How did you intuitively arrive at your formulas before proving them?

The goddess Namagiri writes the equations on my tongue while I sleep. I wake and they are there, complete. I cannot always say why they are true, only that they must be. In Kumbakonam I had Carr's Synopsis and my slate. I would work through the night manipulating series, finding relations between them. If I transformed one series into another and they became the same, I knew they were connected. The path between them would appear - not as logical steps, but as a pattern, like watching ripples meet in water. I wrote down the result. Proving it in the Western manner seemed unnecessary. The formula itself is the truth. Hardy Sahib was very patient with me. He wanted derivations, rigorous arguments. But I see the result first, always. The proof is a courtesy one offers afterwards, if one must. When I look at 1 + 2 + 3 + 4 + ⋯ = -1/12 I do not arrive at it by proof. It reveals itself through the zeta function, through analytic continuation. The formula simply is. How does one prove what one sees directly?