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 $\mathrm{diag}(1,\tfrac12,\tfrac12)$, the algebraic value of $(C_2-J_3^2)/C_2$ at $C_2=2$. Its reading as the Born–Infeld even sector would require an identification between the conditional $3\times3$ model of O30 on $\operatorname{Sym}^2(V_\rho)$ and $(E_\Pi^2)_{\mathrm{even}}$; no such identification is available, so that reading is not used here. 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 carried by the projection-locking axiom A4. The finite locking sector is proved $J_\Pi$-equivariant, so $u_{\mathrm{fin}}=0$, and chirality is shown to be a Lorentzian rather than a finite-fibre datum. Finally, the Schur-transversality branch required by the Born–Infeld genus companion is closed in the present Lorentzian spin stratum: the projected A4 commutator has the exact metaplectic opening \[ \alpha(t,s)=ts\,\frac{r}{\sinh r}=ts+O\!\left((ts)^2\right),\qquad \mu(t,s)\equiv0, \] with $\cosh r=1+\tfrac{ts}{2}$, and its timelike Clifford symbol is not in the transported zero-mode locus. Together with the electric-genus result of the A4 companion, this promotes the existence statement to $u\neq0$ in the present stratum. The remaining open deliverables are the explicit Lorentzian eliminated block $1-P(s)$, the absolute normalisation $|u|$, and the projected Yukawa sector.