Wallis Products from the Four-Dimensional Singular Harmonic Oscillator

We present a variational derivation of the Wallis product and its reciprocal from the four-dimensional singular harmonic oscillator. The inverse-square interaction is absorbed into an effective angular parameter ν, so that the lowest exact energy in a fixed sector is E_4d,exact=hbarω(ν+2). Motivated by the radial Kustaanheimo--Stiefel relation r=ρ² between the four-dimensional oscillator and the three-dimensional Coulomb problem, we use the quartic trial family R_a(ρ)=Nρ^νe^-aρ4. The minimized variational energy yields an accuracy ratio governed by adjacent Gamma functions. In the large-ν semiclassical limit, this ratio approaches unity. Restricting ν to the odd sequence ν=2n-1 gives the standard Wallis product, whereas the even sequence ν=2n gives its reciprocal form. The Coulomb-dual interpretation further relates the two branches to integer and half-integer effective angular sectors in the dual Coulomb/MICZ description. The result shows that Wallis-type infinite products persist under an inverse-square deformation of the oscillator and arise from a common Gamma-function structure in radial variational dynamics.