I'll assume your question was seriously meant and was not merely rhetorical, attempting to make polemical points in support of some position or other. And so I'll not merely repeat my own opinion (or anyone else's.
Taking your possibilities in reverse order:
***Theology -- No. Theologians are concerned with understanding the often obscure issues which arise from Revealed Truth or from sacred Scriptures. This requires close reasoning, but it's all analytical reasoning, not synthetic, for outside information is not allowed, being fundamentally irrelevant. There has been no test, in modern times, to determine whether this or that theological analysis/understanding is wrong or right by the relevant Higher Standard. Which may acocunt for the particular bloodymindness of internicene disputes (eg, in N Ireland between Catholic and Protestant Christians, in Iraq between Sunni and Shia Muslims, in Communist countries between one variant doctrine and another (eg, Trotskyite and Stalinst Communists), or between True Believers (most everywhere) and those who don't also believe (eg, Pol Pot's Khmer Rouge and anyone who could read, wore glasses, or did anything except rural farm work, or non-believers in the Spanish version of Catholicism under the Inquisition, ...), or in parts of Greece / S America / Burma (Myanmar) / Iran / Afghanistan in recent decades between authoritarian regimes and any suggestor of different policy, or in mainland China between officially permitted belief and those holding a non-permited one.
***Psychology -- Depends on the 'psychology' meant. For clinical psychology (and the assorted theoretic structures associated with it), the answer is no. They have, like theology, no method of testing an anaalytical (or clinical) conclusion against reality. Again, close reasoning is involved, but it has no connection to the underlying phenomena, and no good account of whether the phenomena observed are in fact relevant to any clinical problem, including the one under clinical consideration. The other psychology, experimental psychology, is a type of science and the answer for science (below) applies.
***Science -- Yes, certainly it could. Well sort of, science not being in the proof buisness. But...
Science requires an experimentally testable thing, or it can say nothing whatever about that thing. Thus, I can scientifically investigate the predicted water solubility of a new chemical by conducting an experiment. The result of a properly designed and conducted experiment is definitive as to whether some theory should be taken seriously thereafter. In short: Theory's prediction wrong, theory gets dumped. End of discussion, in principle. But no one can currently experimentally investigate any claim about alien life living on Kuiper Belt Objects well beyond the orbit of Pluto in our Solar System. Perhaps someday, but until then, all such claims are speculation, not science nor scientific.
Experimentation with regard to mental states is notoriously difficult to design and conduct. Assorted ad hoc attempted experimental tests of the horse Clever Hans (late 1800s Germany) all foundered, as it were. It took an especially careful series of experiments to demonstrate that Clever Hans was responding to his owner, who could do simple arithmetic, and was somehow pasing clues to the horse. It was vision (established by experimental tests for odor, noise, etc) that was involved, as Hans' math skills consistently deserted him when his owner was not visible, or when he couldn't hear the math problem, if visible. Millenia of close observation of house cats has not advanced human understanding of their mentation much beyond bafflement (string?!, mysterious rushing about?, lust for green pea soup??, ...); see Fritz Lieber's story, Spacetime for Springers for a theory of house cat mentality about as good as any of the current scientific ones, and more enjoyable. Experimental psychology deals with even more awkward and difficult experimental subjects and so much of it, certainly that related to 'higher' human abilities, is on quite uncertain ground. If for no other reason than that another run of an experiment is typically impossible -- humans remember things and make conclusions (helpful or interfering) about them, thus poisoning the experimental conditions.
But the problem with scientific investigation of the nature and existence of a god might perhaps be easier. If some observation were to be made (as none have, in any experimentally accessible way, in recorded history) for which all theories save Divine Existence, were to be experimentally shown to be inadequate, then science would have established, to the extent it ever can, the existence of a god. The theologians would have to carry on from there, I suppose, as most of what remains would be concerned with Divine Intentions and Purposes. Note that history is fundamentally unscientific, as no experimental test is possible in essentially all cases; the situation is complex if for no other reason than that humans are involved, and cannot be repeated. It is on this ground that it's clear the various crime investigator shows are not really about science, in spite of the spiffy technical gear, some perhaps authentic and not props, and the impressive sounding jargon, some of which is probably accurate.
But -- note this VERY carefully -- this is the best science will ever be able to do, and it DOES NOT prove, will not prove, and cannot be proof of any kind for, the existence of a god or indeed of anything else. Science simply cannot prove anything whatsoever, and doesn't try. Anyone who claims something to the contrary is uniformed, flogging some undisclosed agenda (or peraonsl opinion), or both. This is an obscure characteristic of the nature of science for most non-scientists, so it deserves futher comment.
