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

This companion of the Projective Residue Schur reduction (PRS) examines the Lorentzian saturation functional controlling the generation split u. The symmetric square of the J_Π-odd modulus is J_Π-even and cannot carry the orientation sign; the carrier is taken, by hypothesis, to be an odd antisymmetric chiral two-form F_μν(s). B_sat(s) is defined as the Born–Infeld determinantal saturation margin of such a supplied two-form, and its second variation μ_χ² = ½ P_μν P^μν is the Lorentzian genus of the chiral polarisation P. Hypotheses that no source supplies are stated where used: a Lorentzian co-metric, the construction of the two-form, a local-minimum reading of A4, the existence and non-degeneracy of a locked pair, the covariance of the locking family under J_Π, and, from PRS, its generation-reading and first-order transport hypotheses. The scope is that of PRS: under its generation-reading hypothesis the split vanishes for the reference operator, the consequences for a general locking operator are conditional, and no mechanism for a non-zero split is derived; the identification of the compression remainder with the projective endomorphism of Q14 holds only under the symbol identity Π_S D² Π_S* = Lich. The genus is not determined: the central commutator supplies no electric assignment, since the central direction is not temporal in the co-metrics of the corpus and no map from the generation operator to a two-form is available. The quartic coefficient of the radicand depends on the parametrisation and selects no lock; for a collinear purely electric ray the bound is reached at t* = β/|E|. A conditional factorisation of the contact split to first order in s* (Front D) is given under [H-Gen], [H-Mult], [H-Tr], [H-F], [H-Nd] and the JΠ-covariance and diagonal-form premises of Lemma lem:u-odd, and a localisation hypothesis that fails in the available witnesses of the eliminated-block note, which also fall outside [H-Gen](i); no frame satisfying all these hypotheses together is exhibited, and the factorisation is formal; it fixes no magnitude. The magnitude |u| is not predicted.