Closed Surface Integral Geometry A Foundational Formulation Based on Vector Integral Conditions

This paper establishes a foundational framework for closed surface geometry based entirely on global vector integral conditions. Departing from traditional differential-geometric approaches that rely on curvature, metric tensors, or topological classifications, we introduce an integral-based formulation in which closure, symmetry, and rotational neutrality emerge from vector cancellation principles. The first-order integral condition shows that the total outward normal vector over a closed surface necessarily vanishes, providing a global closure criterion. The second-order normal tensor encodes isotropy and geometric distortion, allowing deviation from spherical symmetry to be quantified through tensor invariants. Higher-order tensor generalizations are introduced to construct a complete hierarchy of geometric moments, capturing parity asymmetry and anisotropic structure. Additionally, rotational cancellation identities and surface-volume integral relations are derived, demonstrating that fundamental geometric properties arise from global integral invariants rather than local curvature descriptions. This integral hierarchy provides a coordinate-independent, tensor-based characterization of closed surfaces and suggests a generalized moment-theoretic approach to surface geometry. The framework is extensible to higher-order tensors and potentially to higher-dimensional manifolds.