Infinitesimal Punctures, 4: Infinitesimally Punctured Physics used in Extended Nonstandard Analysis

infinitesimally punctured geometry formulated within Extended Nonstandard Analysis (ENSA). It introduces the MoBiNad manifold as a hyperreal extension of classical differential geometry, where monads, binads, and infinitesimal punctures generate structured micro-deviations in metric, connection, and curvature operators. Through neutrosophic logic, each geometric and physical quantity acquires a triplet valuation (truth, indeterminacy, falsity), allowing curvature, topology, and field dynamics to encode graded logical structure at the infinitesimal level. Building on this foundation, the book develops MoBiNad differential operators, supersymmetric extensions, Smarandache curves and surfaces, and a neutrosophic Gauss–Bonnet theorem, demonstrating how mixed geometric regimes emerge as standard-part projections of a rigorously defined hyperreal substratum. Gauge–gravitational coupling, curvature oscillations, and topological invariants are reformulated within this punctured continuum, providing a deterministic infinitesimal basis for phenomena traditionally attributed to quantum fluctuation. The result is a coherent bridge between differential geometry, topology, logic, and high-energy physics: classical geometry appears as the macroscopic limit of a structured infinitesimal network, while Smarandache geometries arise naturally from local curvature sign alternations induced by monadic asymmetries. This work lays the analytical groundwork for a neutrosophic Chern–Weil theory and higher-dimensional topological invariants in hyperreal spacetime.