Geometric derivation of probability weighting via complex-valued economics: A deductive microfoundation for prospect theory

For decades, behavioural economics has used experimental data alongside inductive and axiomatic reasoning to model choice under risk. While descriptively powerful, the probability weighting function used in prospect theory lacks a deductive microfoundation. Building on the recent framework of complex-valued economics, here we derive an inverse-S function by projecting linear probability $p+q=1$ onto the cognitive unit circle p²+q²=1 in the complex plane. This geometric projection reproduces empirical probability distortion and proves subcertainty as a Pythagorean discount, with zero statistical degrees of freedom at the theoretical baseline. This suggests that behavioural anomalies—long studied as systematic deviations from rationality—may instead emerge from the natural mathematics of optimisation operating in the complex plane, and that the nervous system may implement protogeometric constraints. Grounded in the marginal rate of substitution (MRS), our minimal-assumption PDE framework (β²∂ₓ²U=∂y²U) constitutes a master-key theory, recovering behavioural and classical economics as two transformable regimes of a single optimisation primitive.