The Born–Infeld Saturation Margin of the Chiral Modulus

Core contributions

• Antisymmetry lemma. 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 oriented modulus or the spontaneous $V-A$ branch choice. The primary modulus variable must therefore be an antisymmetric chiral two-form $F_{\chi,\mu\nu}(s)$.
• Sign-locked saturation functional. $\mathcal{B}_{\mathrm{sat}}(s)$ is defined as the Born–Infeld determinantal saturation margin of this two-form, with the overall sign fixed by the admissibility role of $\mathcal{B}_{\mathrm{sat}}$ (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.
• Genus reduction. The second variation reduces to $\mu_\chi := \partial_s^2 \mathcal{B}_{\mathrm{sat}}(0) = \tfrac{1}{2}\,P_{\chi,\mu\nu} P_\chi^{\mu\nu}$, so the existence and stability of the split are governed entirely by the Lorentzian genus of the chiral polarisation $P_\chi$ in the effective metric $g^{\mu\nu} = 2\eta^{\mu\nu}$.
• Schur transversality reduced to a projected commutator. The identification of the Schur projector tilt with the Born–Infeld saturation modulus on the A4 locus rests on a Schur transversality condition that PRS now reduces to the non-vanishing of the projected commutator $[\partial_s P|_0,\, J_\Pi]\big|_{\mathrm{span}(e_+, e_-)}$, with an exhaustive transverse/null dichotomy (including the zero-mode subcase).
• Electric determination of the genus. An index-assignment on the $J_\Pi$-odd outer-weight lift confirms the chiral polarisation is electric in the Schur-transverse branch, so $\mu_\chi^2 < 0$ and the generation split opens spontaneously ($u \neq 0$).
• Quartic radicand and amplitude frontier. The radicand $\mathcal{D}_\chi(s)$ is extended to quartic order: the Born–Infeld pseudoscalar contribution is non-positive and cancels on the electric locus, so the quartic is carried entirely by the cubic backreaction $P_\chi \cdot \mathcal{R}_\chi$ of the chiral two-form trajectory. A coordinate-level boundary-versus-interior dichotomy reduces the locked amplitude $|u|$ to the sign of this backreaction modulo odd reparametrisation.
• Path-invariant saturation on the derived cascade. Under the $\mathrm{su}(2)$ decomposition $\mathrm{End}(C^3_{\mathrm{gen}}) = \mathbf{1} \oplus \mathbf{3} \oplus \mathbf{5}$, the transverse cubic response is pure spin two and an $\mathrm{su}(2)$-equivariant Schur transport leaves it independent of the spin-one (genus) calibration. On the corpus-derived real symplectic cascade ($\gamma = 0$) the spin-two channel vanishes path-invariantly for every ordering and every reference, so the amplitude mechanism is Born–Infeld saturation and the split persists ($u \neq 0$). The sixth-order interior lock is doubly conditional — on a complex metaplectic phase $\gamma \neq 0$ absent from the derived cascade and on a non-prescribed placement of that phase — recorded as a candidate, not a computed value.
• Dictionary status of the magnitude. What remains is the magnitude $|u|$, fixed through the chiral-frontier normalisation $\mathcal{N}_A$, the angular analogue of the radial scale $\kappa = 5/12$ of the exit-deficit dictionary. The derived first-principles front is structural and upstream — ADE case selection, level-to-generation map, projective-resolution growth and the cascade exponent $\beta$, transfer constant $N_{\mathrm{casc}}$ — not a numerical amplitude.

Why antisymmetry is forced

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 saturated branches $\pm s_*$. The modulus must therefore enter through the antisymmetric part — a two-form $F_{\chi,\mu\nu}(s)$ — for the spontaneous $V-A$ branch choice to be representable at all.

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

A Maxwell weak-field matching would only fix the relative magnitude of $\mathcal{B}_{\mathrm{sat}}$ and leave its overall sign undetermined — and the sign of $\mu_\chi$ controls whether A4 selects the symmetric branch ($u = 0$) or the $J_\Pi$-conjugate pair ($u \neq 0$). The note fixes the sign structurally: it is dictated by the admissibility role of $\mathcal{B}_{\mathrm{sat}}$ as the projection-locking functional whose saturated minima are the A4 selection, not by any external matching. This removes the only remaining convention freedom from the PRS deliverable.

Position in the programme

This note belongs to the fermionic matter sub-programme (Presentation Note 6). It is the direct companion of PRS and discharges its first open deliverable. The Lorentzian genus question is closed electric ($\mu_\chi^2 < 0$, $u \neq 0$); the amplitude mechanism is settled on the corpus-derived real cascade — Born–Infeld saturation, with no interior lock reachable without an external complex metaplectic phase and a non-prescribed placement. The remaining quantitative front is the magnitude $|u|$, dictionary-bound through the chiral-frontier normalisation $\mathcal{N}_A$; the genuine 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, Lorentzian Genus, and Electric Determination of the Generation Split. Working paper, 2026. 10.5281/zenodo.20633931