A Spectral Rank-Loss Criterion for Intrinsic Order in Hierarchical Correlative Structures

This manuscript establishes a necessary and sufficient spectral condition for intrinsic structural order in hierarchical sequences of finite positive semi-definite operators. Given a sequence obtained through admissible projections, we prove that strict rank loss is equivalent to a strict reduction of accessible spectral dimension and induces a non-degenerate partial order across levels. The result is purely spectral and does not rely on any metric, probabilistic structure, dynamical law, or external time parameter. The framework is representation-independent and invariant under permutation, orthogonal transformations, admissible block coarse-graining, and Laplacian normalization. A minimal computational illustration is provided using a two-block Laplacian model. The manuscript is self-contained and focuses exclusively on the structural properties of finite-dimensional PSD operators.