I'm trying to find a way to count distinct simple cycles in a graph in order to prove that $\#CYCLE \in \#P$, if I could represent a distinct cycle, then I'll have a witness.
I saw this question: Number of cycles in a graph? they mention the reduction from HAM #CYCLE in Counting Complexity by Arora and Barak, but I don't understand how does that help.
You have some directed graph, no one said it has a hamilton cycle, and the gadget doesn't return how many cycles there in the graph or a way to distinguish between them, it just makes all the cycles longer.
So instead, is it possible to use a topological sort to count cycles? since it can detect every cycle, we can use it to distinguish between them.