Born’s Rule from Contextual Relative-Entropy Minimization

We derive the Born rule from a simple variational principle. Starting from a fixed cover of measurement contexts, we measure how far a quantum state lies outside the non‑contextual polytope via Umegaki relative entropy. Minimizing this “contextual divergence” selects the usual trace‑form probabilities. We assume finite dimensions, full‑rank states, and rank‑1 projective contexts. Our construction exactly reproduces the sheaf‑theoretic definition of contextuality and rests on Petz’s projection theorem. Its chief novelty is to blend three previously separate strands—information‑geometric projections, sheaf‑cohomological contextuality, and categorical classical structures—into a single optimization principle. Unlike Gleason’s theorem, decision‑theoretic reconstructions, or envariance arguments, our approach is less general in scope but richer in its explicit treatment of contextuality and relational viewpoints.