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a 'thought experiment'
Consider the case of any scientific result. The existence of god, perhaps, or Newton's account of the action of gravity and of motion generally including the combined case of the motion of the planets. Assume that both these results are widely accepted by scientists, and presumably by the general public as well. Anyone who proposes a theory which better accounts for the observed phenomena (burning bushes and rods turning into serpents in one case, and the motion of the planet Mercury in the other) would -- after some period of argument and disputation -- replace the previous theory (ie, scientific result) with the new one. In the case of Newton's account of gravity and the motion of the planets, this has already happened. Several times. Einstein's Theories of Relativity are, together, a better account of motion and of gravity than Newton's, and futhermore much better accounted for the observed motion of Mercury than did Newton. There are several alternatives to Einstein's Theories, none of which has, so far, improved on them for experimentally accessible phenomena; in future, perhaps it may be ofund that one of them is better, in some currently unknown, aspect. Newton's theories o fmotion and of gravity are still widely used, as a superb approsimation, for slower speeds na lower gravity fields. But that's because it's convenient, not because anyone believes it's the truth about, or rather the best available theory accounting for, motion and gravity. In the case of our imagined scientific support for the existence of a god, the same possiblity will exist. An Einstein equivalent could propose a better theory (ie, which better accounted for the observed phenomena, whatever they might be) and science would then dump its previous result and adopt the new theory. To the very likely extreme distress of those for whom it was important.
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Best advice is "Beware! tricky footing here"
One should, thus, be very very wary of accepting any scientific proof of anything whatever. A better theory might be uncovered tomorrow. Now, that said, most scientific work has demonstrated there there a considerable interconnection between results in different fields. Eg, between biology and chemistry, chemistry and particle physics, stellar evolution and particle physics, and so on. It is therefore extremely unlikely that there will ever be any scientific revolution which calls into question the major theories of science. Possible, but so is being eaten by a polar bear while standing near the Egyptian pyramids outside Cairo. So exceedingly unlikely that one should effectively behave as though ...
So no scientific establishement of the existence of a god should be relied upon by those for whom certainty in such matters is important. Science provides nothing but liklihoods, highly probable liklihoods when it works well of course, but only those. Not certainty of any kind.
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Scientific proof for non-existence of anything
There is, and can be, no such thing. Non-existent things don't exist (obviously) and so they can't be experimentally investigated. Without experimental test of theories about them, no such theory can be scientific and none has any scientific significance. Thus it will be clear that claims of scientific proof of the non-existence of a god, any god, are purest moonshine. I may be personally convinced by my scientific studies, you may be convinced by the fact the Smith, a scientist, is an atheist, and she may be certain there is no god because one was never mentioned in her school courses on science, but each of these opinions are irrelevant to science. However strenuously and persistnetly these or those attempt to hold science responsible for such opinions. They are, in fact, personal opinions. I also believe in the abominable Snowman, you believe in Ancient Astronauts, and she in Ramtha the Atlantean warrior channeled by J Z Young, None of these beliefs are scientific either, nor science is not responsible for them. Or to be blamed for them either. Except by the uninformed or those pushing some agenda (anti-science or pro-
), most likely.
The only exception is not really one, but is compelling the less. Some theory has survived all experimental testing, has deep connections with much of the rest of science, and is well accepted. If the existence of something would be a gross violation of that theory, the something should probably be taken as impossible in practice. So, Einstein's Special Telativity made predictions about the energy required to accelerate a mass (any mass) to the local speed of light. This does not prove the non-existence of an atomic particle with mass which does move at the speed of light, bu you'd be ill advised to stake your scientific future on it or anything similar. In practice, as it were.
On the question of the non-existence of a god, science has no experimental information, and there is no body of theory which holds that the existence (or non-existence) of a god is central. So far, science has managed to find good theories for much of the phenomena we humans have encounterd, and to establish some considerable correspondence between them. All experimentally. Thsu far, a god has net been required as a part of the best available theories. Perhaps in future, or perhaps something has been missed, but not so far. And that fact says nothing about the existence or non-existence of a god. Science is god neutral for that reason, whatever some may allege for or against. Impressions, understandings, convictions, and opinions on this point are often simply uninformed or wrong. It is the cause of considerable angst among the unscientific, who are taking their views selectively or from those whose views are selectively informed.