Stein normal form and critical points

This report studies the Euclidean distance degree $D_m$ of the standard real symplectic group, for which Baaijens and Draisma suggested the formula $D_m=2^{m^2}+2^{2m-1}$. Building on an earlier regular $m^2$-variable reduction, we identify its denominator exactly and put the full generic critical-point problem on a universal discrete-Lyapunov domain, where the two auxiliary symmetric variables solve explicit Stein equations. We compute the compactly supported Euler characteristic of that domain and prove that every symplectic Euclidean distance degree is even. Computer-assisted exact integer-arithmetic contraction and separation checks, at fixed data matrices, establish 543 distinct simple critical points for $m=3$ and 31\,183 for $m=4$; with the parity theorem this gives $D_3\ge544$ and $D_4\ge31\,184$. The zero-coupling specialization has only $2^m$ regular points. The formula remains neither proved nor refuted: the outstanding step is the generic degree of an explicit rational matrix map.