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If a God interacts with the world in any meaningful way, then those interactions can be measured and proven.
It may be unable to prove god is real but it is also impossible to disprove that god is real.
People all over the world have mystical experiences, especially that of an apophatic nature.
How do you define proof? Current secular science decided long time ago that any data that has the potential to take us beyond what we see will be ignored on the basis that the reality can only be defined as something that can only be observed through senses. God is beyond what we see.
God is a product of belief. Any thing that is proven does not require belief. Thus, proving god's existance is impossible.
If God, as in the traditional defition, is omniscient and omnipotent, that means that he has the power to modify the reality constantly in a way that no proof of his [non-]existence can ever exist. Therefore, leaving a constant mystery.