Exact Positive-Focal Certificates for Fundamental-Matrix Pencils

For an independent rational 3-by-3 matrix pencil, we give an exact procedure that counts its real projective rank-two members compatible with two positive finite focal lengths, or reports that there are infinitely many. Principal points are known, skew is zero and aspect ratio is one. A positive-kernel formulation of the two-view Kruppa relation yields a division-free polynomial sign predicate. Classical Hermite trace forms evaluate this predicate without constructing or isolating determinant roots. Repeated roots, rank-one members, the projective point at infinity and critical calibration families are included. For identically singular pencils, derivative-root sampling also detects bounded feasible intervals. The complementary count identifies incompatible rank-two members. On a guarded generic chart, the count reduces to four Tarski queries.

The construction has polynomial bit complexity for this fixed-size input; no empirical speed advantage over numerical reconstruction is asserted. The exact pencil theorem is complete, but the open-ended AIM Algebraic vision Problem 1.5 is not marked fully resolved. Classical methods and the inherited generic indicator are credited. This AI-assisted, self-audited preprint is unrefereed and makes no absolute priority claim.