Adaptive Gălușcă Pyramidal Vector (AGPV): A Geometric Reduced Formalism with Applications to Gaussian Propagation, Beam Dynamics, and Anisotropic Localization

This record contains the preprint and LaTeX source of the manuscript “Adaptive Gălușcă Pyramidal Vector (AGPV): A Geometric Reduced Formalism with Applications to Gaussian Propagation, Beam Dynamics, and Anisotropic Localization”. The manuscript introduces the Adaptive Gălușcă Pyramidal Vector (AGPV) as a geometric reduced formalism for organizing center, direction, spread, covariance, internal shear, and adaptive axial localization of extended dynamical states. The framework is developed in two stages: a first-order pyramidal state attached to position and velocity, and a second-order phase-space extension carrying covariance structure and an optional bivectorial internal degree of freedom in 3D. The work defines canonical pyramidal invariants, including effective phase-area, normalized shear, and internal shape-energy observables, and studies their propagation under free and harmonic Gaussian dynamics. It also introduces an adaptive localization rule based on projected longitudinal uncertainties, replacing fixed axial localization choices with an uncertainty-balanced estimator. Applications are given to Gaussian wave packets, beam dynamics, and anisotropic tracking/localization. The manuscript includes explicit numerical validation examples in 2D and 3D, showing how the AGPV can improve positional identification when velocity is known accurately and localization is performed along a dynamically adapted kinematic axis. The formalism is presented conservatively as a geometric and operational language for reduced state representation and propagation. It is not claimed to replace standard mechanics or quantum theory, but to provide a compact, readable, and dynamically tractable atlas for extended states and their evolution.