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 EΠ² on a generation triplet C³_gen, through Φ(EΠ²) = diag(1, ½+u, ½−u), with even sector diag(1, ½, ½), the algebraic value of (C₂−J₃²)/C₂ at C₂ = 2; its reading as the Born–Infeld even sector is not used. This note fixes the structural status of u before any explicit construction of EΠ, in two frameworks. For the reference operator P = Π_S* Π_S, under a contraction hypothesis, the compression remainder of the projected Dirac square (the projective endomorphism of Q14 when Π_S D² Π_S* is the Lichnerowicz operator) is EΠ = −N†N ≤ 0; it is zero-order under a symbol condition, which forces EΠ = 0 on irreducible Clifford fibres. Under the generation-reading hypothesis [H-Gen] (with $J_\Pi$-compatible $D$ and ${\Pi_S}$), u = 0 for this operator; the reading of ${E_\Pi}^2$ on the triplet goes through a reading map $\Phi$, and a multiplicative reading of $\tfrac12\pm u$ as eigenvalues is the separate hypothesis [H-Mult], which, with a non-singular even sector, gives $u\equiv0$ or admits no $\Phi$ in the chiral rank-two left-admissible frame (limits stated in Hypothesis hyp:mult). For a general locking operator 0 ≤ P ≤ 1 only the Schur expression EΠ[P] = −Π_S D (1−P) D Π_S* = −N†N and its sign are unconditional, and the consequences for a non-zero split are conditional: a non-zero u requires J_Π P J_Π⁻¹ ≠ P, which on an interpretive reading of the axioms can come only from axiom A4. The antiunitary chiral equivariance defect of the eliminated block is related to u only as a structural reading. The finite sector is conditional on an open identification of the parity c ↔ q−c with J_Π. The J_Π-odd first-order polarisation on the outer generation weights is the transported J_Π-odd tilt of the locking operator; the locking is Schur-null when it vanishes and Schur-transverse otherwise. The oriented metaplectic step has the exact sl₂ opening α(x,y) = xy r/sinh r, cosh r = 1 + xy/2, produced by [E,F] = H; that the transported tilt equals it, up to a non-zero factor, is supplied by no source. Neither the sign of the Born–Infeld genus nor the non-vanishing of u follows, and no mechanism for a non-zero split is derived. Under the chiral splitting and the generation reading, the reference residue is not left-admissible in the sense of Q14 when non-zero; the multiplicative reading of the generation map is a separate hypothesis [H-Mult], which with a non-singular even sector gives $u\equiv0$ or admits no $\Phi$ in the chiral rank-two left-admissible frame. Open: that hypothesis, the eliminated block 1−P(s), |u|, and the Yukawa sector.