A Geometric Dual‑Scale Invariant for the Fine‑Structure Constant

This preprint introduces a dual‑scale geometric framework that yields two structurally distinct expressions approximating the fine‑structure constant α. The first, Honeycomb Alpha, arises at the micro‑scale from the tetrahedral architecture of the Honeycomb Unit (HU) and is explicitly φ‑dependent. The second, Λ‑Alpha, emerges at cosmological scales after extensive coarse‑graining, where octahedral degrees of freedom dominate and φ becomes implicit. Despite their differing internal components, both expressions converge to the same numerical value, suggesting the presence of a scale‑invariant geometric ratio analogous to a renormalization‑group fixed point. The paper contrasts this structural derivation with common numerological α‑approximations, emphasizing that the Honeycomb expressions contain no adjustable parameters and arise from geometric constraints such as the 4:1 tetrahedron–octahedron volume rule, symmetry preservation, and DOF transitions. The result is a non‑numerological, geometry‑driven pathway toward understanding why α takes the value it does, with potential relevance to scale duality, coarse‑graining behavior, and the geometric foundations of physical constants. v1