This work presents a unified spectral framework for analyzing physical structures, detecting gravitational anomalies, and stabilizing machine‑learning models through geometric operators. The core of the approach is the Minimal Energy Operator, a functional filter that suppresses unstable or non‑physical configurations by evaluating the Laplacian energy of a field. The project introduces: a discrete 2D Laplacian (including toroidal and Möbius‑strip variants), a minimal‑energy stability operator for spectral filtering, a spectral anomaly detector for black‑hole‑like signatures based on energy drops and coherence rises in the time‑frequency domain, a machine‑learning formulation where spectral features, Laplacian responses, and energy measures form the input space for classification or regression tasks. The methods are designed to be simple, interpretable, and compatible with both numerical simulations and real‑world signals. The repository includes code, mathematical formulations, and examples demonstrating how spectral operators can be used to detect high‑curvature events, stabilize learning algorithms, and analyze harmonic structures on non‑trivial geometries. This upload provides the theoretical foundations, formulas, and implementation files necessary to reproduce and extend the results.