This repository contains the source of the O23 Cosmochrony paper
Three Stable Directions from Quaternionic Minimality: Derivation of $\Sigma_c(n_3) = 3$ from Born–Infeld Fibre Admissibility.

This work extends the spectral admissibility sub-programme by resolving the second central open problem left by O21 and isolated by O22:

Why is the saturation threshold intrinsically three-dimensional, i.e. why $\Sigma_c(n_3) = 3$ rather than 2 or 4?

Context

O21 established that:

• the physically relevant observable is the canonical fibre-level quantity
  $\sigma_{\mathrm{pair}}^{\mathrm{can}}(n)$
• the admissibility criterion can be reformulated intrinsically through the
  observable rank $n_3^{\mathrm{obs}}$
• the threshold condition
  $\Sigma_c(n_3) = 3$
  defines the physically relevant saturation shell

O22 proved that:

• saturation necessarily occurs on a BFS shell (projection locking)
• shell-alignment is a theorem, not a conjecture

However:

• the value 3 remained an unexplained structural input
• the dimensionality of the stable sector was not derived
• the threshold $\Sigma_c(n_3) = 3$ had no internal justification

This defines the scope of O23.

Core Result

The paper proves that the value 3 is a structural necessity, not an empirical input.

The key mechanism is quaternionic minimality under Born–Infeld admissibility:

• the admissible neutral sector must be non-abelian
• it must be realised within an associative algebra (Weil framework)
• the Hurwitz classification then forces the minimal structure to be $\mathbb{H}$

Central result:

• the admissible neutral traceless subspace is
  $\mathrm{Im},\mathbb{H}$
• its real dimension is exactly 3

Thus:

$\Sigma_c(n_3) = 3$

is derived as the observable signature of this algebraic structure.

Main Structural Results

1. Exclusion of abelian realisations

The neutrality condition cannot be satisfied by any abelian group in a faithful irreducible representation of dimension 2.

Thus the admissible sector is necessarily non-abelian.

2. Exclusion of non-associative realisations

The Weil representation acts in $\mathrm{End}(V)$, which is associative.

Therefore non-associative algebras (e.g. octonions $\mathbb{O}$) are excluded.

3. Hurwitz classification constraint

The only real associative normed division algebras are:

$\mathbb{R}, \mathbb{C}, \mathbb{H}$

Among them:

• $\mathbb{R}$ is too small
• $\mathbb{C}$ is commutative
• $\mathbb{H}$ is the minimal admissible structure

4. Quaternionic minimality theorem

Using the Hurwitz-to-$\mathfrak{su}(2)$ bridge:

• three independent neutral generators span $\mathfrak{su}(2)$
• $\mathfrak{su}(2) \cong \mathrm{Im},\mathbb{H}$

Thus the admissible neutral sector is exactly:

$\mathrm{Im},\mathbb{H} \cong \mathbb{R}^3$

5. Stable directions and Gram–Schmidt span

Each axis of $\mathrm{Im},\mathbb{H}$ corresponds to:

• one independent neutral direction
• one independent contribution to the admissible span

Saturation occurs when all three directions are realised.

Hence:

$\Sigma_c(n_3) = 3$

6. Spectral realisation

The abstract result is realised concretely:

• $Q_8$ provides the minimal discrete prototype (three orthogonal axes)
• ADE binary graphs exhibit exactly three non-trivial spectral levels

The algebraic derivation and spectral observation are:

• consistent
• complementary
• non-circular

Foundational Chain from the Substrate

The derivation is fully internal:

Born–Infeld parity
$\to$ fibre involution
$\to$ non-abelian admissible support
$\to$ $\mathbb{H}$-minimality
$\to$ $\dim \mathrm{Im},\mathbb{H} = 3$
$\to$ $\Sigma_c(n_3) = 3$

No external parameter is introduced.

Mathematical Role of O23

O23 performs the second foundational closure of the fibre-level admissibility programme:

• it removes the last unexplained structural integer
• it derives the dimensionality of the stable sector
• it connects algebraic minimality to observable capacity

More precisely, the paper:

• excludes abelian admissible sectors
• excludes non-associative realisations
• applies Hurwitz classification
• identifies $\mathbb{H}$ as minimal structure
• derives $\mathrm{Im},\mathbb{H}$ as the neutral sector
• derives the value 3 as its dimension
• connects this to $\Sigma_c(n_3)$

Epistemic Structure of the Paper

Established input

• canonical observable (O19–O21)
• projection locking (O22)
• fibre structure (O18)
• Born–Infeld admissibility
• Weil representation framework

New results

• exclusion lemmas (abelian and octonionic)
• quaternionic minimality theorem
• derivation of $\dim \mathrm{Im},\mathbb{H} = 3$
• identification of stable directions
• derivation of $\Sigma_c(n_3) = 3$

Remaining open problems

• determine the numerical value of $n_3$
• extend beyond SU(2)-type structures
• analyse large-$q$ behaviour
• explore possible extended symmetry frameworks

Interpretation of the Result

The conceptual shift is decisive:

• O21: the value 3 is observed
• O23: the value 3 is explained

This transforms the programme:

• from empirical threshold

• to algebraic necessity

• from spectral observation

• to structural derivation

The key insight is:

the number of stable directions is fixed by the minimal admissible algebraic structure, not by geometry.

Structural Role of O23

O23 continues the sequence:

• O18: fibre structure
• O19: canonical normalisation
• O20: persistence criterion
• O21: intrinsic saturation rank
• O22: shell-level locking
• O23: derivation of the threshold dimension

Thus:

• the observable is fixed
• the amplitude is canonical
• the shell is derived
• the threshold value is now explained

This completes the structural closure of the admissibility threshold.

What O23 Adds

• quaternionic minimality as a structural principle
• exclusion of competing algebraic structures
• derivation of stable directions
• identification of $\mathrm{Im},\mathbb{H}$ as neutral sector
• theorem:
  $\Sigma_c(n_3) = \dim \mathrm{Im},\mathbb{H} = 3$
• closure of the second open problem of O21

Outcome

The spectral admissibility framework is now:

• fibre-level grounded (O18)
• amplitude-level canonical (O19)
• saturation-level intrinsic (O21)
• shell-level derived (O22)
• threshold-level explained (O23)

The admissibility condition is now:

• fully structural
• internally derived
• algebraically constrained
• theorem-level established

Residual Open Problems

Shell selection

Determine which shell $n_3$ is selected as a function of the parameters.

Beyond quaternionic minimality

Investigate whether extended symmetry structures modify the neutral sector.

Large-$q$ regime

Study asymptotic behaviour of the admissible structure.

Universality

Test extension beyond SU(2) and Heisenberg graphs.

Native growth law

Any future rate relation must be derived directly on the Heisenberg substrate. The three-direction threshold does not close the refuted cross-substrate reciprocal.

Status

The programme is now:

• structurally closed at the threshold level
• algebraically grounded
• ready for quantitative shell-selection analysis

Repository Structure

paper/
├── out/      # Compiled O23 PDF
├── tex/      # LaTeX sources
└── README.md

Citation

If you reference this work, please cite:

J. Beau Three Stable Directions from Quaternionic Minimality: Derivation of $\Sigma_c(n_3) = 3$ from Born–Infeld Fibre Admissibility Zenodo, 2026.

Acknowledgements

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.

Contributions

This repository is intended as a research reference.

Critical feedback, independent verification, and further analysis of:

• quaternionic minimality
• fibre admissibility
• stable directions
• spectral threshold derivation
• algebraic constraints on admissibility

are welcome.

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