The Power-Law Exponent in Efficient Coding: What q Can and Cannot Identify

Power-law efficient coding is widely used to explain how perception allocates limited resources. In many studies, the power-law exponent q—the exponent on the prior density, such that J(x)∝p_prior(x)^q—is treated as the central result: a larger q is read as a stronger tilt toward likely stimuli, and a change in q is interpreted as a change in what the system optimizes—for example, from accuracy to reward. We show that this interpretation is not licensed by the model itself. The model has three theoretical parameters—a coding objective α , a resource-constraint form β, and a budget c—but produces only two observable quantities: the exponent q and the scale k. Because q=1/(α+β) , the same fitted code corresponds to a one-dimensional family of theoretical interpretations. We call this family a gauge orbit; a simpler name is a family of equivalent interpretations. Along this family, the code and all behavioral predictions remain fixed, while α and c change. Therefore q alone cannot identify what a system optimizes. This redundancy is not merely a caution. It yields three practical tools. A prior-free closure test, c=k^1/q , checks any infomax claim without estimating the stimulus distribution. An independently measured budget reduces the orbit to at most two candidate interpretations, with the residual ambiguity physiological rather than behavioral. And an observed budget below the orbit minimum or at or above the stimulus support measure falsifies the framework at the fitted (q,k) . We illustrate these tools by re-analyzing a reward-manipulation study and show that a null result on k cannot be read as a null result on the budget, because k is dimensionally incommensurable across a change in q. The broader lesson is not that the framework is flawed, but that its central exponent carries less theoretical information than its reporting practice assumes. A model can over-determine its observable predictions relative to its theoretical parameters, and when it does, the excess has an exact structure—one that can be written down, tested, and in part broken. Recognizing that structure is what makes q interpretable rather than merely reportable.