Entropy reduction is not a safety criterion: it can leave predictions unchanged or accompany increasing risk for a protected group. We study entropy-regulated learning under separately specified task, risk, and intervention constraints. A noise-calibrated log-determinant functional yields an exact spectral force formula and a metric decomposition into observable and prediction-null velocities. For scalar feedback, we characterize the complete interval of coefficients preserving a task-descent fraction and meeting an entropy-rate request; restricting intervention to observable directions can make that interval empty. For several protected risks, we derive an exact minimax certificate of feasibility within a specified subspace and norm budget, an exact robust extension for uncertain constraint rows, and a cone-projection formula for the entropy reduction remaining after protection. These formulas use standard convex geometry but identify when apparently active entropy control has no admissible effect. Barrier comparisons, finite-step curvature and gradient-error margins, and fresh-data audits give distinct conditional guarantees for risk thresholds. Human authority enters through the specification and release policy; the mathematics does not identify compression with alignment or certify human well-being. A memorization counterexample separates fresh-input information from sample-level information, and a noisy sample-code assumption supports a conditional generalization bound. Reproducible mathematical checks and a small real-data experiment expose both feasible interventions and failures. The resulting theory characterizes admissible entropy regulation and its limitations, rather than asserting universal accuracy or safety improvements.

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