Energy-Governed Exponent Dynamics in Anisotropic Vector Dirichlet Series: A Self-Regulating Framework for Discrete Expansion

This paper introduces and rigorously analyzes the Proportional Energy Inheritance Series (PEIS), a self-regulating discrete dynamical system in which the decay exponent of a vector Dirichlet series is updated implicitly at each step by the ratio of successive energy norms. The framework proves absolute convergence, positivity, strict monotone drift of the exponent to infinity and the energy to zero, unique solvability of the implicit update, and the exact asymptotic p_n ~ n ln n / (ρ ln k₀). A sub-leading correction, the resonance structure of the oscillatory frequency parameter via Hurwitz zeta decomposition, and a generalization to R^d are also established. The sensitivity parameter ρ maps naturally to physical constants across cosmological, quantum, thermodynamic, and biological domains.