The Figure-Eight Knot Complement as the Unique Admissible Quotient of a Tetrahedral Defect

We study admissible quotients of a cyclic 5-tetrahedron defect sub- ject to explicit combinatorial constraints: orientability, fixed-point-free face pairings, connectedness with a single cusp, and exactly two edge orbits. Within this class, we prove that there exists, up to combina- torial equivalence, a unique quotient triangulation. The admissibil- ity conditions force a reduction to a two-tetrahedron ideal triangula- tion whose associated gluing equations collapse to a single quadratic constraint, uniquely determining a complete finite-volume hyperbolic structure. The resulting manifold is identified as the figure-eight knot complement (m004), and the identification is confirmed both analyti- cally and via computational verification using SnapPy and Regina. This construction provides a concrete instance in which a mini- mal combinatorial defect, when equipped with natural admissibility constraints, rigidly determines a global hyperbolic 3-manifold. While motivated by defect structures arising in higher-dimensional lattice models, all results are established purely within the framework of 3- manifold topology and hyperbolic geometry.