Persistence of exact zero-dispersion values under level splitting

Determine whether exact zero-dispersion values survive after perturbing the balanced counting operator K to the split one-body operator h_ε = K + εV under the complete fermionic positivity hierarchy.

Background

The paper establishes that, for the balanced counting operator K, full order-q positivity admits a continuum of noninteger zero-dispersion values, and that a small one-body splitting h_ε = K + εV leaves a low-dispersion valley of order O(ε²) near this interval. The bounds given do not determine whether the relaxed dispersion remains exactly zero at any noninteger values after the perturbation.

The authors explicitly distinguish the persistence of a low-dispersion valley from the persistence of exact zeros. They note that examples under PQG constraints show that splitting alone need not eliminate a zero interval, but leave the corresponding question for the complete setting unresolved.

References

Whether exact zeros survive requires a separate analysis; the multilevel examples of Ref.~\onlinecite{Nakata2006} show that splitting alone need not eliminate a zero interval under PQG constraints.

— Spurious zero-dispersion continua in full $(2,q)$-positivity: A linear hierarchy-depth lower bound from an exactly solvable counting operator  (2610.09698 - Nakata, 7 Oct 2026) in Section 6.5, “Small perturbations”