From Generalised Trace Dynamics to Jordan Spectra: Normalisation, Vacuum Selection and the Remaining Microscopic Inputs: Paper P8 in a ten-part series P1-P10

Paper P8 of a ten part series P1-P10 on Interaction, Cohesive Emergence, and Fundamental Constants in Generalised Trace Dynamics P8 — Normalised Jordan spectra and dynamical selection Exceptional-algebra proposals for fermion mass ratios require a dynamicalstate and a normalised physical mass operator. We investigate this connectionin finite reconstructions motivated by generalised trace dynamics.An explicit trace isometry maps a spin-one matrix sector to centred Albertinvariants, making its Jordan spectrum calculable once the vacuum is selected.We also match an ordered graded coassociative vertex to pair transfer under adeclared canonical mode hypothesis. For one exact canonical fermion determinanton seven traceless Hermitian $3\times3$ matrices, we prove the identity\[ \Tr\cD^4=144\Tr\!\left(\sum_AQ_A^2\right)^2 +8\sum_{A,B}\norm{[Q_A,Q_B]}_F^2.\]Using this identity and the exact determinant, we prove that the globalminima are equilateral commuting configurations sufficiently close to onset.A competing balanced spin-one branch is locally stable at the tested states,carries the coassociative Jordan spectrum, and becomes energeticallycompetitive deeper in the ordered regime. We calculate a crossing of theexact branch energies, identify a third stationary branch, report unrestrictedsearches and give rigorous energy brackets. A separate finite-parent responsecalculation illustrates normalised current selection and the danger ofintegrating out light modes. The results supply a conditional structurallink between earlier spectral proposals and dynamics. They do not derivethe measured mass ratios or gauge couplings, select physical spacetime, or complete cohesive emergence.