Overview
This article continues the spectral admissibility programme after O17. While O17 established that conjugate Weil blocks $(c,q-c)$ define the correct fibre-level observable, it left open one foundational question: why should the minimal fibre of Π be an involution at all?
O18 answers this at two levels. At the abstract level, it shows that the Born–Infeld action is even in $\chi$, and defines a notion of BI-indiscernability based on equality of effective responses under all BI-admissible perturbations. This implies that $\chi$ and $-\chi$ are projectively indistinguishable.
The central consequence is that every fibre of Π contains the parity orbit \[ \{\chi,-\chi\}. \] Under the explicit conditional hypothesis that no further effective symmetry is present, this orbit is the minimal fibre.
O18 then turns to the Weil realisation on $G_q=\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$ and identifies the abstract involution with the concrete conjugation \[ c \mapsto q-c. \] The conjugate pair observable of O16–O17 is therefore no longer only structurally coherent: it is derived from the Born–Infeld projection structure itself.
Core contributions
- Parity of the Born–Infeld action: $S_{\mathrm{BI}}[\chi]$ depends only on $F^2=(D\chi)^2$, hence \[ S_{\mathrm{BI}}[-\chi]=S_{\mathrm{BI}}[\chi]. \]
- BI-indiscernability: configurations are compared through their effective responses under all BI-admissible perturbations, at fixed resolution scale.
- Fibre constraint: parity implies that every fibre contains the orbit \[ \{\chi,-\chi\}. \]
- Conditional minimality: if parity is the only global effective symmetry compatible with BI-admissible observables, then the minimal fibre is exactly the two-element orbit $\{\chi,-\chi\}$.
- Weil-level realisation: the abstract involution is realised concretely as \[ c\leftrightarrow q-c, \] using the conjugation relation $\rho_{q-c}=\overline{\rho_c}$ and the norm-invariance of Gram–Schmidt orthogonalisation.
- Foundational closure of O16–O17: the fibre identification used in the pair observable is now structurally derived, not only motivated by representation-theoretic consistency.
Interpretation
O18 does not change the pair observable introduced in O16 and analysed in O17. Instead, it explains why this observable class is the correct one.
- O16: the physically relevant observable is the conjugate-pair observable
- O17: conjugate Weil blocks carry identical dynamical information
- O18: the fibre structure itself is derived from Born–Infeld indiscernability
The pair observable is therefore justified at the level of the projection Π itself. The earlier block-level mismatch is no longer only a matter of observable hierarchy: it is traced back to the minimal non-injectivity forced by the effective action.
Relation to the Cosmochrony program
O18 connects the exact Weil analysis of the O-series to the more general Cosmochrony framework. It imports the structural necessity of non-injective projection and the Born–Infeld selection of the effective action, then uses them to derive the minimal parity involution at the abstract level.
The programme now reads: O12–O13 (exact block extraction), O14 (observable mismatch), O15 (block-level derivation failure), O16 (pair observable identified), O17 (pair dynamics derived), O18 (fibre structure derived).
O18 therefore closes the foundational gap left by O17: the observable hierarchy is now not only dynamically and spectrally coherent, but anchored in the Born–Infeld projection structure of the χ-framework.
Current outcome and open directions
O18 establishes that the minimal fibre structure of Π is governed by the parity involution $\chi\mapsto-\chi$, and that in the Weil realisation this becomes \[ c\mapsto q-c. \]
The result is explicitly conditional: if additional global effective symmetries compatible with BI-admissible observables exist in enriched frameworks, larger fibres may occur. A general classification of such fibres remains open.
Three immediate follow-up directions are identified:
- O19: define a canonical normalisation of $\sigma$, independent of block parameters $(b_1,b_2)$
- O20: explain the dynamical selection of the phenomenological window $[7.4,10.6]$
- O21: derive the full fibre-level relation from $\delta_{\mathrm{pair}}$ to $\beta^*$ without pipeline conventions
References
Jérôme Beau. Minimal Fibre Structure of the Non-Injective Projection from Born–Infeld Indiscernability: Derivation of the Parity Involution.