The Projective Residue as a Schur Complement

Core results

• Schur form of the projective residue. Under symbol-compatibility, $E_\Pi = -\Pi_S \, D_{g,A}\,(1 - P)\,D_{g,A}\,\Pi_S^{*} = -M^{\dagger} M$ with $P = \Pi_S^{*}\Pi_S$. Hence $E_\Pi$ is negative semi-definite, zero-order, and vanishes in the injective limit $P = 1$.
• Chiral block reduction. Decomposing $D_{g,A}$ and $1 - P$ in the chiral splitting $\mathcal{S}_\Pi = \mathcal{S}_L \oplus \mathcal{S}_R$ gives diagonal blocks $E_{LL} = -\Pi_S \mathcal{D}^{-}\pi_{RR}\mathcal{D}^{+}\Pi_S^{*}$ and $E_{RR} = -\Pi_S \mathcal{D}^{+}\pi_{LL}\mathcal{D}^{-}\Pi_S^{*}$. The left-admissibility $P_R E_\Pi P_R = 0$ of Q14 is equivalent to $\pi_{LL} = 0$.
• Localisation of $u$. The split $u$ is controlled by the $\mathcal{D}^{\pm}$-transported, generation-projected component of the antiunitary chiral equivariance defect $\Delta_\chi(P) = \pi_{LL} - \tau\,\overline{\pi_{RR}}\,\tau^{-1}$, not by a naive block difference.
• Axiomatic stratification. 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.
• Seeley–DeWitt convention lock. On the gauge-singlet $C^3_{\mathrm{gen}}$ and the flat effective metric, the operator-level definition of $u$ equals the spectral coefficient that enters the effective amplitude; the common $a_4$ normalisation factor cancels in the ratio $(\tfrac12 + u)/(\tfrac12 - u)$.
• No finite chiral label. Chirality $\gamma_5$ exists because of the Lorentzian closure $g^{\mu\nu} = 2\eta^{\mu\nu}$. A cascade edge is a Heisenberg translation, not a symplectic element, so it carries no canonical metaplectic lift; $u$ cannot be extracted from any finite front observable.
• Finite/Lorentzian separation. Finite projection locking is $J_\Pi$-equivariant for $q$ an odd prime, hence $u_{\mathrm{fin}} = 0$. A non-zero generation split can arise only in the Lorentzian completion of the Born–Infeld saturation.

The Schur complement 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 Lichnerowicz block reproduced by $\Pi_S D_{g,A}^2 \Pi_S^{*}$ leaves $E_\Pi = -\Pi_S D_{g,A}(1 - P)D_{g,A}\Pi_S^{*}$, the Schur complement of the eliminated directions. This makes precise the statement of Q14 that $E_\Pi$ is not a postulated finite Dirac datum but is induced by the non-injective projection itself.

The antiunitary equivariance defect

The Born–Infeld spinorial lift $J_\Pi$ exchanges the chiral subbundles via $J_\Pi P_L = P_R J_\Pi$. Equivariance of the locking projector $[P, J_\Pi] = 0$ 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 $C^3_{\mathrm{gen}}$.

Finite/Lorentzian separation

Both constituents of axiom A4 are $J_\Pi$-invariant at finite $q$: the continuous Born–Infeld saturation condition is parity itself, and the discrete shell support is a union of parity orbits with no $c = 0$ fixed point for $q$ an odd prime. The finite locked branch is therefore $J_\Pi$-equivariant and $u_{\mathrm{fin}} = 0$.

Parametrising the $J_\Pi$-odd deformation by $s$ (opened only by Lorentzian complexification), $J_\Pi$ symmetry of the Lorentzian saturation functional $\mathcal{B}$ gives $\mathcal{B}(s) = \mathcal{B}(-s)$, so $s = 0$ is automatically stationary. The locking character is then fixed by the chiral curvature $\mu_\chi^2 := \partial_s^2 \mathcal{B}(0)$: positive stabilises the $J_\Pi$-equivariant branch ($u = 0$); negative drives a $J_\Pi$-conjugate saturated pair $\pm s_*$ and $u \neq 0$. The marginal case $\mu_\chi^2 = 0$ is decided by the leading non-vanishing even coefficient of $\mathcal{B}(s)$.

Status and open deliverable

Established. The Schur form of $E_\Pi$, the chiral block reduction, the stratification on the Cosmochrony axioms, the Seeley–DeWitt convention lock, the no-go result for finite-fibre access to chirality, the finite/Lorentzian separation, and the reduction of the A4 Schur transversality to the non-vanishing of the projected commutator $[\partial_s P|_0,\, J_\Pi]\big|_{\mathrm{span}(e_+, e_-)}$ with an exhaustive transverse/null dichotomy (including the zero-mode subcase).

Open. The magnitude $|u|$ is reduced to the Lorentzian A4 problem: define the saturated Born–Infeld functional $\mathcal{B}(s)$ along the $J_\Pi$-odd modulus and compute the chiral curvature $\mu_\chi^2 := \partial_s^2 \mathcal{B}(0)$, with the marginal case governed by the leading non-vanishing even coefficient. With Schur transversality discharged as above, the active frontier of the sub-programme is now this Lorentzian genus computation (deciding $u = 0$ versus $u \neq 0$) and, beyond it, the Yukawa sector.

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 reducing the magnitude $|u|$ to a single Lorentzian A4-level question, 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