Nonlinear Spectral Filtering in the Entropic Spacetime Field - A Mechanism for Emergent Filament Scale

This work explores the emergence of a characteristic filament scale within the Entropic Spacetime Field (ESF) framework using simplified numerical simulations. We study a governing equation combining relaxation, diffusion, gravitational coupling and nonlinear saturation. Despite the absence of a linear instability or imposed wavelength, the system consistently produces filamentary structures with a characteristic thickness of ~7–9 Mpc. Through 1D, 2D and 3D toy-model simulations, we show that this scale arises from nonlinear spectral reorganisation: amplitude-dependent damping suppresses long-wavelength modes, while diffusion suppresses short-wavelength modes, leaving an intermediate scale as a stable attractor. The result is robust across variations in resolution, solver choice, initial conditions and cosmic evolution (early-to-late time evolution). Fourier analysis reveals a persistent dominant mode at k ≈ 0.12–0.14 Mpc⁻¹, corresponding to the observed filament thickness scale. These findings suggest that filament thickness may emerge intrinsically from ESF dynamics, without requiring explicit wavelength tuning or additional scale-setting mechanisms. While limited to toy models, the robustness of the behaviour motivates further investigation in higher-dimensional and observational contexts.