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

Core contributions

• Antisymmetric carrier. The generation modulus is $J_\Pi$-odd, and the symmetric square $M = F_\chi F_\chi$ is $J_\Pi$-even from birth, so it cannot carry the orientation sign. The carrier is taken, by hypothesis ([H-F]), to be an odd antisymmetric chiral two-form $F_{\chi,\mu\nu}(s)$; that it is the only odd carrier is not shown.
• Saturation margin. $\mathcal{B}_{\mathrm{sat}}(s)$ is defined as the Born–Infeld determinantal saturation margin of such a supplied two-form, with the overall sign fixed once by its admissibility role (projection locking), prior to and independently of the sign of its second variation, and not by matching a Maxwell weak-field expansion; it satisfies $\mathcal{B}_{\mathrm{sat}}(s) = \mathcal{B}_{\mathrm{sat}}(-s)$.
• Genus reduction. The second variation reduces to $\mu_\chi^2 := \partial_s^2 \mathcal{B}_{\mathrm{sat}}(0) = \tfrac{1}{2}\,P_{\chi,\mu\nu} P_\chi^{\mu\nu}$, the Lorentzian genus of the chiral polarisation $P_\chi$, in a Lorentzian effective co-metric supplied as an input ([H-Lor]); only its signature enters.
• The genus is not determined. The central commutator supplies no electric assignment: the central direction is not temporal in the co-metrics of the corpus (the rank-three co-metric of Q5b has a zero central entry, the completion of Q8 is conditional, and the temporal coefficient of Q11 is free), the coefficient of the $J_3$-signal comes from the $\mathfrak{sl}_2$ commutator $[E,F] = H$ and not from the Heisenberg bracket, and no map from the generation operator to a two-form is supplied. Whether the supplied two-form is electric or magnetic is not decided.
• Hypotheses carried. A Lorentzian co-metric, the construction of the two-form, a local-minimum reading of A4 ([H-Min]), the existence and non-degeneracy of a locked pair ([H-Nd]), the coincidence hypothesis [H-Coin] (only to read the lock along the Born–Infeld modulus), the $J_\Pi$-covariance of the locking family and, from PRS, its generation-reading and first-order transport hypotheses. No source supplies any of them. The scope is that of PRS: no mechanism for a non-zero split is derived.
• Quartic radicand. The quartic coefficient of the radicand depends on the parametrisation of the modulus (a shift of the form $\omega\, P \cdot P/\beta^2$) and selects no lock. For an exactly collinear purely electric ray under [H-F], the bound is reached at $t_* = \beta/|\mathbf{E}_{F_0}|$.
• Chart covariance and first contact. The defect rate is reparametrisation covariant, and the first contact, the first zero of the radicand, is chart independent when it exists (its parameter value transforms). No selection between a boundary contact and an interior stationary point is established.
• Front D, conditional and formal. A factorisation of the contact split to first order in $s_*$ is given under [H-Gen], [H-Mult], [H-Tr], [H-F], [H-Nd] and the $J_\Pi$-covariance and diagonal-form premises, together with a localisation hypothesis that fails in the available witnesses of the eliminated-block note. No frame satisfying all these hypotheses together is exhibited, so the factorisation is formal and fixes no magnitude. The magnitude $|u|$ is not predicted.

Why an antisymmetric carrier

The generation modulus $s$ parametrises the $J_\Pi$-odd deformation of the locked eliminated block. Any symmetric square such as $F_\chi F_\chi$ is automatically $J_\Pi$-even, hence blind to the orientation that distinguishes the two branches $\pm s_*$. An odd carrier must therefore enter through an antisymmetric part, and the note takes a two-form $F_{\chi,\mu\nu}(s)$, odd and exactly collinear along the trajectory, as the carrier by hypothesis.

Fixing the sign of $\mathcal{B}_{\mathrm{sat}}$

The determinantal scalar admits an overall sign ambiguity according to the object it names: read as the Born–Infeld action density it recovers a Maxwell-type weak-field expansion, read as the saturation margin it is minimised at the locking event of A4. The two readings differ by an overall sign, which propagates into $\partial_s^2 \mathcal{B}_{\mathrm{sat}}(0)$. The note fixes the sign once, by the admissibility role of $\mathcal{B}_{\mathrm{sat}}$, prior to and independently of the sign of the second variation. With this fixing, the sign of the split is not inferred from the convention: it is carried solely by the Lorentzian genus of $P_\chi$ for a supplied two-form, which is not determined.

Position in the programme

This note belongs to the fermionic matter sub-programme (Presentation Note 6). It is the direct companion of PRS and examines the Lorentzian saturation functional of its open deliverable. The Lorentzian genus of the chiral polarisation is not determined; no amplitude mechanism and no selection of the lock are established; the magnitude $|u|$ is not predicted and remains dictionary-bound through the chiral-frontier normalisation $\mathcal{N}_A$, whose matching with $\mu_\chi^2$ is open. Born–Infeld enters as a conditional completion, not as a derived or unique one. The first-principles work is structural and upstream (ADE case selection, level-to-generation map, cascade exponent $\beta$, transfer constant $N_{\mathrm{casc}}$).

References

Jérôme Beau. The Born–Infeld Saturation Margin of the Chiral Modulus: Antisymmetric Structure and Lorentzian Genus of the Generation Split. Working paper, 2026. 10.5281/zenodo.20633931