The Constrained Jet of the Lorentzian Eliminated Block and the Chiral Defect-Rate Identity: A Moving-Schur Existence Theorem for the Generation Split

Abstract

This note isolates an operator-level lemma in the fermionic-matter sub-programme of Cosmochrony, logically between the Schur reduction of the projective residue and the physical identification of the chiral generator. The Schur reduction writes the projective endomorphism as ${E_{\Pi}} = -{\Pi_{S}}\,D\,(1-P)\,D\,{\Pi_{S}}^{*}$, exhibiting the eliminated block $1-P$ as the controlling object of the three-generation split coefficient $u$; the companion genus analysis fixes the electric sign ${\mu_{\chi}^{2}} < 0$, while the absolute value $|u|$, the Yukawa sector, and the mass spectrum are kept downstream. Writing the self-adjoint projector family as a jet $1-P(s) = Q_0 + s Q_1 + s^2 Q_2 + O(s^3)$ along the modulus $s$, the projector constraints alone force $Q_1 = [A, Q_0]$ to be a purely off-diagonal Grassmann tangent and pin the diagonal blocks of $Q_2$ to $Q_1^2$ with opposite signs, for every smooth family and every smooth unitary transport. The parity grading of the jet is not a consequence of the projector constraints: it holds when the family is equivariant under the antiunitary Born–Infeld parity ${J_{\Pi}}$, ${J_{\Pi}}\,Q(s)\,{J_{\Pi}}^{-1} = Q(-s)$, and then the split-sourcing part of the tangent is the chiral commutator $[A_-, Q_0]$. Propagation to the residue must move the spinorial embedding together with the block, since $P(s) = {\Pi_{S}}(s)^{*}{\Pi_{S}}(s)$. For a ${J_{\Pi}}$-even Dirac operator and a covariant embedding, ${J_{\Pi}}\,{\Pi_{S}}(s)\,{J_{\Pi}}^{-1} = {\Pi_{S}}(-s)$, the residue jet is graded $E_0$ even, $E_1$ odd, and the first-order ${J_{\Pi}}$-odd part of ${E_{\Pi}}^2$ is the anticommutator $\{E_0, E_1\}$. A Hermitian operator that is odd under the antiunitary parity has vanishing $(+,-)$ entry on the generation triplet, so the first-order coefficient of generation mixing vanishes in every covariant family, with or without complex phases, and so does every odd-order coefficient; mixing at orders zero and two lies in the even sector and is not excluded. In the normal form $E_0 = -{\operatorname{diag}}(1, 2^{-1/2}, 2^{-1/2})$ the split rate is the scalar functional $u'(0) = -\sqrt{2}\,(E_1)_{++}$. The contract is realisable: an explicit covariant family on $\mathbb{C}^3 \oplus \mathbb{C}^3$ with a moving embedding has $E_0^2 = {\operatorname{diag}}(1, \tfrac12, \tfrac12)$, $u'(0) = 11 \cdot 2^{1/4}/9 \neq 0$, and neither mixing nor singlet leakage at first order, and its rate differs from the rate of the auxiliary model that freezes the embedding. In the chiral frame the ${J_{\Pi}}$-odd part of the eliminated block has $LL$ block equal to half the transported equivariance defect ${\Delta_{\chi}}(P) = \pi_{LL} - \tau\,\overline{\pi_{RR}}\,\tau^{-1}$, so along the modulus $[{\Pi_{{J_{\Pi}}\text{-odd}}} \dot Q(0)]_{LL} = \tfrac12 \partial_s {\Delta_{\chi}}(P)|_0$; this determines one block of the odd tangent, not a unique generator, which is defined only up to the centraliser of $Q_0$. The identities are checked on exact instances, and the realisation and the counterexamples exactly; the general statements rest on the proofs given. The selection of the family by the programme, the localisation of the odd tangent onto the defect rate, and the normalisation of $\partial_s {\Delta_{\chi}}(P)|_0$ remain open.