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 (PRS) and takes up its open deliverable on the Lorentzian saturation functional: the definition of ${\\mathcal{B}}(s)$ 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 closed in PRS in the present Lorentzian spin stratum, selecting the Schur-transverse branch. 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. On the corpus-derived real symplectic cascade the transverse (spin-two) channel of the cubic response vanishes at the metaplectic phase $\\gamma = 0$ for every ordering and reference, so the derived amplitude mechanism is Born–Infeld saturation while the split persists, $u \\neq 0$; the sixth-order interior lock is doubly conditional, on a non-derived complex phase and on a non-prescribed ordering. With the companion eliminated-block note, in which the left-handed block of the ${J_{\\Pi}}$-odd tangent of the eliminated block is half the transported defect rate $\\partial_s \\Delta_\\chi(P)|_0$, one block of the split-sourcing tangent rather than a unique carrier, we then close the boundary-versus-interior selection by admissibility: the saturation contact $\\Delta_\\chi(s_*) = 0$ is the unique chart-independent A4 lock, the defect rate being reparametrisation-covariant so that the invariant magnitude is $u(s_*)$, with the interior competitor requiring the non-derived phase and ordering. The magnitude $|u|$ itself is not predicted: it is fixed through the chiral-frontier normalisation $\\mathcal{N}_A$, a dictionary-bound quantity.