The Principle of Local Coherence A Geometric Foundation for Learnability Analysis in Classification, Regression, and Anomaly Detection

We present a geometric framework demonstrating that learnability across supervised and mone-class learning paradigms can be characterized by a single principle: the comparison of local versus global structure in the input space. Starting from first principles—specifically, the Locality Axiom and the Information Axiom—we derive three domain-specific metrics: ρclass for classification, ρreg for regression, and e for one-class anomaly detection. We show theoretically and empirically that these metrics instantiate a common geometric pattern: Learning succeeds ⇔ Local structure differs from Global structure Extensive empirical validation on 552 real classification datasets from OpenML (+0.903 Spear- man correlation), 1,452 real regression datasets (identity theorem holds in 87% of cases), and 373 anomaly detection experiments confirms strong predictive accuracy. We further demonstrate algorithm-agnosticity through validation on SVM, Random Forest, Logistic Regression, and ML Pacross 100 datasets. These results suggest the framework captures intrinsic dataset propertiesrelevant to learnability.