Information Geometry of Bounded Rationality: Entropy--Regularised Choice with Hyperbolic and Elliptic Quantum Geometries

Models of bounded rationality include quantum--like (QL) models, which use Hilbert--space amplitudes to represent context and order effects, and entropy--regularised (ER) models, including rational inattention, which smooth expected utility by adding an information cost. We develop a unified information--geometric framework in which both arise from the same structure on the probability simplex. Starting from the Fisher--Rao geometry of the open simplex $\Delta^{n-1}$, we formulate \emph{least--action rationality} (LAR) as a variational principle for decision dynamics in amplitude (square--root) coordinates and lift it to the cotangent phase space $N:=T^*\mathbb R^n$ of unnormalised amplitudes. The lift carries its canonical symplectic form and a para--K\"ahler geometry. For a linear evaluator $\widehat V=\widehat S+\widehat F$ with symmetric part $\widehat S$ and skew part $\widehat F$, the dynamics separate an evaluative channel from a circulatory (co--utility) channel. On a distinguished zero--residual Lagrangian leaf the flow closes as a split--complex (hyperbolic) Schr\"odinger--type evolution, and observable probabilities follow from a quadratic (Born--type) normalisation. When reduced to the simplex, the induced preference one--form decomposes into an exact utility component and a divergence--free co--utility component whose curvature measures path dependence. Context effects, order effects, and interference--like deviations from the law of total probability emerge as projections of this latent rational flow. Finally, standard complex (elliptic) quantum dynamics arises within this real symplectic phase space by imposing an additional K\"ahler polarisation that restricts admissible variations. Unitary evolution is thus a coherent restriction of the underlying least--action framework rather than a primitive postulate.