Unified Topological Field Theory: The Graded C₅ Dark Algebra and Crystal Field Geometries

Unified Topological Field Theory: The Graded C₅ Dark Algebra and Crystal Field Geometries C. J Tully Independent Researcher Engineer https://doi.org/10.5281/zenodo.24098291 https://orcid.org/0009-0007-5661-7332 [email protected] A topological re-foundation of classical crystal-field ratios via a golden-weighted cyclic algebra 10th November 2026 Abstract We present a self-contained topological framework that maps the five d-orbital magnetic quantum states onto a five-fold cyclic manifold C₅ governed by a ℤ₅-graded non-associative algebra 𝒟. An adjacency-weighted golden product is defined on the basis; the central invariant Dark Score R(v) is the grade-zero projection of the golden-symmetrized self-product. Under a structural orbital-to-grade assignment, the classical crystal-field splitting ratios for octahedral (Oₕ), tetrahedral (T_d), and squareplanar (D₄ₕ) geometries emerge as projections of this invariant. In particular, the universal relation Δ_t = (4/9)Δ_o follows from ligand-count and angular factors alone, while the barycentric coefficients of the octahedral field are recovered exactly. The construction is a topological re-expression of known ligandfield geometry rather than a parameter-free derivation of absolute energy scales; the absolute scale is fixed in manifold units by the algebraic combination that yields the ratios. Non-associativity of 𝒟 supplies a topological signature for the grade-coupling boundaries that separate the different coordination geometries.