The Constrained Jet of the Lorentzian Eliminated Block and the Chiral Generator

Overview

This companion note sits 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 $E_\Pi = -\Pi_S\,D\,(1 - P)\,D\,\Pi_S^{*}$, so the eliminated block $1 - P$ controls the three-generation split coefficient $u$; 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 $J_\Pi$-odd modulus $s$, the projector constraints force a constrained form, and the chiral generator is identified — not as a new selector, but as the infinitesimal transported equivariance defect. The split $u \neq 0$ is admissible with $|u|$ left entirely free.

Scope statement. This page provides a structured summary. The authoritative technical reference is the preprint linked above.

Core results

Constrained jet. For the self-adjoint projector family, the projector constraints force the first-order coefficient to be a purely off-diagonal Grassmann tangent, $Q_1 = [A, Q_0]$ with $Q_0 Q_1 Q_0 = 0 = (1 - Q_0) Q_1 (1 - Q_0)$, and pin the diagonal blocks of $Q_2$ to $Q_1^2$ with opposite signs.
Chiral source. Splitting the generator by parity, $A = A_+ + A_-$, the split-sourcing part is $\Pi_{J_\Pi\text{-odd}} Q_1 = [A_-, Q_0]$. The split is dead if and only if $[A_-, Q_0] = 0$, not merely if $Q_1 = 0$: a purely $J_\Pi$-even generator sources nothing.
Split source. Parity propagation grades the residue jet $E_0$ even, $E_1$ odd, $E_2$ even, so $u(s) = s\,\langle J_3, \{E_0, E_1\}\rangle / \langle J_3, J_3\rangle + O(s^3)$. With phase-free data the mixing coefficient $v$ vanishes identically; a non-zero $v$ requires a complex metaplectic phase.
Identification of the chiral generator. Because the antiunitary Born–Infeld parity exchanges chiralities, the $J_\Pi$-odd part of the eliminated block has chiral-diagonal component equal to the transported equivariance defect: $[\Pi_{J_\Pi\text{-odd}}(1 - P)]_{LL} = \tfrac12\Delta_\chi(P)$ and, along the modulus, $[\Pi_{J_\Pi\text{-odd}}\dot Q(0)]_{LL} = \tfrac12\partial_s\Delta_\chi(P)|_0$. The generator $A_-$ is therefore the infinitesimal transported chiral defect rate, not a new selector.
Single remaining target. The value question is reduced to one object: the admissible normalisation of $\partial_s\Delta_\chi(P)|_0$. The magnitude $|u|$ remains dictionary-bound; no value-dictionary object enters the construction.

The constrained jet

For each value of the $J_\Pi$-odd modulus $s$ opened by Lorentzian complexification, the eliminated block $1 - P(s)$ is a self-adjoint projector. Generating the family by conjugation, $1 - P(s) = U(s) Q_0 U(s)^{\dagger}$ with $U = \exp(sA)$, the order-by-order content of $(1-P)^2 = 1-P$ forces $Q_1 = [A, Q_0]$ to be purely off-diagonal with respect to $Q_0$ and pins the diagonal blocks of $Q_2 = \tfrac12[A,[A,Q_0]]$ to $Q_1^2$ with opposite signs. The finite base point $Q_0$ is $J_\Pi$-even, so $u(0) = 0$: the split is a Lorentzian, not a finite-fibre, datum.

The chiral generator

In the chiral splitting $\mathcal{S}_\Pi = \mathcal{S}_L \oplus \mathcal{S}_R$, the Born–Infeld parity $J_\Pi$ exchanges the chiralities. Consequently the $J_\Pi$-odd part of the eliminated block has left-handed diagonal block equal to half the transported equivariance defect $\Delta_\chi(P) = \pi_{LL} - \tau\,\overline{\pi_{RR}}\,\tau^{-1}$ of the Schur reduction. The first-order coefficient therefore carries $\tfrac12\partial_s\Delta_\chi(P)|_0$, identifying the chiral generator $A_-$ with the infinitesimal transported defect rate. Through the split-source formula, the carrier of $u$ is exactly $\partial_s\Delta_\chi(P)|_0$, dressed by the $\mathcal{D}^{\pm}$ transport and projected to the generation triplet as in the localisation of the Schur reduction.

Status and open deliverable

Established. The constrained form of the eliminated-block jet, the off-diagonal Grassmann character of $Q_1$, the opposite-sign $Q_2$ diagonal blocks, the phase-free vanishing of mixing, and the identification of the chiral generator as the transported equivariance defect rate. All identities are verified by exact symbolic and exact-rational computation.

Open. The magnitude $|u|$ is now reduced to a single target: the admissible normalisation of $\partial_s\Delta_\chi(P)|_0$ (the A4-level chiral defect rate that carries the magnitude of the split). The defect rate is the unique structural carrier; its absolute value is not normalised here and remains dictionary-bound.

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 identifies the chiral generator using the transported equivariance defect of that reduction and the electric genus sign $\mu_\chi^2 < 0$ established in the Born–Infeld saturation margin note. 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 Generator. Working paper, 2026. 10.5281/zenodo.20763532