From Null Rejection to Structured Localization: Lambda Star and the Thermodynamics of Frequentist Statistics

Abstract. A statistically significant result shows that a model has detected a departure from the null hypothesis, but not whether it has captured most of the stable structure available to explain. This article develops a thermodynamic and symmetry-theoretic framework for distinguishing these questions. The null hypothesis is treated as a maximally symmetric noise state, whereas alternative hypotheses specify candidate locations for structured departure. Intrinsic noncentrality, λ*, quantifies the total structured departure relative to background variance. Model-captured noncentrality, λH, quantifies the part located by hypothesis H. The capture coefficient, κH = λH / λ*, expresses how completely the hypothesis captures the available structure. This distinction separates null rejection from structure localization: It not only indicates whether the study could reveal some structure, but also how much of it the selected model actually finds. A thermodynamic accounting further connects background variance with temperature, sums of squares with energy, residual degrees of freedom with entropy capacity, measurement precision with resolution, and predictor geometry with volume. The resulting free-energy criterion indicates whether improved fit repays dimension and geometry costs. Five controlled fixed-design simulations recover the predicted distinctions among significance, localization, observed fit, and thermodynamic preference. In a reanalysis of the OECD SSES 2019 older cohort N = 29,130, a 15-skill model was overwhelmingly significant and explained 47.4% of observed variation, yet localized only 54.2% of the estimated stable structure. An exploratory free-energy comparison favored a 14-skill state over the full model. The framework thus helps researchers distinguish statistical detection, observed fit, explanatory completeness, and warranted model complexity.