Dominant Laurent Terms and Gauss Retraction on Analytic Tori

Let q be tropicalization on an analytic torus over a complete non-Archimedean field of residue characteristic zero. We bound the logarithmic loss in the absolute value of an analytic function between the Gauss point and a point of the same fiber where the logarithmic volume form has positive Kähler seminorm. The coefficient depends only on the dominant Laurent exponents at that fiber. For locally uniform limits of finite Fubini–Study weights this gives a bound by the width of the subdifferential of the convex Gauss trace. In particular, the weight is constant on the entire fiber over every point where the trace is differentiable. On the same torus chart a maximum truncation also gives a finite-stage one-sided estimate for the pushforward of the Monge–Ampère measure. For model Fubini–Study weights, we decompose the same measure difference into a base-scheme correction of the reduced linear systems and a term from curvature mass off the Gauss section.