Jordan Spectral Response and Conserved Collective Charges in a Matrix Extension of Generalized Trace Dynamics: Paper P9 in a ten-part series P1-P10

Paper P9 of a ten part series P1-P10 on Interaction, Cohesive Emergence, and Fundamental Constants in Generalised Trace Dynamics P9 — Jordan response and conserved collective charges Exceptional Jordan spectra can acquire physical significance only after a dynamics, a state and a normalization prescription have been specified. We investigate this issue in a finite canonical matrix extension motivated by generalized trace dynamics. Seven Hermitian 3×3 matrices interact through commutator, octonionic cubic and radial quartic terms. At prescribed matrix charge we construct a relative equilibrium, certify consistency of its three-amplitude ansatz and find a locally positive normal Hessian. A principal spin projection defines an explicit Albert-algebra readout. After removal of the charge phase, its 20 independent linear readings miss three collective directions; retaining them permits regular elimination of 39 gapped coordinates and gives a positive induced kinetic matrix. Hidden $SU(2)$ rotations preserve the Jordan element and invariant spectra, although fixed-component responses change. A different effect survives for Jordan-invariant probes: static relaxation and slow closed-system evolution have distinct susceptibilities. We derive the positive collective-charge correction and the bound 0χdynχₛₜₐₜ. At the benchmark the quadratic Jordan invariant has an $8.0702%$ response deficit, and the generalized susceptibility ratios are $(0.916283932,1)$, equally for two independent Jordan roots. Nineteen prescribed nearby cases retain the tested stability and response properties. This supplies a reproducible dynamics-to-Jordan-response calculation in a specified finite parent. It does not derive that parent from microscopic generalized trace dynamics, identify its roots with fermion masses, or determine observed fundamental constants.