Dynamic Spatial Treatment Effect Boundaries: A Continuous Functional Framework from Navier-Stokes

I develop a comprehensive theoretical framework for dynamic spatial treatment effect boundaries using continuous functional definitions grounded in Navier-Stokes partial differential equations. Rather than discrete treatment effect estimators, the framework characterizes treatment intensity as a continuous function $\tau(\mathbf{x}, t)$ over space-time, enabling rigorous analysis of propagation dynamics, boundary evolution, and cumulative exposure patterns. Building on exact self-similar solutions expressible through Kummer confluent hypergeometric and modified Bessel functions, I establish that treatment effects follow scaling laws $\tau(d, t) = t^{-\alpha} f(d/t^\beta)$ where exponents characterize diffusion mechanisms. The continuous functional approach yields natural definitions of spatial boundaries $d^*(t)$, boundary velocities $v(t) = \partial d^*/\partial t$, treatment effect gradients $\nabla_d \tau$, and integrated exposure functionals $\int_0^T \tau \, dt$. Empirical validation using 42 million TROPOMI satellite observations of NO$_2$ pollution from U.S. coal-fired power plants demonstrates strong exponential spatial decay ($\kappa_s = 0.004028$ per km, $R^2 = 0.35$) with detectable boundaries at $d^* = 572$ km from major facilities. Monte Carlo simulations confirm superior performance over discrete parametric methods in boundary detection and false positive avoidance (94\% correct rejection rate versus 27\% for parametric methods). The framework successfully diagnoses regional heterogeneity: positive decay parameters within 100 km of coal plants validate the theory, while negative decay parameters beyond 100 km correctly signal when alternative pollution sources dominate. This sign reversal demonstrates the framework's diagnostic capability---it identifies when underlying physical assumptions hold versus when alternative mechanisms dominate. Applications span environmental economics (pollution dispersion fields), banking (spatial credit access functions), and healthcare (hospital accessibility). The continuous functional perspective unifies spatial econometrics with mathematical physics, connecting to recent advances in spatial correlation robust inference \citet{muller2022spatial} and addressing spurious spatial regression concerns \citet{muller2024spatial}.