Realisation and closure under Cartesian products and disjoint unions

Determine which weighted dual skeletons are realisable by Markovian resetting processes and establish whether the subclass of realisable spectral duality structures is closed under Cartesian products and disjoint unions.

Background

The algebraic theory treats every weighted dual skeleton as admitting an abstract spectral realisation, but the paper distinguishes this formal realisability from realisation by concrete stochastic systems, especially Markovian resetting processes.

The unresolved problem is to develop a realisation theory identifying which spectral duality structures arise from such processes and to determine whether Cartesian products and disjoint unions preserve that concrete realisability. The paper proposes biased random walks on products of intervals as an initial test case.

References

Paper~V showed that these SDS induce a rigid geometry on the simplex, but did not characterise which SDS are realisable by some process. Now that we have defined the operations of Cartesian product and disjoint union, we can ask: is the subclass of realisable SDS closed under these operations? In other words, if \mathcal S_1 and \mathcal S_2 are realisable by resetting processes, is \mathcal S_1\times\mathcal S_2 (or \mathcal S_1\sqcup\mathcal S_2) also realisable? The answer would require a realisation theory for SDS, which is an open problem.

— Algebra of spectral duality structures  (2609.04430 - Coso, 3 Sep 2026) in Section 10.3, subsection “Closure of the realisable class and realisation theory”