Exact enumeration of free polyplets in the {4,5} hyperbolic tessellation

A polyplet in the {4,5} tessellation of the hyperbolic plane is a finite set of faces that is connected when two faces count as adjacent if they share a vertex. Each face has twelve such neighbors. Sequence A390200 of the OEIS counts free polyplets with n cells, and it listed terms only for n ≤ 7. We show that there are 1,261,889 free polyplets with 8 cells. We represent faces as cosets of the stabilizer of a face in the Coxeter group [4,5], and we do all matrix arithmetic exactly in the field Q(√2, √5), so no step of the enumeration depends on floating point numbers. Two programs with separate enumeration and counting layers give the same value. The first program keeps a set of canonical forms, and the second combines Redelmeier's method with an orbit counting formula. A third program in C++ builds the tiling from the hyperboloid model instead of the Coxeter group, and it gives the same counts. We also check the programs against the known terms of A119611, which counts the shapes that are connected through edges. The eighth term reported here was accepted into OEIS sequence A390200. The code, the raw output and the symbolic verifier are deposited separately at doi:10.5281/zenodo.21839653, and the repository is at https://github.com/pragyaangaur/Hyperbolic-Polyplet-Enumerator.