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

We study the finite-dimensional state space of complex Hermitian positive-semidefinite correlation matrices with unit diagonal for n = 4. We identify a closed matrix dynamics that combines a free commutator sector with an intrinsic correlation-dependent response. The flow preserves the admissible state space and exactly conserves the quadratic pair-correlation content Q₂(C) = ½ [tr(C²) − 4], equivalently fixing the second spectral moment p₂(C) = tr(C²). At the same time the dynamics is not generically isospectral: higher spectral moments such as p₃(C) = tr(C³) and p₄(C) = tr(C⁴) remain dynamically mobile. The n = 4 case is the first dimension in which this separation becomes nontrivial. At n = 3 the relevant skew response sector vanishes identically due to a Cayley–Hamilton obstruction — a dimension-specific algebraic fact, not a physically motivated choice of dimension. A concrete experimental target is outlined: a complete 4×4 complex correlation (coherence) matrix, reconstructed by established multi-mode tomography, should exhibit |Δp₂| below a pre-declared equivalence margin while at least one of Δp₃ or Δp₄ remains resolvably nonzero along a controlled non-unitary trajectory. Pure unitary conjugation serves as an exact negative control, since it leaves every spectral moment invariant. This is a preprint of the mathematical construction and its proposed matrix-level experimental test. No physical interpretation beyond the stated matrix dynamics is claimed.