This repository contains the source of the O18 Cosmochrony paper
Minimal Fibre Structure of the Non-Injective Projection from Born–Infeld Indiscernability:
Derivation of the Parity Involution.
This work extends the spectral admissibility sub-programme by resolving the final structural gap left by O17:
Why is the minimal fibre of the projection Π given by conjugate pairs, and can this be derived from first principles?
O17 established that:
- conjugate Weil blocks
$(c, q-c)$ carry identical dynamics - the pair observable
$\sigma_{\mathrm{pair}}(n) = \sigma_c(n),\sigma_{q-c}(n)$ is the correct physical observable - the exponent doubling
$\delta_{\mathrm{pair}} = 2,\delta_c$ is structurally derived
However, O17 left one fundamental open problem:
- why conjugate pairs define the minimal fibres of the projection Π
This identification was:
- structurally consistent
- supported by representation theory
- but not derived from the underlying χ-framework
This defines the scope of O18.
The paper derives the minimal fibre structure of Π from first principles.
It proves that:
- the Born–Infeld action is even:
$S[\chi] = S[-\chi]$ - this implies BI-indiscernability of configurations:
$\chi \sim -\chi$ - therefore every fibre contains the involution:
${\chi, -\chi}$
Under a minimality assumption:
the parity involution is the minimal non-injectivity of Π
Finally, in the Weil realisation:
- the involution is identified with:
$c \leftrightarrow q-c$
Thus, conjugate pairs are no longer a hypothesis:
they are the derived minimal fibres of Π.
The action satisfies:
because it depends only on:
This establishes a fundamental symmetry at the χ-level.
A notion of physical indistinguishability is defined:
- two configurations are BI-indiscernible if all BI-admissible responses coincide
This implies:
Therefore:
the projection Π must identify parity-related configurations.
From BI-indiscernability:
- every fibre satisfies:
${\chi, -\chi} \subset \Pi^{-1}(y)$
Thus:
parity is necessarily contained in every fibre.
Under the assumption:
parity is the only global effective symmetry compatible with BI-admissible observables
it follows that:
- the minimal fibre is exactly:
${\chi, -\chi}$
Any larger identification would:
- either introduce additional symmetry
- or be unstable under admissible perturbations
Using O17:
$\rho_{q-c} = \overline{\rho_c}$ - conjugate blocks are dynamically indistinguishable
This identifies:
Therefore:
the abstract parity involution is realised as
$c \leftrightarrow q-c$ .
O18 proves that:
- fibre structure is not assumed
- but derived from the Born–Infeld framework
Thus:
the identification used in O16–O17 is now a theorem.
With O18, the hierarchy is fully grounded:
- block-level observables (O12–O15)
- pair-level observables (O16–O17)
- fibre structure derived (O18)
The key statement becomes:
observables are defined on fibres induced by Π,
and these fibres are determined by BI-indiscernability.
The central conceptual result is:
non-injectivity of Π is not arbitrary,
it is constrained by the Born–Infeld structure.
More precisely:
- projection fibres correspond to physically indistinguishable states
- indistinguishability is governed by BI-admissible responses
- parity is the minimal such identification
Thus, O18 establishes a direct link between:
- Born–Infeld dynamics
- projection structure Π
- observable definition
O18 completes the full chain:
- O12–O13: exact block extraction
- O14: observable mismatch
- O15: block-level no-go
- O16: pair observable (hypothesis)
- O17: pair dynamics (derived)
- O18: fibre structure (derived)
Thus:
- the observable is justified
- the fibre structure is derived
- the framework is structurally closed
- derivation of parity involution from Born–Infeld structure
- definition of BI-indiscernability
- proof that fibres contain
${\chi, -\chi}$ - conditional minimality of the parity fibre
- identification of
$c \leftrightarrow q-c$ as its Weil realisation - closure of the O16–O17 foundational gap
The spectral admissibility framework is now:
- structurally grounded at the fibre level
- consistent with projection non-injectivity
- derived from the χ-level dynamics
The pair observable is now:
- physically justified
- mathematically derived
- no longer an assumption
The main remaining questions are:
-
Canonical normalisation (O19)
Remove residual dependence on$(b_1, b_2)$ -
Spectral window selection (O20)
Explain the restriction to$[7.4, 10.6]$ -
Full δ → β* derivation (O21)
Derive the relation at the fibre level without pipeline assumptions -
Beyond minimal fibres
Classify fibres in enriched frameworks with additional symmetries
The programme is now:
- observationally well-defined (O17)
- structurally grounded (O18)
- free of unresolved foundational assumptions
Remaining work is:
- technical normalisation
- dynamical selection
- extension to full χ-level derivations
paper/
├── out/ # Compiled O18 PDF
├── tex/ # LaTeX sources
└── README.md
If you reference this work, please cite:
J. Beau, Minimal Fibre Structure of the Non-Injective Projection from Born–Infeld Indiscernability: Derivation of the Parity Involution, Zenodo, 2026.
Portions of the derivations, conceptual synthesis, numerical strategy, and editorial refinement benefited from iterative interactions with large language models used as analytical assistants. All theoretical results, computations, and interpretations remain the sole responsibility of the author.
This repository is intended as a research reference.
Critical feedback, independent verification, and further analysis of:
- fibre-level admissibility
- Born–Infeld-induced projection structure
- Weil-level realisations
are welcome.
Please open an issue to discuss conceptual points, technical details, or possible extensions.