Interacting Generalised Trace Dynamics: Cohesive Emergence, Jordan Spectra and the Problem of Fundamental Constants

We investigate how proposed algebraic relations for particle masses, mixing and gauge couplings can be tested through explicit dynamics. We synthesise nine studies of interacting generalised trace dynamics and its finite matrix extensions, motivated by bosonic exchange between fermionic currents. In a specified physical-current model, two neutral composite condensates appear at a common instability and the global minimum preserves electromagnetism. An exact trace isometry connects a spin-one matrix sector to exceptional Jordan invariants. Vacuum selection places nontrivial restrictions on spectral proposals: one canonical fermion determinant selects commuting equilateral configurations near onset, while a separate polynomial parent favours an associative orientation with a degenerate Jordan doublet rather than equal spacing. In a fixed-charge parent, conserved collective charges produce a calculable difference between relaxed static and closed-system slow spectral response. These results identify dynamical tests for proposed fermion mass relations and the additional physical projection, family frames, kinetic normalisations and matching data needed to obtain masses, mixing and gauge couplings. We also formulate tests of whether the interaction preserves the additive sector's conditional spectral framework at finite coupling. The specified four-dimensional matter representation passes local anomaly tests and, for the stated global group on spin manifolds, the fermionic global test; gauging its family symmetry requires additional dynamical input. The results are established within their declared models; measured fundamental constants and a complete common emergence of spacetime, matter and classical records have not been derived.