Heterogeneous Anisotropic Kac Limits on Rectifiable Resolved Cut Spaces

Abstract

Ambient convergence does not determine a nonlocal energy on a cut body: a cut has zero volume, but a microscopic bond retains which labelled bank its path crosses. We give a direct measure-theoretic resolution of this information and prove the corresponding sharp-interface Gamma-limit. The cut may be a finite or countable family of Borel, countably rectifiable hypersurfaces with finite total surface mass, measurable approximate normals, and measurable binary bank data. No closedness, positive reach, finite atlas, curvature bound, angle condition, or regularity of the branch set is assumed. For heterogeneous Kac interactions generated by a uniformly spanning family of possibly noncommuting flows, the limit is the resolved anisotropic variation plus the outer boundary contact and one coefficient-one contact on each bank. The Gamma-functional contains no compulsory lower-dimensional term where a crack approaches the boundary; constant recovery creates no concentration there. The proof combines an exact spacetime area formula for flow crossings, a one-dimensional split-extension lower bound, and ordered-flow compactness. We also remove compact radial support. A finite first moment yields the same local Griffith functional; at the power-law threshold the limits separate into local, logarithmically renormalized local, and genuinely fractional regimes. Smooth slit and junction detectors are recovered as uniformly equivalent realizations of the direct crossing rule.