Collapse in learned systems is typically diagnosed post hoc, using performance degradation or task failure as proxies. Such criteria are inherently circular and fail to identify regions of structural risk before collapse occurs. Here we introduce the Semantic Information Law, an effective law that formalizes a necessary condition for collapse in supervised learning systems through a closed functional of information capacity, structural separation, spectral concentration, noise extensivity, and rank degeneracy. The law defines a scalar functional, Φ, that is independent of model accuracy and labels performance, and a non-circular collapse criterion based on permutation-induced indistinguishability. A system is said to collapse when its signal-to-noise structure becomes statistically indistinguishable from randomized label assignments. Low values of Φ delimit a region of structural risk in which collapse becomes possible but not inevitable. We validate the law on 5,803 classification datasets, identifying collapse events with 100% recall and a 262× separation between collapsed and non-collapsed systems. A held-out evaluation confirms that the Φ threshold generalizes without tuning, ruling out overfitting. We further introduce a memory coefficient that modulates collapse dynamics without altering the law itself. The Semantic Information Law does not predict collapse deterministically, nor does it assert universality beyond its declared domain. Instead, it establishes an invariant structural boundary for collapse, providing an early-warning signal grounded in information-theoretic and geometric principles.
Benjamín Felipe Pérez Contreras (Sat,) studied this question.
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