A Mathematical Theory of Contradiction

We introduce an information-theoretic framework for quantifying perspectival contradiction—situations where multiple legitimate observational contexts yield data that no single, frame-independent account can reconcile. Starting from six elementary axioms, we prove that any admissible contradiction measure must take the form K(P) = −log₂ α*(P), where α*(P) = max over Q in FI of min over contexts c of BC(p_c,q_c). Here, FI represents the frame-independent polytope and BC denotes Bhattacharyya affinity. The first set of theorems establishes the axiomatic core (minimax representation, Bhattacharyya as the unique kernel, logarithmic formula, additivity, and monotonicity). The second set derives operational laws: +K bits/symbol in compression, type-II testing exponents, ≥2K predictive regret, communication penalties, and a Hellinger-geometry view of composition.

We demonstrate computational tractability through a minimal three-view odd-cycle device yielding K = 0.5·log₂(3/2) bits per observation, alongside a convex minimax program and plug-in estimator with bootstrap confidence intervals implemented in our reference software (contrakit). The framework naturally recovers quantum contextuality as a special case—K is a contextuality monotone when FI is the non-contextual set—while generalizing to arbitrary domains with context-indexed observations. In essence, while entropy quantifies the cost of randomness within a single frame, K measures the fundamental price of incompatibility across multiple frames of reference.

Keywords: perspectival contradiction, contradiction bits (K), frame-independent set (FI), Bhattacharyya affinity, Rényi-1/2, Hellinger distance/, minimax, contextuality, non-contextual models, odd-cycle / KCBS, resource theory, information theory, additivity, data-processing inequality, compression, rate–distortion, hypothesis testing, Chernoff, prediction regret, common message, common reconstruction, Gray–Wyner, convex program, plug-in estimator, bootstrap CIs, ensembles, distillation, multi-context learning, distributed consensus, contrakit, open-source, Python