Does the fusion history of primorial gap cycles contain information beyond the current sieve state?

Consider the reduced residue system modulo a primorial

$$P_k=p_1p_2\cdots p_k.$$

Let

$$R_k=\{r\in\{1,\ldots,P_k\}:\gcd(r,P_k)=1\},$$

ordered cyclically, and let $G_k$ be the cyclic sequence of gaps between consecutive elements of $R_k$.

For example, for

$$P_k=30=2\cdot3\cdot5,$$

the reduced residues are

$$1,7,11,13,17,19,23,29,$$

giving the cyclic gap sequence

$$G_{30}=(6,4,2,4,2,4,6,2).$$

When the next prime $q=7$ is introduced, one can construct the gap cycle modulo $210$ by:

replicating the $30$-cycle seven times; deleting positions divisible by $7$; whenever a position is deleted, fusing its two adjacent gaps.

Thus locally a deletion performs

$$(a,b)\longrightarrow a+b.$$

Now suppose that instead of retaining only the resulting gap $(a+b)$, we retain its fusion history: which gaps produced it, at which primorial stage, and recursively the histories of those parent gaps.

This produces an enriched state

$$M_k=(P_k,G_k,H_k),$$

where $H_k$ is the genealogy of the gap-fusion process.

My question is:

Does this fusion genealogy $H_k$ contain any mathematically non-redundant information once $P_k$ and the complete current gap cycle $G_k$ are known?

More precisely, can two different fusion histories lead to the same current state $(P_k,G_k)$, while producing distinguishable behavior under later primorial-sieve transitions?

Equivalently, is the current gap cycle a sufficient state for all future transitions, or can the genealogy define a genuinely finer dynamical state?

If the genealogy is redundant, I would also be interested in knowing whether this follows immediately from a known result about Eratosthenes' sieve, wheel factorization, reduced residue systems, or cycles of gaps.

My motivation is to understand whether treating the sieve as a dynamical system with memory adds mathematical structure, rather than merely providing another representation of information already contained in the current residue/gap configuration.