A Hierarchical Fractional N-body Framework for Coulomb Stress Evolution in Fault Networks: Exact Analytical Solutions and Falsifiable Seismicity Scaling Laws

Abstract. We develop an exact analytical framework for the hierarchical fractional dynamics of interacting many-body systems and apply it to the nonlinear Coulomb stress transfer that governs seismicity in fractally organised fault networks. The framework rests on a scaling relation α_k= 2 − 2 / (N_k+ 1) that connects the order of a Riemann–Liouville fractional evolution equation at hierarchical level k to the number N_k of interacting bodies at that level, derived in a companion paper (Chishtie, 2026, Physics Open) and applied here to the seismogenic setting. Closed-form solutions are obtained via parametric trigonometric representations σ_k(θ) ∝ sin⁴ θ at each level together with Chebyshev polynomial inversions of the time–parameter relation, and they converge to the classical wave equation as N_k → ∞. We embed the framework into the Time-Dependent Stress Response (TDSR) seismicity model of Dahm and Hainzl (2022, J. Geophys. Res.) to describe the hierarchical Coulomb stress cascade through a fractally organised fault network. Three quantitative, falsifiable scaling laws follow from the closed-form solutions with no free parameters: an Omori–Utsu aftershock decay exponent p_k = α_k/ 2 = 1 − 1 / (N_k+ 1) that stratifies across generations; a spatial Coulomb stress falloff ℓ^−(1+α_k) that departs measurably from the classical elastic ℓ⁻³ law; and a Gutenberg–Richter b-value b_k ≈D_f N_k/ (N_k+ 1) that accounts for the longstanding discrepancy between the fractal-dimension expectation b = D_f≈ 1.5 – 2 and the commonly observed b ≈ 1 as a finite-N_k fractional correction. Numerical verification using the Grünwald–Letnikov scheme against the Mittag-Leffler exact series solution of the fractional relaxation equation confirms the analytical results. The standard TDSR model is recovered exactly as N_k → ∞ at all hierarchical levels and is therefore a special case of the present framework. We present illustrative qualitative comparisons with the 2023 Kahramanmaraş doublet and the 2024 Noto Peninsula earthquake; a systematic generation-stratified analysis across many sequences is set out as the primary observational test of the framework.