The Born–Infeld Saturation Margin of the Chiral Modulus: Antisymmetric Structure, Lorentzian Genus, and Electric Determination of the Generation Split

This note is the companion of the Projective Residue Schur reduction and takes up the first of its open deliverables: the definition of the Lorentzian saturation functional ${\mathcal{B}}(s)$ along the ${J_{\Pi}}$-odd modulus that controls the three-generation split coefficient $u$. We establish three results. First, a structural antisymmetry lemma: the generation modulus is ${J_{\Pi}}$-odd, and the symmetric square $M = {\mathcal{F}} {\mathcal{F}}$ 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 ${\mathcal{F}}_{\mu\nu}(s)$. Second, we define ${\mathcal{B}}(s)$ as the Born–Infeld determinantal saturation margin of this two-form, with the overall sign fixed by the admissibility role of ${\mathcal{B}}$ (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. Third, the second variation reduces to ${\mu_{\chi}^{2}} := \partial_s^2 {\mathcal{B}}(0) = \tfrac12 {\mathcal{P}}_{\mu\nu} {\mathcal{P}}^{\mu\nu}$, so the existence and stability of the split are governed entirely by the Lorentzian genus of the chiral polarisation ${\mathcal{P}}$ in the effective metric $g^{\mu\nu} = 2\eta^{\mu\nu}$. The coincidence lock is discharged to a tangential coincidence; the Schur transversality on which it rests is reduced in the Schur reduction to a projected commutator with an exhaustive transverse/null dichotomy. Assembling the projector-completion grading, the oriented-cascade lift, and the symbol-compatible spin frame, we then determine the genus: in the Schur-transverse branch the chiral polarisation is electric, so ${\mu_{\chi}^{2}} < 0$ and the split opens spontaneously. We then extend the radicand to quartic order: the Born–Infeld pseudoscalar cancels on the electric locus, and the amplitude $|u|$ is reduced to the transverse cubic backreaction of the two-form trajectory, modulo odd reparametrisation, together with the boundary-versus-interior selection of the saturation problem.