Interacting Generalised Trace Dynamics: Cohesive Emergence, Jordan Spectra and the Problem of Fundamental Constants: Paper 10 in a ten-part series P1-P10

Paper P10 of a ten part series P1-P10 on Interaction, Cohesive Emergence, and Fundamental Constants in Generalised Trace Dynamics P10 — Synthesis, physical compatibility and fundamental constants Can the exceptional algebras that organise candidate particle properties alsodescribe the dynamics that selects them? We give a pedagogical researchsynthesis of nine studies of interacting generalised trace dynamics and itsfinite matrix extensions. Interactions between spacetime--matter atoms aremotivated by bosonic exchange between fermionic currents. An orderedcoassociative vertex, its graded selection rules and a positive mediatedkernel make this proposal calculable. A selected physical-current model hasa common onset of two neutral composite condensates, while separate matrixparents exhibit exact ordering bounds, competing geometric configurationsand finite-time record protocols. A trace isometry connects a spin-onematrix sector to exceptional Jordan invariants. For one exact canonicalfermion determinant, a positive-squares identity proves that commutingequilateral configurations, rather than the Jordan-bearing balanced branch,are selected sufficiently close to onset. A complementary exact resultshows that the polynomial parent's cubic selects associative orientationsof the balanced spin-one branch, whose centred Jordan spectrum is adegenerate doublet. Equal spacing is not stationary under thoseunrestricted orientation variations. Massive canonical fermions canprovide positive local orientation curvature, but cancellation of thecubic slope and full vacuum stability are separate conditions. A further fixed-charge parentsupports a stable local branch whose Jordan response requires additionalcollective coordinates. Conserved charges then produce a calculable,coordinate-invariant difference between relaxed static and closed-systemslow spectral response. These results put earlier mass-ratio, mixing andgauge-normalisation proposals into contact with explicit vacuum and responsecalculations. They also identify their remaining inputs: the microscopicphysical projection, state, relative family frames, kinetic residues andmatching data. Full cohesive emergence of spacetime, matter and classicalrecords, and the measured fundamental constants, have not been derived.The interaction is required to preserve the additive STM sector'sconditional spectral framework. Recovery at zero coupling does notestablish this compatibility at finite coupling; we formulate the missingprojection and backreaction tests explicitly.The achievement is a set of constructive bridges and discriminating tests,with a concrete route from algebraic hypotheses to physical predictions.