Minimal Fibre Structure of the Non-Injective Projection from Born–Infeld Indiscernability: Derivation of the Parity Involution

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 provides the foundational derivation of the underlying involution. We proceed in two stages. At the abstract level, we show that the Born–Infeld action $S[{\chi}]$ is even in ${\chi}$, and we define a notion of BI-indiscernability for configurations that produce identical effective responses under all BI-admissible perturbations. Evenness of $S$ immediately implies that ${\chi}$ and $-{\chi}$ are BI-indiscernible, so every projective fibre contains the orbit $\{{\chi}, -{\chi}\}$. We then prove a conditional minimality result: in the absence of any symmetry of $S[{\chi}]$ beyond its parity, the minimal fibre is exactly the involution ${\chi}\mapsto-{\chi}$. At the concrete level, the O17 result ${\rho_{q-c}} = \overline{{\rho_{c}}}$ together with the norm-invariance of Gram–Schmidt orthogonalisation identifies the involution $c\leftrightarrow q-c$ as the Weil-representation instance of the abstract parity. This closes the logical gap in O16–O17 by deriving the fibre structure from the Born–Infeld parity rather than postulating it.