Continuum Limit of Dynamic Gravity and Fluctuation Equations: A Structural Correspondence with Quantum Mechanics

This paper develops the continuum formulation of a discrete, minimal‑length–based theory of dynamic gravity. Building on Papers A–E, where static and dynamic gravity were shown to arise from the expansion and contraction of the minimal length, this work introduces a continuum limit that transforms discrete update chains into differential equations. The study first establishes how the minimal length, update density, and temporal increments become continuous fields in the limit of vanishing update spacing. It then derives fluctuation equations that capture the intrinsic, unavoidable jitter originating from discrete updates. These fluctuations persist in the continuum limit and naturally produce statistical structures—power spectra, correlations, and phase behavior—that mirror the core features of quantum mechanics. By expressing the fluctuation field in complex form, the resulting equations exhibit a structural correspondence to the Schrödinger equation, suggesting that quantum behavior arises not from external randomness but from the intrinsic discreteness of spacetime updates. This work provides the theoretical bridge between discrete dynamic gravity and quantum‑mechanical behavior, forming the foundation for Paper G, where the framework is extended toward quantum field theory and cosmological applications.