A Finite Moment-Cone Criterion for Derivative-Root Configurations

We give a necessary and sufficient finite criterion for prescribed numerical positions of all derivative roots of a real-rooted polynomial-like function. Polynomial-like degree n means that the nth derivative is nowhere zero; the function need not be an ordinary polynomial of degree n. Eliminating integration constants produces m=n(n-1)/2 explicit piecewise polynomial Peano kernels. Realizability is equivalent to zero being interior to their essential-trace convex hull, and to representing an explicit baseline vector by at most m nonnegative trace terms. This gives a finite semialgebraic criterion and a terminating decision procedure for algebraic coordinates. Permitted cross-order coincidences are included. Every feasible array also has an ordinary-polynomial realization of unspecified larger degree. Scope: Shapiro2015 Section8 Problem6, exact numerical positions rather than symbolic ordering or ordinary degree-n polynomial rows. Generic moment and interpolation methods are classical and credited to Kakeya1915, di Dio2019/2025 and Pinkus-Wulbert2005. Whether this explicit application is already-known remains uncertain; an inaccessible1979 Hermite-Birkhoff/monotone-spline paper remains an overlap lead. English, AI-assisted, self-audited, unrefereed preprint. No independent human review, formal proof-assistant certification, practical running-time bound, minimal witness degree or absolute historical-priority claim. Exact reproducibility checks corroborate the formulas; the all-degree result has a full analytic proof. Author: Alper Ferudun, Mercury Software GmbH.