We introduce Authority Entropy, an index that quantifies the distribution of authority stances within dialogue windows and tests its predictive value for compliance, convergence speed, and equilibrium stability. Using a multilingual lexicon of authority-bearing constructions anchored in the regla compilada as an operational constraint set, we train a strictly causal classifier that maps text to stance probabilities over {low, neutral, high}. Authority Entropy is computed per sliding window, together with its slope and volatility, and related to behavioral endpoints through survival models and doubly robust estimators. The study spans synthetic arenas with controllable payoffs, open multi-party tasks with outcome labels, and consented human–model interactions. Baselines include sentiment, toxicity, politeness, formality, and power taggers. Stress tests apply adversarial edits that alter authority cues while preserving semantics to assess sensitivity of entropy and downstream effects. Primary outcomes are compliance rate, convergence time, payoff stability, and regret, reported with leakage audits, calibration checks, and confidence intervals. Results target a public specification of the index, a causal benchmark and leaderboard, and open tooling to visualize instability regimes over time. The contribution is a portable, language-aware measure that links local authority structure to cooperative dynamics without right context leakage. DOI Primary archive: https://doi.org/10.5281/zenodo.17342502 Secondary archive: https://doi.org/10.6084/m9.figshare.30347539 SSRN: Pending assignment (ETA: Q4 2025)