Undecidability of unnormalized univariate trace inequalities

We prove that no algorithm decides whether a rational polynomial in the ordinary traces of powers of one real symmetric matrix is nonnegative for all finite matrix sizes. The proof encodes nonnegative integers as spectral multiplicities. Stationary polynomial interpolation controls deviations from the encoding nodes with an error bound independent of the number of eigenvalues. A polynomial penalty then transfers an undecidable discrete sign problem to an unrestricted pure trace inequality. The construction uses only positive powers and never uses the matrix dimension as a trace variable.