The Projective Residue as a Schur Complement

Core results

• Schur form. For every locking operator $0 \preceq P \preceq 1$, $E_\Pi[P] = -\Pi_S\, D\,(1 - P)\,D\,\Pi_S^{*} = -N_P^{\dagger} N_P \preceq 0$ as a quadratic form, with $N_P = (1-P)^{1/2} D\,\Pi_S^{*}$. For $P_{\mathrm{ref}} = \Pi_S^{*}\Pi_S$, under a contraction hypothesis, it is the compression remainder $D_\Pi^2 - \Pi_S D^2 \Pi_S^{*}$ of the projected Dirac square. It is zero-order under a symbol condition, which forces $E_\Pi = 0$ on irreducible Clifford fibres.
• Relation to the projective endomorphism of Q14. The compression remainder is the endomorphism $E_\Pi$ of Q14 only when $\Pi_S D^2 \Pi_S^{*}$ is the Lichnerowicz operator, an identity that no source states; otherwise nothing is stated about the endomorphism of Q14, whose sign is an explicit hypothesis of Q14.
• Chiral block reduction. In the chiral splitting, $E_{LL} = -\Pi_S \mathcal{D}^{-}\pi_{RR}\mathcal{D}^{+}\Pi_S^{*}$ and $E_{RR} = -\Pi_S \mathcal{D}^{+}\pi_{LL}\mathcal{D}^{-}\Pi_S^{*}$; $\pi_{LL} = 0$ implies the left-admissibility $P_R E_\Pi P_R = 0$ (the converse holds under a faithfulness hypothesis), when $E_\Pi$ is the endomorphism of Q14.
• Conditional localisation of $u$. Under [H-Gen] ($J_\Pi$ antiunitary, compatible with $D$ and $\Pi_S$; a real-linear, $*$-preserving, $J_\Pi$-covariant reading map $\Phi$ applied to zero-order endomorphisms), $u$ is read in the $J_\Pi$-odd part of $\Phi(E_\Pi^2)$. If $J_\Pi P J_\Pi^{-1} = P$ then $u[P] = 0$, which holds for the reference operator; a non-zero $u$ requires $J_\Pi P J_\Pi^{-1} \neq P$. The multiplicative reading of $\tfrac12 \pm u$ as eigenvalues is a separate hypothesis [H-Mult], which, with a non-singular even sector, gives $u \equiv 0$ or admits no $\Phi$ in the chiral rank-two left-admissible frame; nothing else in the note uses it.
• Interpretive stratification. On an interpretive reading of the axioms, a non-zero $u$ can come only from axiom A4. This is a reading, not a derivation.
• Left-admissible branch of Q14. Under the chiral splitting and [H-Gen](i), a $J_\Pi$-commuting locking operator has $E_\Pi[P] = 0$ on the left-admissible branch; a non-zero left-admissible residue breaks $J_\Pi$-compatibility and is not the reference residue. These are necessary conditions only.
• Chirality is not a finite-fibre label. A cascade edge is a Heisenberg translation, not a symplectic element, so it carries no canonical metaplectic lift; $u$ cannot be extracted from a finite front observable.
• Finite sector, conditional. The finite locking data are invariant under the parity $c \leftrightarrow q - c$, whose identification with $J_\Pi$ is open; under that identification, an equivariant construction of the finite locking operator and a finite reading, $u_{\mathrm{fin}} = 0$.
• Schur-null and Schur-transverse loci. The transported $J_\Pi$-odd first-order tilt of the locking operator defines the Schur-null and Schur-transverse loci; a dichotomy is proved under explicit hypotheses, and which locus holds is open.
• Exact $\mathfrak{sl}_2$ opening. The oriented metaplectic step has the exact opening $\alpha(x,y) = xy\, r/\sinh r$, $\cosh r = 1 + \tfrac{xy}{2}$, produced by the $\mathfrak{sl}_2$ commutator $[E,F] = H$ (Q14, Proposition 6.5); in the $\mathrm{SL}(2)$-on-$V$ model of Q14 the Heisenberg bracket does not enter. That the transported tilt equals it up to a non-zero factor is the transport hypothesis [H-Tr], supplied by no source.

