Hierarchies within segment types and modulo loops

Collatz procedure can be analyzed in many moduli, but, for practical reasons, I tend to use mod 16 (tuples) and mod 12 (segments).

The analysis of some phenomena requires higher moduli, for instance, two phenomena that are related: hierarchies within segment types and modulo loops.

Definition (Modulo loop): Modulo loops occur when a given modulo is applied to numbers (e. g. “loop mod 16”). The focus here will be on short loops that play a significant role in the procedure, one by segment type: in mod 12, they are 4-2-1(-4) for S2EO (yellow), 4-8(-4) for S2E (blue), 10-11(-10) for SEO (green), 12(-12) for S3EO (rosa).

For mod 96, the corresponding numbers are: 4-2-1(-4) for S2EO (yellow), 64-32(-64) for S2E (blue), 94-95(-94) for SEO (green), 96(-64) for S3EO (rosa). It is easy to see that they occupy similar positions within the range: Beginning (yellow), 2/3rd-1/3rd (blue), antepenultimates (green) and ultimate (rosa).

These loops occupy the top of a hierarchy within each segment type, as visible in the figure below. For instance, a number 94 mod 96 (green) will iterate into several other green numbers before merging into a number from another segment type (on the left), located at different levels in their own hierarchy.

In mod 12, a long green sequence would appear as [10-11]-10-5, the brakets indicating a loop. These are visible in the post about convergent and divergent series of preliminary pairs, triangles and walls. The hierarchy is already at work, on a small scale: 10-11-10-5 can occur, but not 10-5-10-11.

Possible iterations by segment type mod 96