Compactification and the Four-Dimensional Limit Stable Geometry, Exact Family Breaking, the Singlet-Quadrupole Flavour Boundary, and Finite BV/BRST Closure

This paer derives an exact four-dimensional Minkowski compactification of the canonical CP1 x CP2 carrier with a positive homogeneous moduli Hessian. The external spin-two tower contains one massless graviton, the universal vector, tensor and diagonal scalar harmonic sectors are non-tachyonic in the selected gauge complex, and the chiral and weak spectral ranks persist on open gap-protected neighbourhoods. The CP2 metric and Berry colour connections are one SU(3)c field. The CP1 isometry SU(2) is instead a family group acting on F=H^0(CP1,O(2)) congruent to the spin-one triplet, while weak SU(2)W is a separate rank-two spectral bundle. Canonical Toeplitz compression identifies the complete l<=2 family band with End(F)=1+3+5 and transports the CP1 Laplacian to eigenvalues 0,4,12. Two noncommuting finite-Dirac arms generate a stable fully broken family vacuum, and every positive factorising fermionic tower strengthens the conservative condensation test. The zero-heavy audit supplies a genuine positive result at the operator level: the Toeplitz family truncation is not closed under multiplication by the CP1 coordinate functions, and a calculable fraction of each arm leaks into the heavy complement. The isotropic total leakage is exactly one half of the family identity. For one arm the Schur complement of that leakage is a family singlet plus an axial quadrupole: it changes eigenvalue spacing without rotating the arm eigenbasis. In the unbroken phase the up- and down-sector heavy spaces remain separated by their weak and hypercharge quantum numbers, so no common heavy mediator connects the two outer nodes. Paper boundary. The carrier has exact flavour capacity but no derived dynamical bridge to the observed hierarchy and small CKM mixing. Further progress requires an enlarged finite incidence justified independently of the desired flavour result. Keywords: stable compactification; family zero modes; Bergman projector; Toeplitz leakage; Schur complement; heavy-mode mixing; axial quadrupole; CKM hierarchy; finite BV/BRST. Version 2.8 closes the remaining mediator class. Vector-like partners of the left node evade species superselection, but for couplings linear in the singlet-plus-adjoint vacuum the induced left metric carries a quadrupole of the exact form (C/3)I-N with N positive semidefinite, giving the singlet-quadrupole bound S>=2 lambda_max(Q) and the scale-invariant reachability condition R>1. Every left metric that reproduces the six quark masses and three CKM angles is biaxial, misaligned by forty-five degrees from the arms, and has R between 0.264 and 0.879. The flavour boundary of Paper 8 is therefore a single statement: the data require independent l=2 symbol content that the derived vacuum does not excite.