The Semantic Tolerance Law A Statistical Law Part 1: The Manifesto

We present the Semantic Tolerance Law, the first statistical law for predicting dataset structural collapse in machine learning. Like other statistical laws (Zipf's law for language, Benford's law for numerical distributions), this law holds empirically with high probability rather than as an absolute deterministic rule. From the foundational axiom that a learnable structure is captured by mutual information I(X;Y), we derive a composite metric Φ combining information-theoretic bounds and geometric properties. Validated across 13,018 classification datasets (from 54,370 scanned) spanning 21 domains, the law predicts collapse with 98.4% success rate on curated data (OpenML) and 66.1% adjusted success on wild data (Kaggle)—higher reliability than most established statistical laws (Benford ~85%, Zipf ~90%). The framework combines: 40% rigorous theoretical foundation (established theorems) 30% theoretically motivated components (concentration principles) 25% empirical validation 5% engineering design choices Like thermodynamic laws (statistical mechanics) and information laws (Shannon's theorems), this is a probabilistic law with known boundary cases (<6% prevalence). It provides machine learning with a systematic framework for pre-training quality assessment. Threshold Φ_c ≈ 0.007 shows domain-dependence (range 0.003-0.012) but the core principle appears universal across tested domains.