Papers · Preprint
Six Birds Verified II: The Memory Monoid
Loops that change nothing visible now can still change the future. This sorts them into five types.
In plain words
A system can remember something its present state does not show. Picture a device with a hidden bit. Flipping it changes nothing you can see now, but some later measurement will read it. This paper takes all the loops that leave the present reading unchanged and asks what they do together to the future. Their combined actions form a finite algebraic object, which the paper calls the memory monoid. A monoid here is just a set of actions that can be chained, with a do nothing action included.
The main result is that every interface has exactly one of five types. Flat: nothing is hidden. Inert: something is hidden but no loop touches it. Aperiodic active: loops act, but every action eventually settles. Transiently group bearing: some action cycles, but the cycling dies out under maximal compression. Kernel coherent: cycling survives even then. A four question decision tree reads the type off the monoid together with the map from future classes to present ones. Measuring single loops, such as how many states one loop moves, cannot decide it.
There are surprises. Two actions that each settle after one step can together produce reversible, cycling behavior, and four states is the smallest system where this happens. Every finite monoid that respects the present reading is the memory monoid of some system. Merging states can destroy cycling but never create it. And one monoid can need words of length two or three to reach full compression, depending on which generators are chosen. Most results are checked in Lean 4, with some finite counts done by exact enumeration outside it.
What it shows
- A five type classification of hidden memory, decided by a four question tree.
- Single loop measurements cannot tell the types apart.
- Two settling actions can generate a reversible cycle, and four states is the minimum.
- Every finite monoid that respects the present reading actually occurs as a memory monoid.
What it does not claim
Everything is about finite systems, and exhaustive finite checks are not theorems about infinite ones. The algebra of finite monoids is classical and credited; what is new is its exact use at this interface.
Cite
Tsiokos, I. (2026). Six Birds Verified II: The Memory Monoid. Zenodo. https://doi.org/10.5281/zenodo.23097912