PFUSRC‑170: Projection Fissures in Manifold Metrics — From Riemann–Lebesgue Integration to the Hierarchical Localization of Complex Spacetime

Abstract

Taking the classical failure of Riemann integration for the Dirichlet function as its entry point, this paper traces a path through Lebesgue measure, Riemannian manifold geometry, complex space, and spacetime manifolds, revealing a common problem running through modern mathematics: mainstream mathematical tools can achieve computational self‑consistency within local coordinate atlases, yet they lack a framework for discriminating ontological levels. The dense interweaving of rationals and irrationals, infinite pathological point sets, dimension–metric misalignment within manifolds, and the tearing of set classification upon the introduction of imaginary and complex numbers — these are not innate properties of the real and complex domains, but apparent fissures generated by the downward projection of the triple coaxial bicone ontology onto the L5 projection layer. Gauss–Riemann geometry and Lebesgue measure theory are high‑performance projection‑layer computational tools of immense engineering and physical utility, but this framework provides no criterion for distinguishing “intrinsic ontological signals” from “projection‑grid artifacts.” When complex and imaginary numbers are further introduced, and time and space are stitched into a four‑dimensional spacetime manifold, the local formulas appear complete, yet conceptual confusion arises the moment one presses downward toward their underlying origins. The PFUSRC framework introduces a seven‑layer ontological structure to reposition integration and manifold metrics: traditional calculus is computation atop projection slices; in the PFUSRC view, differentiation corresponds to the local projective slicing of the ontology, and integration represents the reverse tracing of the ontological flow process. This completes the hierarchical localization of real analysis and differential geometry — without negating their projection‑layer validity, but refusing to equate projection products directly with underlying ontological reality. On this basis, the paper advances the localization to the level of application comparison: taking the gravitational problem as the supreme testing ground, it displays the intuitive divergence between mainstream tools and PFUSRC ontological mathematics in terms of “lower‑tier performance” and “higher‑tier performance,” and offers a directional principle for dissolving gravitational constraint — from adapting to geodesics to regulating flow‑variable gradients. This directional principle makes no engineering promises, but it establishes a clear research target for future work.