A4-Note — The Born–Infeld Saturation Margin of the Chiral Modulus

Antisymmetric Structure and the Reduction of the Generation Split to a Lorentzian Genus

J. Beau, Independent Researcher, France

Status

Preprint, v1.0. DOI: 10.5281/zenodo.20633931

Abstract

This note is the companion of the Projective Residue Schur reduction (PRS) and takes up the first of its open deliverables: the definition of the Lorentzian saturation functional $\mathcal{B}{\mathrm{sat}}(s)$ along the $J\Pi$-odd modulus that controls the three-generation split coefficient $u$.

Three results are established.

1. Antisymmetry lemma. The generation modulus is $J_\Pi$-odd, and the symmetric square $M = F_\chi F_\chi$ is $J_\Pi$-even from birth, so it cannot carry the oriented modulus or the spontaneous $V-A$ branch choice; the primary modulus variable must be an antisymmetric chiral two-form $F_{\chi,\mu\nu}(s)$.

2. Sign-locked saturation functional. $\mathcal{B}{\mathrm{sat}}(s)$ is defined as the Born–Infeld determinantal saturation margin of this two-form, with the overall sign fixed by the admissibility role of $\mathcal{B}{\mathrm{sat}}$ (projection locking; axiom A4 selects saturated minima), not by matching a Maxwell weak-field expansion. This neutralises the convention sign-trap that would otherwise make the split sign a free choice.

3. Genus reduction. The second variation reduces to $\mu_\chi := \partial_s^2 \mathcal{B}{\mathrm{sat}}(0) = \tfrac{1}{2},P{\chi,\mu\nu} P_\chi^{\mu\nu}$, so the existence and stability of the split are governed entirely by the Lorentzian genus of the chiral polarisation $P_\chi$ in the effective metric $g^{\mu\nu} = 2\eta^{\mu\nu}$.

The identification of the Schur projector tilt with the Born–Infeld saturation modulus on the A4 locus is discharged to a tangential coincidence, conditional only on Schur transversality; the Lorentzian genus computation is deliberately deferred.

Position in the programme

This note belongs to the fermionic matter sub-programme (Presentation Note 6). It is the companion to PRS (Schur form of $E_\Pi$, A4 stratification, finite/Lorentzian separation) and discharges its first open deliverable: the definition of the saturated Born–Infeld functional along the $J_\Pi$-odd modulus, with the sign convention locked by the admissibility role of the functional. The remaining open deliverable — the computation of the chiral curvature $\mu_\chi^2$ — is thereby reduced to a single Lorentzian genus question on the chiral polarisation two-form.

Compilation

bash compile.sh

Runs pdflatex → bibtex → pdflatex → pdflatex on tex/A4Note.tex and produces out/A4Note.pdf.