Complete parameterization and parameter-space topology of discrete Wigner representations on $d \times d$ phase space

Abstract: Toroidal discrete Wigner functions (DWFs) for finite-dimensional systems are non-unique. We classify the labeled single-qudit ${d\times d}$ representations whose phase-point operators are Hermitian, unit-trace, Hilbert-Schmidt orthogonal, and Weyl-Heisenberg covariant. Our previously established stencil theorem expresses this family as descents of a doubled ${2d\times2d}$ parent. Here, we solve its projected-stencil admissibility conditions explicitly. In the symplectic-Fourier representation, admissibility fixes the modulus, leaving phase data on a ${d\times d}$ base cell satisfying a parity-dependent twisted-oddness relation. This gives the parameter space ${(S<sup>1)<sup>{(d<sup>2-1)/2}}$ for odd ${d}$ and $${(S<sup>1)<sup>{(d<sup>2-4)/2}\times\mathbb{Z}_2<sup>3}$$ for even ${d}$. Canonical horizontal and vertical marginals reduce these spaces to ${(S<sup>1)<sup>{(d-1)<sup>2/2}}$ and ${(S<sup>1)<sup>{d(d-2)/2}\times\mathbb{Z}_2}$, respectively. The same phase data parameterize validity-preserving rephasings of the doubled Weyl-Heisenberg displacement operators. Requiring these operators to have order dividing the Hilbert-space dimension ${d}$ replaces each ${S<sup>1}$ factor by a discrete ${\mathbb{Z}_d}$ factor. The associated symplectic-Fourier characteristic functions encode the same operator information, with pointwise magnitudes independent of the valid stencil convention. This classification separates the freedom intrinsic to DWF validity from that selected by marginal and displacement-algebra requirements and makes explicit the structural distinction between odd and even dimensions.