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

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 form of the projective residue writes the residue of a locking operator as EΠ = -ΠS\,D\,(1-P)\,D\,ΠS*, exhibiting the eliminated block $1-P$ as the controlling object of the three-generation split coefficient u. The companion notes do not settle the two upstream questions: the Schur transversality of the locking is stated there as a dichotomy under explicit hypotheses, with the branch open, and the Lorentzian genus is not determined; neither is used here, and 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) = Q₀ + s Q₁ + s² Q₂ + O(s³) along the modulus s, the projector constraints alone force Q₁ = [A, Q₀] to be a purely off-diagonal Grassmann tangent and pin the diagonal blocks of Q₂ to Q₁² 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 JΠ, JΠ\,Q(s)\,JΠ⁻¹ = Q(-s), and then the split-sourcing part of the tangent is the chiral commutator [A₋, Q₀]. Propagation to the residue must move the spinorial embedding together with the block, since P(s) = ΠS(s)*ΠS(s). For a JΠ-even Dirac operator and a covariant embedding, JΠ\,ΠS(s)\,JΠ⁻¹ = ΠS(-s), the residue jet is graded E₀ even, E₁ odd, and the first-order JΠ-odd part of EΠ² is the anticommutator ₀, E₁\. 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 E₀ = -diag(1, 2-1/2, 2-1/2) the split rate is the scalar functional u'(0) = -√2\,(E₁)₊₊. The contract is realisable: an explicit covariant family on C³ ⊕ C³ with a moving embedding has E₀² = diag(1, 12, 12), u'(0) = 11 · 21/4/9 ≠ 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. This finite witness has J² = +1 and lies outside hypothesis [H-Gen](i) of the Schur note: it meets the identity-reading clauses of its separate hypothesis [H-Mult] and conditions (ii)–(iii) of [H-Gen] (reading JΠ as J on its finite space), and illustrates a moving-embedding family without being an instance of [H-Gen]. In a chiral frame in which JΠ = Σ_τ ∘ conj with τ\,τ̄⁻¹ = ± 1, an assumption of this note, the JΠ-odd part of the eliminated block has $LL$ block equal to half the transported equivariance defect Δχ(P) = πLL - τ\,πRR\,τ⁻¹, so along the modulus [Π_JΠ-odd Q(0)]LL = 12 ∂ₛ Δχ(P)|₀; this determines one block of the odd tangent, not a unique generator, which is defined only up to the centraliser of Q₀. 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 ∂ₛ Δχ(P)|₀ remain open.