From BMG to DHL to Black-Hole Horizon Dynamics: A Boundary Field Equation for Accretion and Hawking Quantum Depletion

Abstract Black holes represent one of the most compelling settings in modern physics for the convergence of geometry, thermodynamics, quantum effects, and boundary-based reasoning. Einsteinian gravitation determines horizon structure, Bekenstein’s entropy-area relation assigns thermodynamic significance to the boundary, Hawking radiation introduces a quantum depletion channel, and holographic thinking suggests that boundary quantities may encode physically meaningful bulk structure. Yet these ingredients are typically expressed through global laws, semiclassical balance relations, or effective analogies rather than through a single local boundary-dynamical equation governing both horizon growth and depletion. In this paper, I propose a boundary-centered framework that develops a conceptual and mathematical progression from Boundary-Mediated Growth (BMG) to the Dynamic Holographic Law (DHL), and from there to black-hole horizon dynamics. The aim of the present work is not to replace general relativity, black-hole thermodynamics, Hawking’s semiclassical framework, or holographic theory. Rather, it is to introduce a complementary boundary-dynamical perspective in which black-hole horizon evolution is described through a local field equation defined on the horizon. In this formulation, accretion acts as an inward source of boundary evolution, Hawking radiation acts as a quantum depletion channel, and the horizon carries a local entropy-information density whose dynamics reflect the competing effects of influx, redistribution, and loss. The proposed framework builds on the previously developed BMG law, in which bulk growth is governed by sustained flux across a boundary, and on DHL, which promotes boundary information from a static scaling quantity to a dynamical field. In the black-hole setting, this logic leads naturally to the sequence: incoming flux → mass growth → horizon expansion → entropy growth, while also admitting an opposing depletion contribution associated with Hawking radiation. The present paper therefore treats black holes as a distinguished limit case of boundary-mediated dynamics, in which mass-energy accumulation, horizon geometry, and entropy are unusually tightly coupled. The result is a candidate boundary field equation for black-hole horizon dynamics, intended as an effective semiclassical law that unifies accretion, entropy production, boundary information, and quantum depletion within a single interpretive structure. The framework is presented cautiously and explicitly as a proposed extension compatible with established gravitational and thermodynamic theory, not as a substitute for it.