Axiomatic Void Geometry: Viscoelastic Bridge Curves, Singularity Scanning Operator, and Dynamics of (N, K)-Clusters (Part I)

This paper presents an axiomatic model and research program for Void Geometry—a mathematical framework describing the shape, evolution, and inertial mass of ma terial continua through a system of internal non-material singularities (imaginary points) of zero density. A self-consistent model is constructed in four-dimensional spacetime M4 by separating the manifold of singularities from tangent vector spaces. We introduce a two-level cluster classification (N, K) comprising parent metric anchoring centers and optional child symmetry axes. Based on Green’s second identity, we derive a universal integral void-scanning operator Vˆw[Φ] capable of in variantly determining the number and spatial coordinates of hidden parent centers solely from the outer boundary of a body. Equations of motion in a viscoelastic void medium are formulated for regimes (1, 0), (1, 1), and lemniscate trajectories (1, 2), alongside the deformation equation of a smooth bridge curve for N ≥ 2. Finally, we state the central open problem regarding cluster dynamics for N > 2.