Criterion for factorizations without orbit factors

Determine an explicit criterion for reducibility of fixed-point-free skeletons with a non-admissible number of orbits, including factorizations that do not contain an orbit factor, by characterizing the relevant sum-and-difference sets.

Background

For a skeleton with no fixed points and an admissible number of orbits—namely 8, 16, a prime, or twice a prime—Theorem 5.6 gives an exact reducibility criterion in terms of the existence of an orbit factor and partitions of the ratio multiset into pairs with a common center. The theorem also shows that when the orbit count is non-admissible, reducible examples without orbit factors exist.

The paper provides a finite-search procedure for deciding reducibility and enumerating factorizations, but does not derive a general structural criterion in the non-admissible case. The unresolved issue is therefore to replace the finite search with an explicit characterization formulated through the associated sum-and-difference sets.

References

Proposition~\ref{prop:general} reduces reducibility to a finite search, but an explicit criterion is a question about sum and difference sets.

— The semiring of spectral duality skeletons: cancellation, characters and factorization  (2610.23684 - Coso, 16 Oct 2026) in Section 7, item 1 of the Open problems section; Theorem 5.6 and Proposition 5.3

What is the structure of the reducible locus beyond these cases?

— The semiring of spectral duality skeletons: cancellation, characters and factorization  (2610.23684 - Coso, 16 Oct 2026) in Section 7, item 6 of the Open problems section