The Spectral Geometry of Ternary Gamma Schemes:Sheaf-Theoretic Foundations and Laplacian Clustering

Abstract: This article develops a self-contained affine $Γ$-scheme theory for a class of commutative ternary $Γ$-semirings. By establishing all geometric and spectral results internally, the work provides a unified framework for triadic symmetry and spectral analysis. The central thesis is that a triadic $Γ$-algebra canonically induces two primary structures: (i) an intrinsic triadic symmetry in the sense of a Nambu--Filippov-type fundamental identity on the structure sheaf, and (ii) a canonical Laplacian on the finite $Γ$-spectrum whose spectral decomposition detects the clopen (connected-component) decomposition of the underlying space. We define $Γ$-ideals and prime $Γ$-ideals, endow $\SpecG(T)$ with a $Γ$-Zariski topology, construct localizations and the structure sheaf on the basis of principal opens, and prove the affine anti-equivalence between commutative ternary $Γ$-semirings and affine $Γ$-schemes. Furthermore, we demonstrate that the triadic bracket on sections is invariant under $Γ$-automorphisms and compatible with localization. The main spectral theorem establishes the block-diagonalization of the Laplacian under topological decompositions and provides an algebraic-connectivity criterion. The theory is verified through explicit computations of finite $Γ$-spectra and their corresponding Laplacian spectra