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.