Proof? I don't need no stinkin' proof.
In all seriousness, Goedel's theory says that there is no system we can devise that can contain all the truths of mathematics. There will always be some theorem that can neither be proven or disproven outside the system. God as defined by the SGGS is infinite and timeless. Therefore since God IS the infinite system and mathematics IS the language of science, there's no way to prove God scientifically. In other words, there is nothing beyond God to prove, and mathematics will always leave something outside the system to prove.
Additionally, the SGGS states that God can only be obtained "through the Guru's grace." There is no other way to approach God. Science can't do it. Logic can't do it. Only faith, meditation/prayer and service, depending on how your spiritual code defines its approach to God.
Another approach: I cannot directly prove that my spouse loves me. That is, I cannot put some kind of "probe" in her brain that would output "S. loves J." Instead, I rely on effect instead-- she takes care of me. She holds me and kisses me. She will be there when I am sick, she will be there when I am well. She tells me in words, she shows me in acts. That is all I can rely on for scientific proof.
Same thing with Waheguru-- I can feel WGji's love for me in the ambrosial hours. I feel psychologically stronger when I meditate on the True Guru. I feel calmer. I feel forgiven, loved, taken care of when I submit to the grace of the True Guru. It is far more subtle than the love of my spouse, but it is real, and it is deep, and it is incredible.
I return my spouse's love by returning all the care she shows for me. I return my Waheguru's love by doing what WGji asks of us in the SGGS.