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

TL;DR

This work constructs a self-contained affine $Γ$-scheme theory for commutative ternary $Γ$-semirings, unifying geometry and spectral analysis through a canonical triangulation of algebraic and geometric structures. It shows that a triadic $Γ$-algebra induces a Nambu–Filippov-type bracket on the structure sheaf and a canonical Laplacian on the finite $Γ$-spectrum, whose eigenstructure encodes the clopen decomposition of the underlying space. The development proceeds through Phase I (affine $Γ$-schemes, spectra, localization, and affine anti-equivalence), Phase II (triadic symmetry and invariance), Phase III (canonical Laplacian and spectral decomposition), and Phase IV (explicit finite examples); together, these establish an algebraic and geometric framework for triadic symmetry and spectral clustering. The results enable exact factorization via block-diagonal Laplacians corresponding to clopen components and relate the second Laplacian eigenvalue to geometric connectedness, with potential applications in spectral clustering pipelines and robust analysis of triadic systems.

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

Theorems & Definitions (54)

• Definition 1: Commutative semiring
• Definition 2: Ternary $\Gamma$-semiring
• Remark 1
• Definition 3: $\Gamma$-ideal
• Definition 4: Generated $\Gamma$-ideal
• Definition 5: $\Gamma$-radical
• Lemma 1
• Definition 6: Prime $\Gamma$-ideal
• Remark 2
• Lemma 2: Radical as intersection of primes