As I was reasoning about this puzzle, I hypothesized that the main difference between Cycles and Period would be fractional P. So I started approaching the the puzzle on TI-ish terms.

In the end, those early fractional P ideas fizzled because they completely deconstructed the input. That meant they needed a bonder, and there just wasn't space.  But this solve lineage went a different direction, and achieved fractional P by "mistiming" arm activations in an otherwise mundane algorithm.  No TI tech at all.

This machine processes one product per loop, but alternates which product gets built. 1/2P if we consider a pair of outputs as one, or 1P if we consider them separate.