A Necessary and Sufficient Spectral Criterion for Intrinsic Structural Order in hierarchical PSD correlation operators

This article establishes a minimal spectral principle for the emergence of intrinsic structural order in hierarchical systems of positive semi-definite operators. We show that intrinsic order appears precisely when the accessible spectral dimension of a system decreases strictly along an admissible hierarchical projection. Formally, if C_k denotes a sequence of positive semi-definite correlative operators and Cₖ₊₁ = P_k C_k P_k^T is an admissible projection, intrinsic structural ordering occurs if and only if rank(Cₖ₊₁) < rank(C_k). This strict spectral rank loss corresponds to the disappearance of at least one independent correlative mode and induces an irreversible reduction of the accessible correlation space. The result is purely spectral and does not rely on metric, probabilistic, or temporal assumptions. It therefore provides a minimal structural mechanism for the emergence of order in hierarchical systems. This work contributes to the mathematical foundations of Correlative Structure Theory (CST).