A Newton-face criterion for finite-state categorical Bernoulli factories

Let a deterministic finite-state machine read independent samples from an unknown categorical distribution with full support and stop almost surely with a binary output. We characterize the rational target probabilities that such a machine can realize. Write the target as P/Q, where P and Q are coprime homogeneous polynomials of the same degree with rational coefficients and 0<P<Q on the positive orthant. A finite-state factory exists if and only if every nonempty Newton-face polynomial of both P and Q-P is strictly positive on the positive orthant. Whenever this condition holds, a fixed-block restart factory exists. The criterion is decidable by real quantifier elimination, and a successful test leads to an exact construction. The proof combines a Markov-chain tree representation of the two output probabilities, Handelman's positive-multiple theorem and the block construction of Mossel and Peres. Examples explain why the complementary numerator is essential and why powers of the coordinate sum do not exhaust the possible multipliers. No bound on decision complexity, block length or state count is asserted. This record contains the standalone PDF preprint and a source archive with the LaTeX and Markdown manuscript, PNG figures, figure-generation code, bibliography and build instructions. The results are theoretical; no experimental dataset or numerical computation is used in the proofs. The figures depict the stated constructions and exponent sets. AI assistance is disclosed in the manuscript. The preprint has not undergone journal peer review.