Infinitesimal Punctures, 1: Infinitesimally Punctured Geometry. Foundations of Weak–Strong Manifolds and Distributional Curvature

For more than a century, singularities and ultraviolet divergences have stood at the frontiers of modern theoretical physics, marking points where our most successful theories cease to be mathematically well defined. Infinitesimal Punctures proposes a structural shift in perspective: instead of inserting point-like sources into smooth manifolds, matter and physical attributes are interpreted as intrinsic geometric defects—measure-zero punctures—within spacetime itself. In this framework, curvature, charge, and quantum behavior arise not as external additions but as distributionally supported features of geometry. The series develops this idea systematically, moving from foundational geometry, through dynamical physical laws, to a unified S-MultiSpace and S-MultiStructure structural formalism. This volume establishes the geometric and analytic foundations of infinitesimally punctured manifolds. It introduces the weak–strong decomposition of geometry, in which smooth differential structure coexists with defect-supported distributional curvature on measure-zero sets. Classical singularities are reinterpreted as punctures carrying finite, integrable curvature, replacing divergent invariants with well-defined distributional objects. The book develops codimension-based classification of defects, holonomy effects, regularisation via smooth metric families, and the structural Neutrosophic triad as a geometric—not yet dynamical—decomposition. A substantial portion of the volume is devoted to operator theory on punctured manifolds, including self-adjoint extensions of Laplace-type operators, spectral stability, defect-induced bound states, and scattering without external potentials. Explicit low-dimensional toy models—punctured circles, conical planes, and point defects in three dimensions—demonstrate how geometry alone generates observable analytic consequences. This book contains no field equations and no cosmological models; its purpose is to provide a rigorous, self-contained mathematical groundwork upon which the subsequent volumes build.