Proof of God's existence does not exist ie A theist can never prove God's existence

A common atheistic objection is that what's the proof that God exists? And people have spent and grappled with this idea for a loong period of time. But I feel, a proof for god's existence can never exist.

There's roughly 3 ways to prove God's existence-

Experiential-Ones own subjective mystical experiences with God.

Empirical- which can be roughly divided into scientific proof for god's existence AND historical proof.

Philosophical-Philosophical arguments showcasing god exists.

And NONE of them works to prove god definitively. There's not a single full stop argument one can create out of this.

Experiential's main problem is that it is...subjective and hard to truly verify.

A christian couple prays to jesus as their child battling cancer is on her deathbed but suddenly, in few days, the child is healed.

A hindu wife prays to vishnu to save her husband, who just had a nasty accident and the husband survives.

Yet none of them can definitively prove that either Jesus or vishnu exists.

Although, oddly enough...experiential is the best shot given in Hinduism, God does respond to prayers(even though, the likelihood is so low as we are not Prahlada or Draupadi).

Second is empirical. God by itself is defined as someone outside the domain of sciences(I would plead people to read Vishnu Purana Book 1 ch 2 that gives a succinct view of god's nature). Hes completely independent, not made of "matter" and by definition is a scientific evidence not possible for god.

It's like asking what's the scientific evidence for the laws of logic. It maynot be a perfect analogy, so treat it more as a pedagogical rhetorical example to prove why scientific evidences fail.

A sub section of empirical is archaeological or historical evidences. And...the thing is- We can't prove god exists from there too. Muhammad exists as a real historical figure yet we cant infer that Allah existed. Similarly, even if we prove that bhagwan Rama or krishna existed, we still cannot definitively prove that they were vishnu avatara, there lilas were true etc etc.

It's simply not possible to prove that x exists--->x is divine.

Third is philosophical. Arguments such as fine tuning, contingency argument, "objective morality", kalam cosmological argument yet none can prove god exists simply because- A logical consistency doesn't imply it must exist in reality. It's simply not possible.

Any philosophical arguments of God shall exist necessarily can simply be countered by "brute fact".

Even the Advaita vedantins iirc, have made it clear that brahman can't be accessed by strictly logic alone.

And hence, I believe a proof for god existence can simply just not...exist.

But this DOESNOT mean that a belief in God's existence is irrational. Well that is for another post exploring why a belief in God is...rational.