The Anta-Rai Framework: Centered Spectral Stability and Harmonic Kernel Structures in Hodge-Type Geometric Operators

Abstract / Description: This paper introduces the Anta-Rai framework, a differential-geometric operator model designed to bridge the gap between topological invariance and systemic stability. At the core of the framework is the centered operator family: LAR(σ)=ΔH+α(σ−c)2,α>0 where ΔH represents the Hodge Laplacian on a compact Riemannian manifold M. Key Contributions: Spectral Centering: The work establishes the conditions under which a stable harmonic kernel emerges, specifically at the spectral center σ=c. Topological Link: It is shown that the existence and dimension of the kernel are governed by the cohomology of the underlying manifold (Hk(M,R)), providing a rigid structural foundation for complex systems. Structural Slack: The framework formalizes the concept of "Structural Slack"—the necessary margin of variability (σ−c) that allows a system to maintain its core topological identity while absorbing perturbations. Significance: While rooted in Riemannian geometry and Hodge theory, the Anta-Rai framework is designed for interdisciplinary application in Cybernetics, Network Theory, and Theoretical Biology. It provides a formal language to describe how viable systems balance structural persistence with adaptive flexibility.