The Schur form

The projected Dirac square $D^2_{\Pi,g,A} = \Pi_S D_{g,A} P D_{g,A} \Pi_S^{*}$ is the Feshbach reduction of the lifted square along the eliminated sector $1 - P$. Subtracting the block $\Pi_S D_{g,A}^2 \Pi_S^{*}$ leaves the compression remainder $-\Pi_S D_{g,A}(1 - P)D_{g,A}\Pi_S^{*}$, the Schur complement of the eliminated directions. It is the projective endomorphism $E_\Pi$ of Q14 when $\Pi_S D_{g,A}^2 \Pi_S^{*}$ is the Lichnerowicz operator; that identity forces $\Pi_S \Pi_S^{*} = \mathrm{id}$ at symbol level, its converse is not claimed, and no source states it.

The antiunitary equivariance defect

The Born–Infeld spinorial lift $J_\Pi$ exchanges the chiral subbundles via $J_\Pi P_L = P_R J_\Pi$. Under [H-Gen](i), equivariance of the locking operator $[P, J_\Pi] = 0$ forces on the diagonal chiral blocks the transported, complex-conjugated equivalence $\pi_{LL} = \tau\,\overline{\pi_{RR}}\,\tau^{-1}$. The relevant obstruction to $u = 0$ is therefore the antiunitary defect $\Delta_\chi(P) = \pi_{LL} - \tau\,\overline{\pi_{RR}}\,\tau^{-1}$, reduced to $\pi_{LL} - \tau \pi_{RR} \tau^{-1}$ on the real structure fixed by $J_\Pi^{(2)}$ on the generation triplet.

A failure of $J_\Pi$-equivariance at projection locking is a necessary but not sufficient condition for $u \neq 0$: the commutator may be supported outside the generation triplet, or in components that do not survive the Dirac transport and the final projection to $C^3_{\mathrm{gen}}$. The defect is related to $u$ only as a structural reading.

Finite sector and the Born–Infeld modulus

The finite locking data are invariant under the parity $c \leftrightarrow q - c$, whose identification with $J_\Pi$ is open. Under that identification, an equivariant construction of the finite locking operator and a finite reading, finite projection locking does not generate $u$. A non-zero split would then be a $J_\Pi$-odd datum outside that construction, located in the Lorentzian completion as a hypothesis.

Parametrising the $J_\Pi$-odd deformation by $s$, $J_\Pi$-symmetry of the saturation functional $\mathcal{B}$ of the A4 companion note gives $\mathcal{B}(s) = \mathcal{B}(-s)$, so $s = 0$ is stationary and the local locking character is read from the chiral curvature $\mu_\chi^2 := \partial_s^2 \mathcal{B}(0)$ of a supplied two-form. The genus of that two-form is not determined, and whether a symmetry-breaking minimiser carries $u \neq 0$ is not shown.

Status and open deliverable

Proved or structural. The Schur form of the compression remainder and its sign; its zero-order character under the symbol condition; the chiral diagonal blocks; $u[P] = 0$ for a $J_\Pi$-commuting $P$ under [H-Gen]; the exact $\mathfrak{sl}_2$ opening $\alpha(x,y) = xy\, r/\sinh r$ of the oriented step, produced by $[E,F] = H$; and the Schur-null and Schur-transverse loci with a dichotomy proved under explicit hypotheses.

Conditional. The identification of the compression remainder with the endomorphism of Q14 (symbol identity); the localisation of $u$ ([H-Gen]); the multiplicative reading ([H-Mult]); the finite sector ($u_{\mathrm{fin}} = 0$); the interpretive reading that only A4 can break $J_\Pi$-equivariance; the Schur-transverse locus, which holds under the transport hypothesis [H-Tr].

Open. Which locus holds, the transport hypothesis [H-Tr], the explicit Lorentzian eliminated block $1 - P(s)$, the sign of the Born–Infeld genus, the non-vanishing of $u$ (no mechanism for a non-zero split is derived), its magnitude $|u|$ (dictionary-bound through the chiral-frontier normalisation), and the projected Yukawa sector with its mass matrices.

Relation to the Cosmochrony programme

This note belongs to the fermionic matter sub-programme (Presentation Note 6). It refines the Q14 open deliverable on the inter-generation splitting by localising $u$, conditionally, in the antiunitary chiral equivariance defect and defining the Schur-null and Schur-transverse loci of the A4 direction, and identifies the front observables of the companion oriented-frontier and angular-amplitude notes as null controls rather than carriers of $u$.

References

Jérôme Beau. The Projective Residue as a Schur Complement: Reduction of the Generation Split to the Chiral Asymmetry of Projection Locking. Working paper, 2026. 10.5281/zenodo.20601040