Finite Fibers of Mixed Fourier Ratios for Cyclic pq-Roots

Let p and q be distinct primes. For each prescribed nonzero table of mixed Fourier ratios, there are only finitely many complex cyclic pq-roots with those ratios. The ratio map is a finite morphism. We prove this in a reciprocal tensor system whose two matrices have coprime sizes and all square minors nonzero. A support estimate rules out simultaneous degeneration in the two scaling directions. If only one direction degenerates, the row and column sums of an intermediate product matrix give a divisibility obstruction.