A Simpson–type Decomposition of the Euler–Mascheroni Constant (γ) as a Sum of Two Irrational Numbers

An elementary and self-contained approach to the Euler–Mascheroni constant γ is presented, based solely on Simpson's quadrature rule and the convexity of the function \( f(x)=1/x \). Using Simpson--type weighted harmonic sums, we approximate local logarithmic increments by simple finite linear combinations of reciprocal integers. Exploiting the monotonic and convex nature of \( 1/x \), sharp two-sided inequalities are established that relate these numerical approximations to the exact logarithmic increments. These inequalities imply that the accumulated quadrature errors form a convergent series, yielding a simple proof of the classical limit defining γ without recourse to the Euler–Maclaurin formula.A central structural observation of the paper is that γ admits a Simpson--type decomposition as a sum of two irrational numbers. More precisely, we show that \( \gamma = ( \log{[2]} + 1 ) / 3 + \delta \), where both \( ( \log{[2]} + 1 ) / 3 \) and δ are irrational. The constant δ arises naturally as the limit of a rational sequence derived from a Simpson--type approximation, and its irrationality is established by an elementary rigidity argument based on prime divisibility in rational approximations.