sorrge wrote:You need to provide some arguments/thoughts/proof sketches, otherwise it is just a collection of, at best, observations, and at worst, random statements.
Well, a few come to mind.
For the last one, say, for a p5, there are 5 states 3 bits (rotor of 3 cells). However, this must be dependent on the environment because it would be dependent on a particular stator: basically, the highest number of combinations is required for an idealised p8 in 3 cells (which I think has not been discovered anyway, but it serves as an abstraction).
Since the survival & birth conditions must depend on the surroundings and
1) in a 3-cell rotor p8 all states are covered;
2) in a 3-cell rotor, say, p5, more than 8/2 (more than 4) states are covered;
any change in surroundings does not allow the rotor to exist anymore in the following way:
1) for a "saturated" 3-cell p8: all states are covered, so if a change in the environment allowed any combination for a 3-cell rotor, this implies that all states are accepted, which would, in principle, not be possible, because every state would require a precisely defined stator environment to follow the sequence.
2) for an "unsaturated" 3-cell p5, it gets more complicated, because the environment technically has degrees of freedom, but since there is no possible lower boundary for the number of rotor cells, this really should not be possible and should, in some way, be related to the previous case. The main idea is that for this case, since 5 > 2^(3-1), if the stator changes, the rotor cannot change, since it allows a maximum of 8 combinations, so any other p5 would share one state with the previous p5, which can only allow one p5 to exist.
Hence, the corollaries would be:
- for any given totalistic rule, there is one single environment that stabilises an n-cell period 2^n rotor.
- for any given totalistic rule, and for each period P there is only ONE possible oscillator with an n-cell rotor for P > 2^(n-1) [trivial]
EDIT:
I think this probably also happens for non-totalistics and rotor cell count > 1.
Actually, this could potentially imply that there exist no "saturated" oscillators for periods > 2 in any isotropic rule at all (I might well be wrong).