The Projective Residue as a Schur Complement: Reduction of the Generation Split to the Chiral Asymmetry of Projection Locking

The fermionic sub-programme of Cosmochrony locates the three-generation mass split in the $J_3$-odd part of the squared projective endomorphism $E_\Pi^2$ restricted to the gauge-singlet generation triplet ${C^3_{\mathrm{gen}}}$, parametrised by a single real number $u$ through $E_\Pi^2|_{{C^3_{\mathrm{gen}}}}=\mathrm{diag}(1,\tfrac12+u,\tfrac12-u)$, with the even sector $\mathrm{diag}(1,\tfrac12,\tfrac12)$ already closed by Born–Infeld parity . This note fixes the structural status of $u$ before any explicit construction of $E_\Pi$. First, the projected Dirac square admits a universal Feshbach/Schur form $E_\Pi=-{\Pi_S}\,D\,(1-P)\,D\,{\Pi_S}^{*}=-M^{\dagger}M$ with $P={\Pi_S}^{*}{\Pi_S}$, exhibiting $E_\Pi$ as the Schur complement of the spinorial directions eliminated by the non-injective projection, negative semi-definite and vanishing in the injective limit. Second, the chiral block decomposition shows that $u$ is controlled by the ${\mathcal D}^{\pm}$-transported, generation-projected part of the antiunitary chiral equivariance defect \[ \Delta_\chi(P)=\pi_{LL}-\tau\,\overline{\pi_{RR}}\,\tau^{-1} \] of the eliminated block $1-P$, rather than by a naive block difference. Third, the minimal non-injectivity $c\leftrightarrow q-c$ is chirally symmetric, so $u=0$ at the level of axioms A1–A3; a non-zero $u$ requires a chiral symmetry-breaking that can originate only in the projection-locking axiom A4. The Seeley–DeWitt conventions are locked so that the operator-level and spectral-level definitions of $u$ coincide on the flat effective metric, and a no-go result establishes that $u$ is not accessible to any finite front observable because chirality is a Lorentzian rather than a finite-fibre datum. The remaining open deliverable is reduced to three sharp questions about A4.