Exhaustion of s-points by finite-order s-points

Determine whether every s-point has finite order, specifically whether the full s-point set satisfies \(\Upsilon_q=\bigcup_{m\geqslant 1}\Upsilon_q^m\).

Background

The paper organizes s-points into subsets Υqm\Upsilon_q^m according to the order of the associated finite-energy q-polyharmonic unreachable state. Although finite-order s-points arise from q-polyharmonic states, the paper does not establish that every s-point must arise in this way. The unresolved question is therefore whether the union of all finite-order subclasses exhausts the entire set of s-points.

References

Does the finite order s-points exhaust the set of all s-points, i.e., is the equality $\Upsilon_q=\cup_{m\geqslant 1}\Upsilon_qm$ valid?

— Controllability, returning waves and scattering without reverberation in 3D acoustic dynamic system  (2609.03536 - Belishev et al., 3 Sep 2026) in Comments, bullet beginning “Does the finite order s-points...”