α and Λ as a UV/IR Duality: A Discrete Geometric Perspective

This paper proposes a discrete geometric interpretation of the apparent duality between the fine-structure constant (α) and the cosmological constant (Λ), framing them as manifestations of a UV/IR correspondence in a relational spacetime lattice. Using a self-similar tetrahedral–octahedral honeycomb as a toy model, we explore how local geometric excitation and global combinatorial residue give rise to two closely related geometric approximations of α. One formulation emphasizes local structural geometry, while the other captures a global, combinatorial deficit, suggesting that α encodes both microscopic coupling strength and a residual large-scale geometric imprint. Within this framework, Λ is reinterpreted not as a local vacuum energy density, but as an emergent global curvature scale arising from the cumulative structure of the lattice. The tetrahedral–octahedral duality provides a natural geometric analog for ordinary matter, dark matter, and dark energy, while the UV/IR correspondence offers a conceptual bridge between local quantum couplings and cosmological curvature. Although presented as a toy model, the construction illustrates how rigid local geometry combined with high global configurational freedom can naturally separate ultraviolet and infrared behavior without fine-tuning. The work does not claim a derivation of α or Λ from first principles, but offers an alternative geometric perspective on the cosmological constant problem and on the possible relational origin of dimensionless constants. v1