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 \(\mathbb R^3\) 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.