in a systematic investigation they do not need a clear idea of the person to eliminate other possibilities.
That can't be right, since "God" can only be a possibility if the concept "God" is the coherent concept of a real being, one that's not imaginary, one that has objective existence like your cat (I don't actually wish a dentist's bill on you). Otherwise any word that is meaningless, or denotes only something imaginary can be used "to eliminate other possibilities" ─ 'this patient is suffering from grofgrof', for example, or 'the murder was committed by a pixie'.
Your error is in saying well we don't know anything about the person so we cannot eliminate a natural cause of death.
Your error is thinking there's any coherent concept of a real person associated with the word "God". In fact only an imaginary entity, wholly unpersonlike, bald of real characteristics like a nervous system, a metabolism, sex, organs of sense, can be discerned, and to it are attributed wholly imaginary characteristics like omnipotence, omniscience, magic, or meaningless ones like 'perfection' and so on.
Cats are finite and the finite is dependent on the infinite
There are no known examples of 'the infinite' in reality. The infinite is only found in your imagination, and that includes the mathematical concept. though that at least is logically coherent.
If you think the infinite is from the finite then you are stuck with everything from nothing
As I said, there's nothing real that's either infinite or 'the infinite'. (As for everything from nothing, where do you imagine your imagined infinite came from?)
Don't sit here and say you need a definition of an Infinite Being in order to eliminate a natural cause for the universe and life here. You don't.
But the infinite being purely imaginary, I can and do instantly eliminate it from explanations of reality.
Is your existence unscientific because it cannot be falsified by you?
This conversation demonstrates my existence to you (and even if you think I'm a Turing machine, you think I'm a real Turing machine.)
Well, they are written as history, not stories. Events in space-time. The problem here is not with evidence because it is all abundant and clear to reasonable persons.
Evidence of what real thing, exactly?
Or to go back to the start, if we find a real suspect, what objective test will tell us whether [he/she/it/they/other] is
god / a god or not? And the answer is, because there's no coherent concept of a real god, there's no such test, and no such suspect.