A Spectral Rank-Loss Criterion for Intrinsic Structural 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. We further show that, under natural invariance and monotonicity requirements, rank is the unique scalar spectral invariant compatible with admissible compressions, up to monotone reparametrization. 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.