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Classical solid geometry recognizes cones (point apex) and cylinders (planar apex) but lacks systematic classification of intermediate cases where solids taper to line segment apices. We introduce Phylum Apicalia, organizing all apex-convergent solids (where the topological dimension of the apex set is less than that of the base) by apex dimensionality, and establish Class Apexia for solids satisfying a monotonicity constraint (cross-sections never exceed the base footprint). We establish mathematical rigor through four core theorems at the genus level (symmetry-breaking, volume hierarchy, curvature classification, projective maximum maximality), supported by two foundational observations at the Phylum level (apex dimension as topological invariant, cylindrical constraint dichotomy). The vertical path (Phylum Apicalia → Class Apexia → Order Apextoid → Family Hewel → Genus Skwedge → Species: curvum, convexum, projectivum, minimalis) demonstrates that apex dimensionality provides a natural organizing principle for geometric classification, with the resolution of Genus Skwedge (shown to lack uniqueness in the sense of Hadamard, with a volumetric disparity of 13.13% relative to the projective maximum) serving as proof-of-concept for the framework's rigor and extensibility.