Measure Residual Calculus

Abstract The historical evolution of calculus has continuously oscillated between continuity and discreteness, often relying on abstract concepts (such as the limit) to mask the underlying ontological contradictions of infinitesimals. This paper proposes Measure Residual Calculus (MRC) to resolve the logical vacuum between macroscopic spacetime and the Planck scale. By introducing a non-fractal "polygonal disk" as the logical atom and anchoring the minimum measure fidelity to a global constant , we reconstruct the classical derivative into a "Jump Operator" (). In this framework, never collapses to absolute zero , generating an inevitable geometric residual . By introducing the Measure Residual Calculus (MRC) framework, this paper resolves the long-standing conflicts between discrete spacetime hypotheses and Lorentz covariance, while ensuring the continuity of the energy-momentum tensor through the accumulation of second-order measure residuals. We mathematically demonstrate that the macroscopic accumulation of these measure residuals—previously discarded as higher-order infinitesimals in classical limits—is the ontological origin of Dark Matter (gravitational carry-loss) and Dark Energy (vacuum measure drift ground noise) , while also providing a logical resolution to the Hubble tension. Furthermore, addressing the temporal dilemma of infinite iteration, we argue that the discrete "jump" acts as a necessary logical circuit breaker against computational irreducibility . Ultimately, MRC reveals that classical calculus is merely a highly efficient "high-frequency sampling" tool, and the true mysteries of the universe are hidden within the logical remainders it rounds off. Keywords: Measure Residual Calculus, Ontology of Infinitesimals, Dark Matter, Dark Energy, Computational Irreducibility, Non-standard Analysis Comments This paper introduces Measure Residual Calculus (MRC) to resolve the ontological contradictions of classical limits. It mathematically demonstrates that geometric measure residuals—previously discarded as infinitesimals—accumulate macroscopically to provide a purely topological origin for Dark Matter and Dark Energy, eliminating the need for unobserved particles or arbitrary cosmological constants.