A Simpson-Type Decomposition of the Euler-Mascheroni Constant and the Irrationality of δ

An elementary and self-contained proof of the existence of the Euler-Mascheroni constant γ is presented, based solely on the Simpson quadrature formula and the convexity of the function f( x mapsto 1/x ). The local logarithmic increments are approximated as follows: ( int_{2n-1}^{2n+1} frac{dx}{x} ) Using Simpson’s rule, a discrete approximation expressed as a finite linear combination of reciprocal integers is constructed. Exploiting the monotonic and convex nature of the function ( 1/x ), sharp two-sided inequalities relating the numerical approximation to exact logarithmic increments are established. These inequalities imply that the accumulated quadrature errors form a convergent series. Consequently, the following classical limits ( gamma = lim_{N to inf} left( sum_{k=1}^{N} frac{1}{k} – log{[N]} right) ) are proven to exist. This approach provides a conceptually simple alternative to traditional proofs based on the Euler-Maclaurin formula, highlighting the direct connection between numerical integration, convexity, and the analytical nature of γ. I further show that λ can be expressed as ( (log{[2]}+1)/3 + delta ), where both ( (log{[2]}+1)/3 ) and ( delta ) are irrational, and where ( delta ) arises as the limit of a rational sequence derived from as Simpson-type approximation.