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 B (s) along the J_{}-odd modulus that controls the three-generation split coefficient u. We establish three results. First, a structural antisymmetry lemma: the generation modulus is J_{}-odd, and the symmetric square M = F F is J_{}-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_ (s). Second, we define B (s) as the Born–Infeld determinantal saturation margin of this two-form, with the overall sign fixed by the admissibility role of 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 _{^2}: = ₛ² B (0) = 12 P_ P^, so the existence and stability of the split are governed entirely by the Lorentzian genus of the chiral polarisation P in the effective metric g^ = 2^. 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 _{^2} < 0 and the split opens spontaneously, leaving only the amplitude |u| open.
Jérôme Beau (Fri,) studied this question.