Is there any theorem that puts limits on how irregular the cycles can be in a general case of tape length L with N options corresponding to N outputs from each graph node?

What has to be constant is the cycle length for (node, tape instruction #) pairs, because all the state to determine every future step is based on the current node and tape position; if both are the same, the path will be the same as before. I think, at least without advanced math, the only thing to rely on for "true" loops is identical pairs of (instr #, node), not just instr # or tape steps or node individually.