The Depth of Almost Nothing IV: Algebraic Detection of a Near-Null Spectral Hierarchy

This is the fourth in a series studying an explicit prime-Archimedean spectral construction. The finite real symmetric matrix examined throughout is built from the same aggregate as Volumes I-III of this series, and at every size examined its low spectrum is sharply separated: the smallest eigenvalue sits far below the next level, itself well separated from the third. This volume shows that both scales of this finite hierarchy — the absolute depth of the smallest eigenvalue and the relative separation between the next two levels — can be recovered directly from the characteristic coefficients of the matrix, without diagonalization and without selecting which eigenvalue carries the hierarchy. The absolute depth is recovered through an exact algebraic identity. The relative separation is recovered through a controlled finite-dimensional expansion, derived from the elementary symmetric functions of the reciprocal spectrum, with an explicit leading-order error term. The two algebraic observables are combined into a joint certificate and tested against structured controls that preserve the arithmetic data entering the matrix while altering its specific pairing: in every tested configuration, none of the sampled controls reproduces either coordinate of the unperturbed certificate. A further, independent finite-dimensional fact concerns the sign of the determinant, which is positive across all tested sizes for the unperturbed construction while control trajectories lose this sign much earlier. No eigenvector is constructed from these algebraic certificates, no asymptotic law is established for either spectral scale or for the determinant's sign, and no implication for the location of the zeros of the Riemann zeta function is claimed. The results are strictly finite-dimensional.