dorsal/arxiv
View SchemaThe Spectral Geometry of Ternary Gamma Schemes:Sheaf-Theoretic Foundations and Laplacian Clustering
| Authors | Chandrasekhar Gokavarapu |
|---|---|
| Categories | |
| ArXiv ID | 2601.09268vv2 |
| URL | https://arxiv.org/abs/2601.09268 |
| License | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
Abstract
This article develops a self-contained affine $\Gamma$-scheme theory for a class of commutative ternary $\Gamma$-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 $\Gamma$-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 $\Gamma$-spectrum whose spectral decomposition detects the clopen (connected-component) decomposition of the underlying space. We define $\Gamma$-ideals and prime $\Gamma$-ideals, endow $\SpecG(T)$ with a $\Gamma$-Zariski topology, construct localizations and the structure sheaf on the basis of principal opens, and prove the affine anti-equivalence between commutative ternary $\Gamma$-semirings and affine $\Gamma$-schemes. Furthermore, we demonstrate that the triadic bracket on sections is invariant under $\Gamma$-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 $\Gamma$-spectra and their corresponding Laplacian spectra
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"abstract": "This article develops a self-contained affine $\\Gamma$-scheme theory for a class of commutative ternary $\\Gamma$-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 $\\Gamma$-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 $\\Gamma$-spectrum whose spectral decomposition detects the clopen (connected-component) decomposition of the underlying space. We define $\\Gamma$-ideals and prime $\\Gamma$-ideals, endow $\\SpecG(T)$ with a $\\Gamma$-Zariski topology, construct localizations and the structure sheaf on the basis of principal opens, and prove the affine anti-equivalence between commutative ternary $\\Gamma$-semirings and affine $\\Gamma$-schemes. Furthermore, we demonstrate that the triadic bracket on sections is invariant under $\\Gamma$-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 $\\Gamma$-spectra and their corresponding Laplacian spectra",
"arxiv_id": "2601.09268",
"authors": [
"Chandrasekhar Gokavarapu"
],
"categories": [
"math.RA"
],
"license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
"title": "The Spectral Geometry of Ternary Gamma Schemes:Sheaf-Theoretic Foundations and Laplacian Clustering",
"url": "https://arxiv.org/abs/2601.09268",
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