A constructive inverse theory for symmetric tropical characteristic polynomials

We give a constructive inverse criterion for the full coefficient sequence of a symmetric tropical characteristic polynomial in every dimension, including absent coefficients and arbitrary Newton polygons. Canonical zero factors reduce cover completion to three local states. Occurrence incidences then give a finite family of balanced inequalities, with explicit bounds on certificate size, and admissible data yield a matrix of the original order. Principal-witness compression makes the rational inverse fixed-parameter tractable in the support endpoint; a specialized data-parameterized algorithm handles full-span horizontal polygons. For horizontal polygons in even dimension, every realizable profile has a zero perfect matching, and an uncrossing argument reduces all odd coefficients to primitive two-port circuits. Exact assignment potentials transport every Newton face to edge costs and uncovered-vertex penalties; the global criterion decides their simultaneous compatibility. Explicit inverse slices and quantitative obstructions show why graph support alone is insufficient. A generic real symmetric lift proves that nonsaturated indices are isolated, settling Conjecture 6.1 of Kiani and Tavakolipour in all dimensions. We also realize every isolated pattern with prescribed depths and derive sharp spectral compression bounds.