Parity Decomposition and Logarithmic Obstructions in the Erdős–Moser Equation

This work develops a structural analytic framework for the Erd\H{o}s--Moser equation $\sum_{n=1}^{m-1} n^k = m^k$. Using parity decomposition and logarithmic projection, we identify a stable first-order defect of size $C_k/(m\ln m)$ for all $k\ge 2$. This defect arises from the nonlinear collapse of the parity progression and appears as a spiral geometric drift. The resulting continuous--discrete mismatch produces a logarithmic residue incompatible with the integer lattice, yielding a unified obstruction to non-linear solutions.