Cohesive emergence and the exceptional Jordan algebra: an exact dictionary at $N=3$, a $\mathbb Z_2$-even Jordan prior, and what it costs: Paper P4 in a ten-part series P1-P10

Paper P4 of a ten part series P1-P10 on Interaction, Cohesive Emergence, and Fundamental Constants in Generalised Trace Dynamics P4 — The matrix–Albert interface and flavour structure The octonionic matrix model of cohesive emergence and the exceptional Jordan algebra $J_3(\mathbb O)$ serve different purposes in the same programme: the first produces a fuzzy sphere and its records, the second carries the conditional fermion-mass ansatz of arXiv:2508.10131 --- a real coassociative configuration with spectrum $(s-\delta,s,s+\delta)$, $\delta^2=3/8$, centres as inputs, square-root masses from degree-3 monomials of the eigenvalues, and a stated list of what a dynamical completion must determine. We show that at $N=3$ the invariant sector $Q_A=\sum_kv_{Ak}J_k$ of the matrix model is the $(\mathbf7,\mathbf3)$ of the traceless Albert algebra under $F_4\supset G_2\times\mathrm{SO}(3)$, that on it the model's mass term and octonionic cubic are the two $F_4$-invariant forms $q(X)$ and $\det X$, and that the fuzzy sphere is the element with Jordan spectrum $(\kappa,\kappa,-2\kappa)$, a point of the Cayley plane whose degenerate generation pair is split only by massive modes (the five shape modes and four coassociative modes), never by a flat direction. We then give a symmetry prior that produces the ansatz's spectrum as a vacuum: an $F_4$-invariant potential whose self-interaction is even under $X\to-X$ has as its global minimum the $F_4$-orbit of the coassociative configuration, with equal spacing forced by the $\mathbb Z_2$, one $\delta$ for all sectors forced by $F_4$ invariance of the self-interaction (and preserved by cross-couplings only if these are sector-symmetric), centres from sector sources, and the earlier version's relation $X_R=-X_L+\mathrm{const}$ as the $\mathbb Z_2$ itself; the value $\delta^2=3/8$ is a coupling ratio. The price is established as theorems: on the Jordan sector there is no totally antisymmetric invariant cubic, hence no Myers term and no geometry, and the un-merged model with Jordan-valued matrices has only commuting vacua --- geometry and the generation ladder exclude each other in one field. For the $F_4$-covariant fermion bilinears on the $\mathbf{27}$ the octet masses are polynomials in the state's own eigenvalue with coefficients symmetric in the spectrum and the singlet block is permutation-covariant; the ansatz's selected monomials $(a^2b,abc,c^3)$, $(a^2b,abc,b^2c)$ are therefore not the spectrum of any polynomial covariant bilinear --- only the raw-cube law is --- which is the $F_4$ form of the ansatz's own statement that its selection and lift are symmetry-breaking data. Finally, every $F_4$-invariant coupling between two Jordan copies is a function of the frame-overlap matrix $P_{ij}=\operatorname{tr}(E_i\circ F_j)$, a complete invariant of their relative position; the realisable $P$ are the unistochastic matrices (two generic Albert elements generate a $J_3(\mathbb C)$); all couplings of the form $\operatorname{tr}(f(X_L)\circ g(X_R))$ are linear in $P$ and align the frames; the one further coupling at bidegree $(2,2)$ undoes this and yields either a degenerate vacuum or a minimiser on the boundary of the unistochastic set --- a real orthogonal mixing with one or three adjustable angles and no CP-type phase --- so that three-generation mixing can arise at this order but nothing fixes it. The two strands are faces of one algebra but of different elements; a model with both needs a $\mathbb Z_2$-odd geometry copy and a $\mathbb Z_2$-even mass copy, and of the completion list the symmetry fixes the form of the vacuum and proves that the selection and the lift cannot come from any $F_4$-covariant bilinear, while the mixing orientations it can supply are real and adjustable.