The Dynamic Holographic Law: A Boundary Field Equation for Flux-Driven Growth and Instability

Abstract We introduce the Dynamic Holographic Law (DHL), a boundary field equation for the evolution of boundary information in open systems. The framework is motivated by two broad observations. First, many physical descriptions with holographic character suggest that boundary measures can encode bulk structure. Second, across natural growth systems, accumulation in the interior is constrained by transport across boundaries. What has remained missing is a dynamical law that directly links boundary information, incoming flux, and bulk growth within a single formal structure. To address this gap, we formulate a continuity equation on evolving system boundaries in which flux acts as a source term for boundary information density. In this formulation, boundary information is not treated as a static geometric bookkeeping quantity, but as a dynamical field whose evolution reflects transport, redistribution, and local amplification or smoothing along the interface. Under geometric scaling, the formalism yields the Boundary-Mediated Growth (BMG) law as a derived consequence rather than an independent postulate. In this sense, the present work is not a replacement for holographic ideas, but a dynamical extension built on boundary-based scaling intuition and explicitly connected to open-system growth. The theory further produces an instability condition showing when amplification of boundary flux overcomes smoothing and leads to interface growth, roughening, or pattern formation. This places the framework in direct conceptual dialogue with classical free-boundary and interfacial instability theories, while extending the analysis toward biological and tumor-relevant systems in which perfusion, metabolic exchange, and boundary irregularity play important roles. The result is a unified description in which geometry, transport, and boundary information are coupled through a single dynamical law.