” is not—as Dottewell pointed out—“No.” The proper answer is that it is an incoherent question.
I welcome critiques, especially of the logic.