Exact Finite-Radius Reading of Logarithmic Terms in Irregular Singular Germs: A Matrix Pencil Approach

Logarithmic terms in Hukuhara–Turrittin germs at irregular singular points are classically treated as asymptotic obstructions to contour integration. This work completely overturns this view by establishing an exact, finite-radius reading law. We show that the logarithmic axis forms a free abelian group occupying the bottom coordinate of a rank-2 lexicographic valuation. By identifying the reading map as a discrete shift operator—the exact analogue of the logarithmic derivative on the \(z\)-axis—we reformulate the recovery of the order and the logarithmic polynomial as a generalized eigenvalue problem. Our matrix pencil approach yields exact numerical results to machine precision from a single circle, bypassing traditional root-finders and slow asymptotic expansions. A reference implementation is included, demonstrating why standard trapezoidal quadrature fails due to monodromy and how geometric convergence is restored.