A Simpson–Type Decomposition of the Euler–Mascheroni Constant

An elementary and self-contained approach to the Euler--Mascheroni constant $\gamma$ is presented, based solely on Simpson's quadrature rule and the convexity of the function $f(x)=1/x$. By introducing Simpson-type weighted harmonic sums, local logarithmic increments are approximated by simple finite linear combinations of reciprocal integers. Sharp two-sided inequalities, derived from monotonicity and convexity, yield explicit control of the quadrature error and provide a purely numerical proof of the classical limit defining $\gamma$, without recourse to the Euler--Maclaurin summation formula.A key structural outcome of this framework is a decomposition $\gamma = ( \log [2] + 1 )/3 + \delta$, where the constant $\delta$ arises naturally as the limit of a rational sequence associated with Simpson-regularized harmonic sums. This formulation isolates a dominant oscillatory component and leads to a sequence with markedly faster convergence.This work highlights an unexpected connection between elementary numerical quadrature and one of the fundamental constants of analysis.