Tetrahedral Defects and Hyperbolic 3-Manifolds: A Geometric Framework from Lorentzian Quasicrystals

We present a geometric framework in which local tessellation failures in the irrational cut-and-project construction from the Lorentzian lattice II9,1 are resolved into hyperbolic 3-manifold structure. Specifically, the ubiquitous 5-tetrahedron clusters (angular deficit δ ≈ 7.36◦ ) carrying A4 symmetry admit explicit face identifications whose Thurston gluing equations are consistent with the complete hyperbolic structure of the figure-eight knot complement S 3\41. Using SnapPy manifold classification and explicit combinatorial mappings, we verify this correspondence to machine precision. Harmonic phason displacements in the perpendicular space are proposed as a smoothing mechanism that yields a continuous Riemannian 3- manifold whose core carries constant negative scalar curvature (3 )R = −24a 2 . All geometric and topological results are constructive or computationally verified. Broader connections to discrete curvature models are discussed only at a heuristic level and are in- tended to motivate future investigation rather than establish definitive results. This paper is offered as Part I of a research program exploring how discrete geometric defects can generate continuous geometric structure.