A General-Dimensional Origin-Invariant Cram'er-von Mises Test for Complete Spatial Randomness

We introduce a general-dimensional origin-invariant Cramér–von Mises statistic for testing complete spatial randomness (CSR), defined as the average of corner-oriented Cramér–von Mises functionals over all $2^d$ corners of the unit cube. The statistic admits a closed-form $O(n^2)$ computing formula and is therefore directly implementable in arbitrary dimension. In dimension one, it reduces to the classical rank-based Cramér–von Mises statistic; in dimension two, it agrees with Zimmerman’s origin-invariant statistic. Under the CSR null hypothesis, the finite-dimensional distributions of the normalized empirical process converge jointly to a mean-zero Gaussian process with an explicit cross-corner covariance kernel. We establish the existence of the limiting process, construct a continuous modification, and prove asymptotic equicontinuity using Bernstein chaining. The resulting quadratic limit is characterized through the covariance operator on the corner–location product space and has a Karhunen–Loève representation. An ordered-list construction and restriction law connect the conditional CSR model with the homogeneous Poisson point-process formulation. Monte Carlo null percentiles are reported for dimensions one through five and agree with the exact identity $\mathbb{E}[n{\bar{\omega }}_d^2]=2^{-d}-3^{-d}$. The principal foundational results are formally verified in Lean 4 and mathlib4.