We derive a closed-form expression for the Wigner distribution function (WDF) of kaleidoscope-Gauss (KG) beams, a discrete Helmholtz¿Gauss (HzG) family formed by sampling the spectral ring at N equally spaced angles with arbitrary complex weights. We apply the expression to obtain the transverse intensity and power spectrum, the screen marginal, and the local flux field, as well as general formulas for the higher-order moments of position and momentum in terms of generating functions. We also introduce two new marginals, a radial projection and an angular projection, which can be mapped onto a torus, and establish the conditions for discrete rotational invariance of the WDF. Finally, we show how to map the results to the N ¿ ¿ limit, in which the KG phase-space framework converges to the general HzG theory. In all the analytical expressions of the WDF and its marginals, we confirm that the Wigner distribution consists of a superposition of functions that oscillate in space and are monotonic in momentum.
