Additive–Multiplicative Competition and Corridor Regimes: A Portable PCA Diagnostic

This technical note introduces a portable diagnostic for identifying structural regimes in bivariate data arising from the interaction of additive and multiplicative mechanisms. Using the eigenvalue ratio of a two-variable Principal Component Analysis (PCA), we distinguish three regimes: diffuse (structures independent), rank-1 (one structure dominates), and an intermediate corridor regime in which additive and multiplicative contributions compete. We show that the corridor regime corresponds to moderate correlation between additive and multiplicative axes and is characterized by PCA eigenvalue ratios in the range R_{\mathrm{PCA}} \in [1.3, 3.0]. This regime emerges consistently across disparate domains, including Goldbach partition statistics (number theory), network growth models, and synthetic constructions with explicit additive–multiplicative coupling. The key result is that corridor behavior is not domain-specific but arises generically when neither additive accumulation nor multiplicative compounding dominates. The PCA eigenvalue ratio provides a simple, interpretable diagnostic for detecting this competition. The framework is intended as an exploratory tool for structural analysis and does not replace domain-specific modeling or inference.