Diagnostic Map of Constraint-Coupled Vacua in String-Inspired Models

We present a diagnostic framework for identifying when additive and multiplicative contributions to physical observables are structurally coupled by constraints in string-inspired vacuum ensembles. Using principal component analysis (PCA) applied to pairs of additive observables and logarithmic multiplicative observables, we define a “corridor regime” characterized by intermediate eigenvalue ratios R_{\mathrm{PCA}} \in [1.3, 3.0], corresponding to moderate correlation. We show that this regime emerges if and only if constraints directly couple the signed values of additive and multiplicative sectors, as in superpotential stabilization conditions, and does not arise from unsigned bounds or decoupled hierarchies. The diagnostic is applied to several classes of toy models motivated by string compactifications, including flux superpotential ensembles, KKLT-type stabilization, large-volume scenarios, and multi-moduli models with D-terms. We find that KKLT-like and multi-instanton constructions exhibit corridor signatures, while purely topological data and simple large-volume models remain diffuse. The framework is explicitly non-predictive and does not select vacua; rather, it provides a structural map of constraint coupling across vacuum ensembles. This diagnostic may serve as a tool for landscape cartography and for comparing stabilization mechanisms in large statistical scans.