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 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 _{^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 = 0 for every ordering and reference, so the derived amplitude mechanism is Born–Infeld saturation while the split persists, u 0; the sixth-order interior lock is doubly conditional, on a non-derived complex phase and on a non-prescribed ordering. Using the companion eliminated-block note, which reduces the split to the normalisation of the single carrier ₛ _ (P) |₀, we then close the boundary-versus-interior selection by admissibility: the saturation contact _ (s_*) = 0 is the unique chart-independent A4 lock, the carrier 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 NA, a dictionary-bound quantity.
Jérôme Beau (Fri,) studied this question.