Functional Decoupling Framework for P-adic Hamiltonian Models of Granular Field Interactions: Addressing Continuous Temporal Evolution versus Discrete Physical Granularity

The reconciliation of continuous temporal evolution with discrete physical granularity remains a central challenge in theoretical physics, requiring novel mathematical approaches to model localized action within unbroken fields (Crespo & Pelayo, 2025). Standard Hamiltonian models assume smooth Archimedean spaces, which fundamentally fail at the Planck scale where non-Archimedean, p-adic geometries appear to dictate energetic interactions (Peterson, 2025). This study introduces a functional decoupling framework that divides the total Hamiltonian into localized workers, delocalized signals, and a mathematically rigorous interaction term. We employ p-adic heat equations and subhyperbolic dynamics to model continuous time over discrete space, a method validated through Python-based numerical simulations of 1D p-adic random walks and tripartite energy transfer protocols. Analytical proofs demonstrate strict energy conservation ($\frac{d\mathcal{H}_{total}}{dt} = 0$) within the tripartite structure. Revised computational simulations, which model energy transfer using a discrete approximation of the Vladimirov operator and stochastic quantum fluctuations, confirm this theoretical foundation by showing constant total energy ($E=10.0$) during signal-worker coupling. Furthermore, worker trajectories exhibit bounded fractal diffusion with a mean p-adic displacement of 0.693 (variance $\sigma^2 \approx 0.122$), contrasting sharply with Euclidean divergence. These results offer a preliminary framework for resolving the smooth evolution paradox and addressing the mismatch between macroscopic models and Planck-scale granularity. As a candidate for Planck-scale modeling, the framework provides a rigorous mathematical bridge for integrating discrete wave equations with continuous signaling fields, offering profound implications for Einstein causality and the deterministic correlated history of non-Archimedean spaces.