Equivariant Born–Infeld Parity and the Fibre-Identification Problem of the Non-Injective Projection

Abstract

The identification of conjugate Weil blocks $\{c,\, q-c\}$ as the fibres of the non-injective projection ${\Pi}:{\chi}\to{\mathcal{O}}$ was stated as a structurally motivated hypothesis in O16 and used in O17 to derive the pair-level observable and the exponent doubling ${\delta_{\mathrm{pair}}} = 2\,\delta_{c} \approx 7.44$. The present paper determines exactly which part of that identification the Born–Infeld structure settles. We prove that the Born–Infeld action $S[{\chi}]$ is even in ${\chi}$, and that parity acts equivariantly on the family of its response functionals: under the transported-probe comparison $({\chi},\eta)\mapsto(-{\chi},-\eta)$, a response of derivative order $k$ transforms with the character $(-1)^{k}$, so even-order responses (the action itself among them) are preserved while odd-order responses — including the Born–Infeld constitutive response — reverse sign and generically separate ${\chi}$ from $-{\chi}$. We also prove, by an explicit countermodel, that evenness does not force the stronger fixed-probe (pointwise) indiscernibility of ${\chi}$ and $-{\chi}$: an even response can separate ${\chi}$ from $-{\chi}$ under one and the same perturbation. Consequently, the parity orbit $\{{\chi},-{\chi}\}$ is a candidate fibre only relative to a supplied restriction of the observable family to even-order responses, and is not forced by the action alone under either comparison. We then state, as a typed classification problem, the data that a genuine fibre derivation must supply: the configuration space with its boundary conditions, the group of candidate transformations, the prior quotients, the separating family of responses, and a factorisation theorem connecting response equivalence to the fibres of ${\Pi}$. The absence-of-further-symmetries hypothesis (H-rank) is one clause of that problem, not a theorem and not yet a classifiable statement on its own. At the Weil level, the conjugation ${\rho_{q-c}} = \overline{{\rho_{c}}}$ proves a spectral equivalence of the blocks $c$ and $q-c$; the identification of this equivalence with a physical projection fibre remains an open bridge. The pair-level observable of O16–O17 therefore keeps the status O16 gave it: a structurally motivated hypothesis, spectrally supported, awaiting a fibre theorem.