Finite metric fibres for exact self-tilings of convex pentagons

For every realizable cyclic pentagon angle tuple admitting no nonnegative integer combination equal to π, we prove that only finitely many similarity classes satisfy the side-length relations forced by exact self-tiling. The relations arise from full vertex stars in a locally finite rectangle extracted from an arbitrary exact countably infinite tiling. Their positive solution set is finite by a parity argument and a five-line divisor calculation on normalized projective curves. For the angle tuple (4,6,6,8,9)π/11, we determine the entire solution set: it consists of two explicitly described algebraic metrics. We verify Euclidean closure and all five label relations at their selected real embeddings and give an exact certificate verifier requiring no root search. MSC 2020: Primary 52C20; Secondary 05B45, 13P15. The accompanying files contain the manuscript PDF and a curated LaTeX source archive.