Description
This monograph develops a unified geometric field theory of probability and decision-making. Departing from classical probability weighting curves, probability is treated as a structured fieldendowed with density, gradients, curvature, and global topology. At the local level, decision behavior is described by the full differential system where gradients generate decisional forces and curvature governs stability. A Figure–Background decomposition introduces two primitive geometric quantities - Tension and Congruence - from which local forces, stabilization, and bifurcations arise naturally. Local heuristics are formalized as kernel systems that form an atlas over the probability domain. Their global inconsistency is measured by a functional S2_Index, which quantifies the excess of destabilizing tension over stabilizing congruence across scales and charts. System I corresponds toregimes of local geometric stability, while System II emerges as a global regulatory processminimizing this inconsistency. This regulation is shown to follow a natural-gradient flow in the geometry of information, admitting deterministic, stochastic (Langevin), and population-level (Fokker–Planck) formulations. A central contribution is the reinterpretation of the Arrow–Pratt risk aversion measure as a localcurvature invariant of the decision field rather than a preference parameter. Sign changes of thisquantity mark bifurcation boundaries between stable and unstable decision regimes andaccumulate into global geometric and energetic effects, including negative Ricci curvature and polarization. These effects are unified within a free-energy framework, where System II minimizes cognitive free energy balancing inconsistency (energy) and diversity (entropy). The theory subsumes classical Prospect Theory and variational Bayesian inference as limiting casesand yields clear, falsifiable empirical predictions. A complete experimental protocol is provided for reconstructing the decision field, estimating curvature-based quantities, detecting System II activation, and testing the theory against behavioral data. Overall, the monograph reframes decision-making as a geometric and energetic process unfolding in probabilistic fields, offering asingle mathematical language for heuristics, deliberation, risk, and cognitive control.