Angle-Data Reconstruction for Triangulated Polyhedral Surfaces and a Genus Deficit

An AIM problem asks for a characterization of the face-angle and dihedral-angle data of a triangulated polyhedral surface in ℝ³ and conjectures that the realizable data have dimension E−1 in every genus. We use the realization convention later made explicit by Hempel: a labeled, simplexwise-linear, simplexwise-injective map of a consistently oriented triangulated closed surface, modulo similarities. The combined face-angle and oriented-dihedral map is injective and has a local real-analytic left inverse at every nondegenerate realization. Its actual image therefore has dimension 3V−7 = E−1−6g. Thus the AIM formula is correct for the sphere and fails in every positive genus. The embedded Császár torus gives the concrete count 14 rather than 20. We also give a finite necessary-and-sufficient realizability test: intrinsic sine-law compatibility followed by vertex and non-tree-hinge closure in a dual-spanning-tree development. At generic realizations, Fogelsanger rigidity shows that face angles alone already have rank E−1−6g, identifying the missing 6g as global extrinsic closure codimension. This note is unrefereed and makes no absolute priority claim. Research status: Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-use disclosure: AI-assisted tools supported literature search, computation, proof auditing, and manuscript preparation. The author remains responsible for all claims and the final text.