This repository contains the source of the O24 Cosmochrony paper
Observable-Rank Stability under Vertical Non-Injectivity: Closing the Fibre-Structure Conditionality in the Born–Infeld-to-Cascade Chain.

Version 1.2.0. The verticality lemma and observable-rank stability theorem are unchanged. The paper now separates their fibre-structure closure from the independent Heisenberg capacity-to-rate step.

This work extends the spectral admissibility sub-programme by resolving the residual fibre-cardinality condition inherited from O18 after the closure achieved by O22 and the conditional carrier result of O23:

Does the admissibility mechanism depend on the cardinality of the fibres of the projection $\Pi$, or only on the structure of its image?

O21–O23 provide the inputs:

• the proposed pair-level observable is the canonical pair quantity $\sigma_{\mathrm{pair}}^{\mathrm{can}}(n)$
• saturation occurs intrinsically on a BFS shell (O22, projection locking)
• the threshold value
  $\Sigma_c(n_3) = 3$
  is a supplied selection rule: O23 proves (Theorem 3.1) that the traceless sector of a supplied spinor carrier $V_\rho \cong \mathbb{C}^2$ is $\mathfrak{su}(2)$, of real dimension exactly 3, while the carrier selection and the identification of $\Sigma_c$ with that dimension remain open

Moreover:

• the clause that parity is the only symmetry of $S_{\mathrm{BI}}$ is part of the open fibre-identification problem of O18 (Problem 2.8, Remark 2.9), not a theorem
• if established, it would make parity fibres minimal: ${\chi, -\chi}$
• the fibre-side observable-rank argument depended on this assumption

This defines the scope of O24.

The paper proves that the size of the fibres is irrelevant to the stated observable-rank result.

The key mechanism is the verticality of admissible symmetries:

• any symmetry preserving Born–Infeld admissibility preserves the admissible sector
• any new direction it generates must lie in this sector
• under the supplied spinor carrier of O23, the admissible neutral traceless sector has real rank exactly 3 (O23 Theorem 3.1)

Therefore:

• no transversal symmetry is admissible
• all admissible symmetries act within the fibres of $\Pi$

Thus:

• fibre cardinality can vary freely
• observable rank is invariant

Lemma. The real rank of
$\operatorname{Im}\Pi \cap \mathcal{N}_{\mathrm{trl}}$
is independent of $\ker \Pi$.

Thus:

• enlarging fibres does not change observable directions
• microscopic multiplicity is decoupled from effective structure

Lemma. Any $g \in G_{\mathrm{BI}}$ compatible with admissibility either:

• acts within the fibres (vertical), or
• generates a new admissible neutral traceless direction

The second case is impossible:

• admissibility is preserved by $g$
• any generated direction lies in the admissible sector
• under the supplied carrier, O23 (Theorem 3.1) bounds this sector at real rank 3

Thus all admissible symmetries are vertical.

Any transversal symmetry would:

• produce a fourth independent admissible direction
• contradict the rank-three bound of the supplied carrier (O23 Theorem 3.1)

Thus:

• transversal non-injectivity is excluded
• only vertical non-injectivity is compatible with admissibility

Theorem.
Under the supplied spinor carrier of O23, the real rank of the admissible neutral traceless observable sector is:

$\dim_{\mathbb{R}}(\operatorname{Im}\Pi \cap \mathcal{N}_{\mathrm{trl}}) = 3$

independently of the cardinality of the fibres of $\Pi$.

Corollary. The observable-rank condition supporting the fibre-side $c_{\mathrm{BI}} \to \delta_{\mathrm{pair}}$ segment is insensitive to the fibre structure of $\Pi$, with no assumption on fibre size; it remains conditional on the supplied carrier and $\Sigma_c$ identification of O23.

The separate $\delta_{\mathrm{pair}} \to \beta^*$ step is not derived natively on the Heisenberg substrate. Using 1ドル/(\delta_{\mathrm{pair}}+\tfrac12)$ therefore remains a conditional cross-substrate phenomenological prescription.

Within the stated Born–Infeld premise and the supplied rank-three carrier of O23, the fibre-side argument is internal:

Born–Infeld admissibility
$\to$ admissible sector invariance
$\to$ rank-three carrier (O23 Theorem 3.1, supplied carrier)
$\to$ exclusion of transversal directions
$\to$ verticality of symmetries
$\to$ rank invariance
$\to$ fibre-side rank stability

No minimal-cardinality assumption on fibres is required for that result. The chain does not produce a native pair-capacity growth carrier.

