The Honeyverse: A Relational Geometric Toy Model

This preprint presents the Honeyverse, a relational geometric toy model built on an invariant tetrahedral–octahedral honeycomb lattice. The model introduces a geometric analog of the fine‑structure constant, Honeycomb alpha (αH = √2 / 120φ), derived from the volume structure and 10‑node relational adjacency of the Honeycomb Unit (HU). Through algebraic transformation and coarse‑graining, this local φ‑explicit expression yields a global φ‑implicit counterpart, Honeycomb Lambda (Λα = (√10 − √2) / 240). Despite their differing internal components, both expressions remain numerically invariant, suggesting a dual‑scale geometric ratio analogous to a renormalization‑group fixed point.

The Honeyverse framework emphasizes recursive topology, scale‑dependent interpretation, and the preservation of the HU’s 4:1 tetrahedron–octahedron volumetric rule. Under coarse‑graining, tetrahedral rigidity gives way to octahedral connectivity, offering geometric analogs for localized structure (matter) and large‑scale expansion behavior (dark energy). Curvature is interpreted as global topological compensation rather than stress‑energy, providing an alternative relational lens through which expansion and structure may be viewed.

This model is not proposed as a replacement for general relativity or ΛCDM, but as a conceptually coherent geometric analog that highlights how scale‑invariant ratios and structural recursion may illuminate the persistence of certain dimensionless constants across scales.

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