Hyperbolic EM Symmetry and Metrological Closure of Vacuum Impedance with Links to Topological Response in Metals and Alloys

We develop a symmetry-based reconstruction of the vacuum impedance and the fine-structure constant. Hyperbolic geometry and discrete sectorization of the electromagnetic field plane are the only input assumptions. The construction identifies a unique integer-square hyperbolic selector that fixes the electric–magnetic partition without adjustable parameters. This yield the geometric part of the vacuum impedance when combined with the quantum scale $h/e^{2}$. The same discrete structure provides a normalization for the fine-structure constant through a universal sector angle $\pi/24$, connecting topological quantization phenomena in metals and alloys, including Berry phases, Zak phases, and quantized Hall responses. The resulting framework places electromagnetic constants within a unified geometric–topological setting and suggests experimentally accessible consequences in systems with discrete rotational or modular symmetry.