Born's Rule from Contextual Relative-Entropy Minimization

We give a variational characterization of the Born rule. For each measurement context, we project a quantum state ρ onto the corresponding abelian algebra by minimizing Umegaki relative entropy; Petz's Pythagorean identity makes the dephased state the unique local minimizer, so the Born weights pC(i)=Tr(ρPi) arise as a consequence, not an assumption. Globally, we measure contextuality by the minimum classical Kullback-Leibler distance from the bundle {pC(ρ)} to the noncontextual polytope, yielding a convex objective Φ(ρ). Thus, Φ(ρ)=0 exactly when a sheaf-theoretic global section exists (noncontextuality), and Φ(ρ)>0 otherwise; the closest noncontextual model is the classical I-projection of the Born bundle. Assuming finite dimension, full-rank states, and rank-1 projective contexts, the construction is unique and non-circular; it extends to degenerate PVMs and POVMs (via Naimark dilation) without change to the statements. Conceptually, the work unifies information-geometric projection, the presheaf view of contextuality, and categorical classical structure into a single optimization principle. Compared with Gleason-type, decision-theoretic, or envariance approaches, our scope is narrower but more explicit about contextuality and the relational, context-dependent status of quantum probabilities.