Conserved Quadratic Content and Higher-Spectral Mobility in a Closed Relational Matrix Dynamics

Preprint / submitted manuscript. Currently under consideration at Foundations of Physics (Springer Nature). This version has not yet undergone journal peer review. Can a relational system reorganize its higher-order collective structure while preserving the totalquadratic strength of its pairwise correlations? We study this question on the finite-dimensionalstate space of complex Hermitian positive-semidefinite correlation matrices with unit diagonal. Forn = 4, we identify a closed relational dynamics consisting of a free commutator sector and an intrinsiccorrelation-dependent response. The flow preserves the admissible correlation state space and conserves Q2 (C) =tr(C 2 ) − 4 equivalently fixing the second spectral moment p2 (C) = tr(C 2). At the same time, the dynamics is notgenerically isospectral: higher spectral moments such as p 3 (C) and p 4 (C) remain dynamically mobile.This separates conserved quadratic correlation content from higher-order spectral reorganization. Theexecutable n = 4 law has been subjected to exact structural checks, blind-constructor provenance controls,external red-team review, and independent reconstruction from upstream mathematical authoritieswithout material executable mismatch or theory repair. A prospectively frozen numerical demonstrationprovides a direct visualization of the effect, and established complete four-mode tomography platformsprovide realistic experimental routes to a corresponding matrix-level test.Central result. A closed n = 4 correlation-matrix dynamics preserves p 2 exactly while allowinghigher spectral moments to change. In the prospectively fixed demonstration trajectory,Δp 2 = 0, Δp3 = −0.0373817585, Δp 4 = −0.1455242558at the fixed endpoints.