Observable-Rank Stability under Vertical Non-Injectivity: Closing the Fibre-Structure Conditionality in the Born–Infeld-to-Cascade Chain

Abstract

O18 establishes Born–Infeld parity as a covariance of the response family of ${S_{\mathrm{BI}}}[\chi]$ and states the identification of the parity orbit with a fibre of the non-injective projection $\Pi$ as an open problem; the hypothesis that parity is the only global symmetry of ${S_{\mathrm{BI}}}$ is a clause of that problem, not a theorem. O22 proved the shell-alignment conjecture of O21. O23 proves that the traceless sector of a supplied spinor carrier $V_\rho \cong \mathbb{C}^2$ is $\mathfrak{su}(2)$, of real dimension exactly $3$; the carrier selection and the identification of $\Sigma_c$ with that dimension remain open, so the threshold $\Sigma_c(n_3) = 3$ is a supplied selection rule. Together, these results locate the integer $3$ in $\Sigma_c(n_3) = 3$ as a property of $\operatorname{Im}\Pi \cap {\mathcal{N}_{\mathrm{trl}}}$, not of the cardinality of the fibres. The programme then requires only that any symmetry of ${S_{\mathrm{BI}}}$ acts vertically with respect to $\Pi$, i.e.\ does not enlarge the real rank of the admissible neutral traceless sector. The present paper proves this verticality condition from Born–Infeld admissibility and the rank-three carrier supplied by O23, without requiring a classification of $\mathcal{C}_{\mathrm{eff}}$. As a corollary, conditional on the supplied carrier and fibre hypotheses, the fibre-structure conditionality in the ${c_{\mathrm{BI}}} \to {\delta_{\mathrm{pair}}}$ segment reduces to those supplied inputs: the observable rank is independent of fibre cardinality. This does not establish the separate ${\delta_{\mathrm{pair}}} \to {\beta^*}$ capacity-to-rate step, for which no native Heisenberg growth carrier is defined in the corpus. Non-injectivity may increase microscopic multiplicity, but cannot enlarge the admissible observable structure.