The Jagged Boundary of Asymmetric Divergence: Quantifying the Metric Topology and Geometric Origins of the Euler-Maclaurin Remainder

Abstract Standard numerical integration methodologies and the Euler-Maclaurin summation formula abstract the discrete-continuous interface into scalar remainder bounds, effectively obscuring the physical topology of the transitional boundary. This paper recontextualizes the local truncation error of the Trapezoidal Rule as a strict geometric architecture rather than a purely algebraic artifact. By parameterizing the local error function between a continuous logarithmic field and a discrete linear interpolation over a single step, the abstract Taylor coefficients are translated into explicit spatial deformations. The resulting divergence is strictly quantified by two primary geometric features: a symmetric quadratic sagitta and an asymmetric cubic skew. The cubic skew is identified as the lowest-order non-vanishing metric deformation that physically separates the continuous plane from the discrete harmonic chord. Integration of these isolated geometric deviations across the localized interval rigorously reconstructs the classical algebraic remainder. Extending this spatial framework across the global domain, the derivation ultimately decodes the Euler-Mascheroni constant (γ). Rather than treating γ as a numerical difference between a discrete harmonic accumulation and a continuous field, this paper characterizes the constant as the infinite global accumulation of these localized asymmetric cubic skews and higher-order deformations. Ultimately, this spatial derivation demonstrates that the abstract Bernoulli constants governing global summations are not algebraic artifacts and scalar remainders artifacts, but the aggregated metric remnants of the inherently jagged boundary bridging discrete sequences and continuous curves.