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.