Cohesive emergence and the exceptional Jordan algebra: an exact dictionary at $N=3$, a Z₂-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 J₃( 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-δ,s,s+δ), δ²=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 QA=∑ₖvAkJₖ of the matrix model is the (7,3) of the traceless Albert algebra under F₄⊃ G₂(3), that on it the model's mass term and octonionic cubic are the two F₄-invariant forms $q(X)$ and X, and that the fuzzy sphere is the element with Jordan spectrum (κ,κ,-2κ), 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 F₄-invariant potential whose self-interaction is even under X→-X has as its global minimum the F₄-orbit of the coassociative configuration, with equal spacing forced by the Z₂, one δ for all sectors forced by F₄ 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 XR=-XL+const as the Z₂ itself; the value δ²=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 F₄-covariant fermion bilinears on the 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 (a²b,abc,c³), (a²b,abc,b²c) are therefore not the spectrum of any polynomial covariant bilinear --- only the raw-cube law is --- which is the F₄ form of the ansatz's own statement that its selection and lift are symmetry-breaking data. Finally, every F₄-invariant coupling between two Jordan copies is a function of the frame-overlap matrix Pᵢⱼ=tr(Eᵢ∘ Fⱼ), a complete invariant of their relative position; the realisable P are the unistochastic matrices (two generic Albert elements generate a J₃( C)); all couplings of the form tr(f(XL)∘ g(XR)) 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 Z₂-odd geometry copy and a Z₂-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 F₄-covariant bilinear, while the mixing orientations it can supply are real and adjustable.