A Mathematical Formalization of Legal Decision-Making Incorporating Structural Inequality via Stochastic Dynamics and Tensor Analysis

We propose a mathematical framework to formalize legal decision-making by integrating structural inequality into stochastic dynamical systems and tensor analysis. The legal system is modeled as a dynamic entity evolving under internal and external forces, where judicial outcomes are influenced by both deterministic processes and inherent uncertainties. The framework introduces a Hilbert space representation of legal norms, capturing their discontinuous genesis through a sovereign decisionist act, then traces their interpretive trajectory via a stochastic differential equation that accounts for institutional and societal pressures. A key innovation is the Structural Inequality Tensor, which quantifies the differential impact of economic, social, and symbolic capital on legal outcomes, transforming raw power into effective force within a Riemannian manifold of legal space. The probability of judicial decisions is derived from a logistic function that explicitly incorporates these inequalities, providing a predictive tool for analyzing bias in legal processes. The proposed method bridges normative legal theory with empirical sociopolitical dynamics, offering a rigorous yet interpretable model for studying how structural disparities shape legal outcomes. Moreover, the framework enables systematic comparisons of legal systems across jurisdictions by quantifying the permeability of institutional boundaries to external influences. This approach advances interdisciplinary research at the intersection of law, mathematics, and social science, with potential applications in policy design, judicial reform, and inequality studies.