Hyperbolic Electromagnetism 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 yields the geometric part of the vacuum impedance when combined with the quantum scale h/e2. The same discrete structure provides a normalization for the fine-structure constant through a universal sector angle π/24. We discuss how discrete rotational and modular symmetries, commonly realized in crystalline metals and layered heterostructures, provide experimentally accessible settings to probe the proposed geometric structure through electromagnetic and transport measurements.