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 J₃-odd part of the squared projective endomorphism E_Π² restricted to the gauge-singlet generation triplet C³gen, parametrised by a single real number u through E_Π²|_C³gen=diag(1,12+u,12-u), with the even sector diag(1,12,12), the algebraic value of (C₂-J₃²)/C₂ at C₂=2. Its reading as the Born–Infeld even sector would require an identification between the conditional 3×3 model of O30 on Sym²(V_ρ) and (E_Π²)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_Π. First, the projected Dirac square admits a universal Feshbach/Schur form E_Π=-ΠS\,D\,(1-P)\,D\,ΠS*=-M^M with P=ΠS*ΠS, exhibiting E_Π 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 D±-transported, generation-projected part of the antiunitary chiral equivariance defect \[ Δ_χ(P)=πLL-τ\,{πRR}\,τ⁻¹ \] of the eliminated block $1-P$, rather than by a naive block difference. Third, the minimal non-injectivity c↔ 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_Π-equivariant, so ufin=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 \[ α(t,s)=ts\,r/ r=ts+O\!((ts)^2), μ(t,s)≡0, \] with r=1+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≠0 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.