Smyth's conjecture and a non-deterministic Hasse principle

In a 1986 paper, Smyth proposed a conjecture about which integer-linear relations were possible among Galois-conjugate algebraic numbers. We prove this conjecture. The main tools (as Smyth already anticipated) are combinatorial rather than number-theoretic in nature. For instance, the question can be reinterpreted as a question about the possible eigenvalues of a specified linear combination of permutation matrices. What's more, we reinterpret Smyth's conjecture as a local-to-global principle for a "non-deterministic system of equations" where variables are interpreted as compactly supported K-valued random variables (for K a local or global field) rather than as elements of K.