The Born–Infeld Saturation Margin of the Chiral Modulus: Antisymmetric Structure and the Reduction of the Generation Split to a Lorentzian Genus

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, leaving only the amplitude $|u|$ open.