The Constrained Jet of the Lorentzian Eliminated Block and the Chiral Defect-Rate Identity

Core results

• Constrained jet. For every smooth self-adjoint projector family and every smooth unitary transport $U(s) = 1 + sA + \tfrac{s^2}{2}(A^2 + B) + O(s^3)$, the projector constraints force $Q_1 = [A, Q_0]$ with $Q_0 Q_1 Q_0 = 0 = (1 - Q_0) Q_1 (1 - Q_0)$, and pin the diagonal blocks of $Q_2 = \tfrac12[B, Q_0] + \tfrac12[A,[A,Q_0]]$ to $Q_1^2$ with opposite signs.
• Parity under equivariance. A family equivariant under the antiunitary parity, $J_\Pi Q(s) J_\Pi^{-1} = Q(-s)$, has alternating jet parities, and its odd tangent is the chiral commutator $[A_-, Q_0]$. With a moving embedding, a vanishing odd tangent does not by itself kill the first-order split.
• Moving embedding. Since $P(s) = \Pi_S(s)^{*}\Pi_S(s)$, the residue jet carries the derivatives of the embedding, $E_1 = -(G_1 Q_0 G_0^{*} + G_0 Q_1 G_0^{*} + G_0 Q_0 G_1^{*})$ with $G = \Pi_S D$. For a $J_\Pi$-even Dirac operator and a covariant embedding, $E_0$ is even, $E_1$ is odd, and the first-order odd part of $E_\Pi^2$ is $\{E_0, E_1\}$. Freezing the embedding gives a different rate.
• Mixing. A Hermitian operator odd under the antiunitary parity has zero $(+,-)$ entry on the generation triplet. The first-order mixing coefficient therefore vanishes in every covariant family, with or without complex phases, and more generally every odd-order coefficient vanishes. Mixing at even orders is not excluded, and it is non-zero at second order in the explicit family. In the normal form $E_0 = -\mathrm{diag}(1, 2^{-1/2}, 2^{-1/2})$ the split rate is $u'(0) = -\sqrt{2}\,(E_1)_{++}$.
• Moving-Schur existence theorem. An explicit covariant family on $\mathbb{C}^3 \oplus \mathbb{C}^3$ has $E_0^2 = \mathrm{diag}(1, \tfrac12, \tfrac12)$, $u'(0) = 11 \cdot 2^{1/4}/9 \neq 0$, and neither mixing nor singlet leakage at first order; the frozen-embedding model gives $79 \cdot 2^{1/4}/90$ instead.
• Chiral defect-rate identity. In the chiral frame, $[\Pi_{J_\Pi\text{-odd}}(1 - P)]_{LL} = \tfrac12\Delta_\chi(P)$ and $[\Pi_{J_\Pi\text{-odd}}\dot Q(0)]_{LL} = \tfrac12\partial_s\Delta_\chi(P)|_0$. This determines one block of the odd tangent, not a unique generator: $A$ is defined only modulo the centraliser of $Q_0$.

The constrained jet

For each value of the modulus $s$ opened by Lorentzian complexification, the eliminated block $1 - P(s)$ is a self-adjoint projector, unitarily transported from its base point. The order-by-order content of $(1-P)^2 = 1-P$ fixes the off-diagonal tangent and the diagonal blocks of the second-order coefficient for every smooth family; a constant-generator orbit $U(s) = \exp(sA)$ is only a special case. The alternating parity of the jet is not a consequence of these constraints: it holds when the family is equivariant under the antiunitary parity. For a covariant family the finite base point is even, so $u(0) = 0$: the split is a Lorentzian, not a finite-fibre, datum.

The chiral defect-rate identity

In the chiral splitting $\mathcal{S}_\Pi = \mathcal{S}_L \oplus \mathcal{S}_R$, the Born–Infeld parity exchanges the chiralities through the intertwiner $\tau$. In a chiral frame where it acts as $J_\Pi = \Sigma_\tau \circ \mathrm{conj}$, the $J_\Pi$-odd part of the eliminated block has $LL$ block equal to half the transported equivariance defect $\Delta_\chi(P) = \pi_{LL} - \tau\,\overline{\pi_{RR}}\,\tau^{-1}$ of the Schur reduction, and the first-order tangent carries $\tfrac12\partial_s\Delta_\chi(P)|_0$ in that block.

The identity has three boundaries. It determines one block of the odd tangent, not the generator, which is defined only up to the centraliser of $Q_0$. The $LL$ block does not recover the full odd tangent, so no injective localisation onto $\partial_s\Delta_\chi(P)|_0$ is claimed. And the passage from the tangent to $u'(0)$ goes through the transport, the moving embedding, and the projection to the generation triplet.

Status and open questions

Established. The constrained jet for every smooth family; the parity grading under equivariance; the moving-embedding propagation to the residue; the vanishing of every odd-order generation-mixing coefficient in covariant families; the moving-Schur existence theorem; and the chiral defect-rate identity. The identities are checked on exact instances, and the realisation and the counterexamples exactly; the general statements rest on the proofs in the note.

Open. The selection, by the programme, of a covariant family among those the Schur contract admits; the uniqueness of the split carrier and the injective localisation of the odd tangent onto its $LL$ block; the normalisation of $\partial_s\Delta_\chi(P)|_0$, hence the magnitude $|u|$, which remains dictionary-bound; and generation mixing at even orders.

Relation to the Cosmochrony programme

This note belongs to the fermionic matter sub-programme. It takes up the first open deliverable of the Projective Residue Schur reduction, the explicit Lorentzian eliminated block $1 - P(s)$, and relates its odd tangent to the transported equivariance defect of that reduction. The electric genus sign $\mu_\chi^2 < 0$ of the Born–Infeld saturation margin note orients the physical reading but is not used in the proofs. The absolute value $|u|$, the Yukawa sector, and the mass spectrum remain downstream, dictionary-bound questions.

References

Jérôme Beau. The Constrained Jet of the Lorentzian Eliminated Block and the Chiral Defect-Rate Identity: A Moving-Schur Existence Theorem for the Generation Split. Working paper, 2026. 10.5281/zenodo.20763532