Establish a collapse theorem for the stencil formalism

Determine whether the stencil formalism admits an analogous uniqueness-or-nonexistence collapse theorem under stronger coordinate-marginal and determinant-one integer lattice-covariance requirements, yielding a unique representation in odd dimensions and no representation in even dimensions.

Background

The paper compares its parameterized family with the result of Horibe et al., who obtain uniqueness in odd dimensions and nonexistence in even dimensions under stronger coordinate-marginal and determinant-one integer lattice-covariance assumptions.

The authors explicitly state that they have not proved an analogous result within the stencil formalism. Establishing such a theorem would clarify how the broad admissible parameter space collapses when those additional covariance and marginal requirements are imposed.

References

We do not prove an analogous collapse theorem within the stencil formalism here; the comparison shows why uniqueness or nonexistence must be assessed relative to the additional requirements imposed.

— Complete parameterization and parameter-space topology of discrete Wigner representations on $d \times d$ phase space  (2609.39119 - Antonopoulos et al., 30 Sep 2026) in Section 6, Discussion and conclusions

How do the independent parameters and topology change when this requirement is relaxed? This would distinguish features specific to Hermiticity from those that persist in a broader stencil framework.

— Complete parameterization and parameter-space topology of discrete Wigner representations on $d \times d$ phase space  (2609.39119 - Antonopoulos et al., 30 Sep 2026) in Section 6, Discussion and conclusions