Observable-Rank Stability under Vertical Non-Injectivity: Closing the Fibre-Structure Conditionality in the Born–Infeld-to-Cascade Chain

Current Zenodo release: version 1.3 (2026-08-15). Official title: Observable-Rank Stability under Vertical Non-Injectivity: Closing the Fibre-Structure Conditionality in the Born–Infeld-to-Cascade Chain.

Overview

This article continues the spectral admissibility programme after O23. O22 showed that admissible saturation must occur on a BFS shell, and O23 showed that the relevant threshold is \[ \Sigma_c(n_3)=3. \] The remaining question was whether the mechanism still depended on the strong fibre condition inherited from O18.

Within those supplied hypotheses, the central objective of O24 is: to prove that observable rank does not depend on fibre cardinality, but only on the stability of the rank of \(\mathrm{Im}\,\Pi \cap \mathcal{N}_{\mathrm{trl}}\), fixed to 3 by the quaternionic maximality established in O23.

The paper introduces a clean distinction between the two structural levels of the projection: \[ \ker \Pi, \] which encodes microscopic multiplicity and projection residuals, and \[ \mathrm{Im}\,\Pi, \] which encodes the effective observable structure. A larger fibre may increase fluctuations, but it cannot create new admissible directions as long as every Born–Infeld-admissible symmetry acts vertically.

The key structural statement is therefore a rank-rigidity theorem: non-injectivity may increase microscopic degeneracy, but it cannot enlarge the admissible observable structure.

Scope statement. This page summarises the conditional structural content of O24: rank–kernel decoupling, the verticality lemma, exclusion of transversal actions, and observable-rank stability with respect to fibre cardinality. O24 does not derive a first-principles identification of the supplied conjugate-pair structure with the physical fibre of \(\Pi\). It also does not derive the downstream continuation to \(\beta^*\), which the span-growth no-go (SGN) subsequently shows has no native Heisenberg carrier.

Core contributions

Rank–kernel decoupling: the real rank of \[ \mathrm{Im}\,\Pi \cap \mathcal{N}_{\mathrm{trl}} \] is independent of the cardinality of \[ \ker \Pi. \]
Verticality lemma: every symmetry of \(S_{\mathrm{BI}}\) compatible with admissibility preserves the admissible sector and therefore cannot generate new admissible directions beyond those already allowed by O23.
Exclusion of transversal actions: any transversal action would generate a fourth independent direction in the neutral traceless sector, contradicting quaternionic maximality.
Observable rank stability theorem: the admissible observable sector therefore satisfies \[ \dim_{\mathbb{R}}(\mathrm{Im}\,\Pi \cap \mathcal{N}_{\mathrm{trl}})=3 \] independently of fibre size.
Interpretation within the supplied model: larger fibres change microscopic multiplicity without changing the proved observable rank.
Conditional rank closure: given the stated fibre and Weil data, O24 replaces a minimal-cardinality condition by verticality. The physical identification of those data is not proved.
Consequence for the admissibility chain: the segment \[ c_\chi \to \delta_{\mathrm{pair}} \] is stable under changes of fibre cardinality within the stated model. This does not make the physical fibre identification unconditional. The continuation to \(\beta^*\) is not covered by this result and is refuted as a native derivation by the span-growth no-go.

Interpretation

O24 separates two questions that must not be conflated.

Rank stability: proved under the supplied Weil and fibre hypotheses
Physical fibre identification: not derived and still open

The crucial point is that non-injectivity is not itself a threat to the programme. What matters is not how many microscopic configurations project to the same observable, but whether any admissible symmetry can add a new independent direction in the observable sector.

In other words, the programme moves:

from a condition on fibres
to an invariant of the image
from preimage minimality
to observable rank rigidity

Relation to the Cosmochrony program

O24 occupies a decisive position in the O-series. After the construction of the fibre-level observable (O16–O19), the persistence and intrinsic saturation criterion (O20–O21), shell locking (O22), and the derivation of the threshold dimension (O23), O24 shows that none of this depends on a strong assumption about minimal fibres.

The programme now reads: O16 (pair observable), O17 (pair dynamics), O18 (minimal fibre structure), O19 (canonical normalisation), O20 (persistence criterion), O21 (intrinsic saturation rank), O22 (projection locking and shell condition), O23 (derivation of the threshold dimension), O24 (rank stability under non-injectivity).

O24 closes only the dependence on fibre cardinality inside its supplied model. The identification of the conjugate-pair construction with the physical admissible fibre, as well as the native capacity-to-rate bridge, remains open.

Current outcome and open directions

O24 establishes that \[ \dim_{\mathbb{R}}(\mathrm{Im}\,\Pi \cap \mathcal{N}_{\mathrm{trl}})=3 \] remains true regardless of fibre cardinality under the stated Weil-realisation and fibre hypotheses. This is not a first-principles derivation of the physical fibre.

Remaining directions include:

Large-\(q\) numerical campaign: quantitatively confirm the shell-alignment prediction across multiple conjugate pairs.
Effective shell selection: derive not only the threshold value, but the actually selected shell \(n_3\).
Symmetry extensions: test whether enriched frameworks preserve verticality and rank rigidity.
Asymptotic regime: study the behaviour of the admissible structure in the large-graph limit.
Status of the downstream comparison: the quantitative continuation \[ \delta_{\mathrm{pair}} \to \beta^* \] is closed negatively by the span-growth no-go: it has no derived carrier on the Heisenberg measurement substrate, and the numerical agreement with the charged-lepton window survives only as a cross-substrate phenomenological check.

References

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