I sound like a dessert or bodily waste depending on how you pronounce me. I am important in geometry. I have a day named in my honor. I have a very famous specific set of numbers.

Used in math but not a number. I can be golden, inverse, or direct. I remain constant when my parts change size.

I ate the one after the next - who was three by three, The count of the rainbow's arcs, The steps on music's marks.

I am a thousand threads of what is possible. When it is threaded into string, my many becomes one, undeniable possibility and the rest impossible. I am always ahead, but my opposite is always behind. What am I?

A stint of warmth for dough not risen, A clue that sends the thief to prison. The reasoning behind why math is true, Or the stiffness of the drink you brew.