O24 performs the final closure of the fibre-level admissibility framework:

• it removes the absence-of-further-symmetries clause (part of O18's open Problem 2.8)
• it replaces fibre minimality with rank invariance
• it establishes verticality as the correct structural condition

More precisely, the paper:

• proves rank–kernel decoupling
• proves verticality of admissible symmetries
• excludes transversal admissible actions via the rank-three bound
• establishes observable rank rigidity
• removes dependence on fibre cardinality
• closes the fibre-cardinality conditionality in the $c_{\mathrm{BI}} \to \delta_{\mathrm{pair}}$ segment

• Born–Infeld parity equivariance (O18; the fibre identification is open there)
• canonical observable (O19–O21)
• projection locking (O22)
• conditional rank-three carrier (O23 Theorem 3.1, supplied spinor carrier)
• Born–Infeld admissibility
• Weil framework

• rank–kernel decoupling lemma
• verticality lemma
• exclusion of transversal admissible symmetries
• observable rank stability theorem
• fibre-structure closure corollary

• numerical determination of $n_3$
• large-$q$ behaviour
• extension beyond SU(2)-type structures
• dynamical modelling of fluctuations
• full numerical closure of the pipeline
• a native growth process carrying the pair observable

The conceptual shift is decisive:

• absence-of-further-symmetries clause (O18 Problem 2.8): fibres must be minimal
• O24: fibres can be arbitrarily large

provided that:

• they act vertically
• they do not increase observable rank

Thus:

• the mechanism depends on image structure
• not on preimage multiplicity

The key insight is:

observable physics is controlled by rank, not by microscopic multiplicity.

O24 completes the fibre-structure sequence:

• O18: parity equivariance, fibre identification stated as open
• O19: canonical normalisation
• O20: persistence criterion
• O21: intrinsic saturation rank
• O22: shell-level locking
• O23: conditional threshold dimension (supplied carrier)
• O24: rank stability under non-injectivity

Thus:

• the observable is fixed
• the shell is derived
• the threshold is a supplied selection rule with a conditional dimension theorem
• the fibre-cardinality dependence is removed (carrier and fibre hypotheses supplied)

This closes the dependence on fibre cardinality, not the capacity-to-rate bridge.

• verticality as a structural principle
• rank–kernel decoupling
• exclusion of transversal admissible symmetries
• observable rank rigidity
• independence from fibre cardinality
• theorem-level fibre-structure closure

The spectral admissibility framework is now:

• parity-equivariant at the Born–Infeld level (O18), with the fibre identification open
• amplitude-level canonical (O19)
• saturation-level intrinsic (O21)
• shell-level derived (O22)
• threshold-level conditional on the supplied carrier (O23)
• fibre-cardinality independent (O24)

The admissibility condition is now:

• structural with respect to fibre cardinality
• algebraically constrained
• rank-invariant
• insensitive to the fibre structure of $\Pi$ (conditional on the supplied carrier of O23)

Determine which shell $n_3$ is dynamically selected.

Study asymptotic behaviour and scaling.

Investigate possible higher symmetry frameworks.

Test extension beyond SU(2) and Heisenberg graphs.

Construct a native growth process carrying the pair observable, or state an explicitly cross-substrate hypothesis for $\delta_{\mathrm{pair}} \to \beta^*$.

The programme is now:

• fibre-structure conditionality reduced to the supplied carrier and fibre hypotheses
• independent of fibre-cardinality assumptions
• explicit about the unresolved native capacity-to-rate step

paper/
├── out/ # Compiled O24 PDF
├── tex/ # LaTeX sources
└── README.md

If you reference this work, please cite:

J. Beau Observable-Rank Stability under Vertical Non-Injectivity: Closing the Fibre-Structure Conditionality in the Born–Infeld-to-Cascade Chain Zenodo, 2026.

Portions of the derivations, conceptual synthesis, structural organisation, 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:

• vertical non-injectivity
• observable rank rigidity
• admissible symmetry structures
• fibre-level dynamics
• spectral admissibility

are welcome.

Please open an issue to discuss conceptual points, technical details, or possible extensions.