The Intersection Euler Characteristic Profile: Euler Calculus and Stability for Topological Interaction of Ball Unions

The Intersection Euler Characteristic Profile (Intersection ECP) of k colored point clouds X1, ldots, Xk subset ℝd is the Euler characteristic chi(bigcapi=1k U(Xi; ti)) of the overlap of their ball unions---an integer-valued, multiparameter invariant of their topological interaction across scales. Its organizing framework is the Euler calculus on constructible functions: the profile is equally the Euler integral int prodi=1k 1U(Xi; ti) , dchi of the product of the k data-dependent offsets, and this identity---our Intersection Theorem---is a commuting square interchanging geometric intersection and algebraic product. The invariant is rigid-motion invariant, scale-equivariant, and L1-stable, and it is canonical: among pointwise-Euler interaction profiles it is the one forced by separation and normalization, the top floor of a spectrum of descriptors graded by how many clouds meet. For n points a single sorted Alpha-complex sweep computes it in O(nlceil d/2 rceil log n) time with no persistence reduction, worst-case optimal in even dimensions. Where the Euler characteristic cancels, a relative-homology refinement resolves the finer interaction and is stable in the two-parameter interleaving distance. Finally, for increasingly dense samples the profile and its refinement are consistent, recovering the (relative) homology and Euler characteristic of the underlying shapes---in the inverse limit for compact sets, and, under positive reach, persistently and with explicit sample complexity.