Proof.

Why hasn't God demonstrated his existence in "terms" which are incontrovertible?

God could present himself on the earth or in the heavens in all his glory for all to see and speak to us making his existence unquestionable. Carl Sagan (an atheist) asked why God does not just put a huge cross in the sky if he is real. Why doesn't he? It wouldn't be difficult for him. He created the heavens and the earth. Why does God not make himself obvious?

Love. Love is the answer. God is love (1 John 4:16). Love cannot be coerced. It cannot be forced. The only way for people to love God is if they have the opportunity to not love God. When I first met my wife, if I pointed a gun at her and said you will love me at this moment or I will shoot, she would have been unable to love me even to save herself. We cannot truly love God unless we have the opportunity to reject him. God loves us and wants us to love him. This has to be our choice by definition. In order for it to be our choice, God could not present the evidence for himself in incontrovertable terms.

Having said that, God has revealed himself to all people in atleast amounts that are enough to result in belief in him. Man has no excuse for rejecting him for the heavens declare the glory of God (Romans 1:20, Psalm 19:1).