This repository contains the source of the Topological Invariants Cosmochrony paper
Topological Invariants of Admissible Configurations: Charge Quantisation, Absence of Magnetic Monopoles, and Projective Chirality in Cosmochrony.

This work establishes the topological layer of the Cosmochrony framework, complementary to the dynamical and spectral analyses developed in previous papers.

While earlier works derived:

• the emergence of charge from non-injective projection
• the bounded-flux constraint and its dynamical consequences
• the spectral admissibility hierarchy

they left open a fundamental question:

what topological invariants remain admissible under the projection constraint, and what physical consequences do they impose?

The present work answers that question.

The paper establishes three unified structural results:

• Charge quantisation arises from
  $\pi_1(S^1) \cong \mathbb{Z}$

• Magnetic monopoles are excluded on the canonical admissible model via
  $\pi_2 = 0$

• Parity violation arises from a structural mechanism of
  projective chirality

These are not independent phenomena, but consequences of a single principle:

admissibility under non-injective projection determines both the available topological invariants and their orientation.

The paper proves three central points.

The U(1) fibre of the Hopf fibration supports a winding number:

$$ \pi_1(S^1) \cong \mathbb{Z} $$

This winding number is identified with electric charge.

Unlike the Dirac mechanism, charge quantisation here:

• does not require magnetic monopoles
• follows directly from the topology of the fibre

In the unconstrained case:

$$ \pi_2(S^2) \cong \mathbb{Z} $$

supports Dirac monopoles and their non-abelian generalisations ('t Hooft–Polyakov, classified by $\pi_2(G/H)$).

Under the bounded-flux constraint, the admissible base becomes:

$$ S^2_{\mathrm{adm}} \simeq S^1 \times (0,1) $$

which implies:

$$ \pi_2(S^2_{\mathrm{adm}}) = 0 $$

Thus:

• isolated magnetic monopoles are topologically excluded
• their empirical absence is explained structurally

The non-injective projection introduces an orientation dependence:

$$ \Gamma(Q) \neq \Gamma(-Q) $$

This defines projective chirality, which:

• breaks parity structurally
• selects one orientation of winding invariants
• provides a mechanism for the observed V−A asymmetry

The paper establishes a sharp contrast with the standard paradigm:

• Standard gauge theory:

  • monopoles are topologically admissible
  • their absence requires a dynamical explanation

• Cosmochrony:

  • monopoles are topologically excluded
  • their absence follows from admissibility constraints

This inversion is central to the result.

To avoid conflating distinct layers, the paper separates:

• topological layer: classification of admissible homotopy invariants

• dynamical layer: relaxation processes and flux constraints

• spectral layer: admissibility selection and mass hierarchy (O-series)

These layers are complementary:

• topology defines what exists
• dynamics defines how it evolves
• spectral structure defines how it is organised

What is rigorously established:

• $\pi_1(S^1) \cong \mathbb{Z}$ → charge quantisation (canonical model)
• $\pi_2 = 0$ → absence of monopoles (canonical model)
• projective chirality as a structural mechanism

What remains open:

• extension of $\pi_2(C_{\mathrm{eff}}) = 0$ to the full configuration space
• derivation of maximal chirality (strict V−A)
• identification of weak-interaction sectors within Weil blocks

This distinction is explicit throughout the paper.

This work is a structural clarification of fundamental phenomena.

It shows that:

• charge
• absence of monopoles
• parity violation

are not independent inputs of the theory, but consequences of a single topological structure.

It converts empirical facts into topological constraints.

This paper complements the existing Cosmochrony programme:

• White paper: emergence of projection and gauge structure
• Dynamical papers: bounded flux and relaxation
• Spectral programme (O-series): admissibility and hierarchy
• Topological invariants (this work): classification of admissible sectors

Together they provide:

• classification (topology)
• mechanism (dynamics)
• quantification (spectral)

The central conceptual outcome is:

• the absence of monopoles is not contingent
• parity violation is not an external symmetry breaking
• charge quantisation is not derived from monopoles

Instead:

👉 all three arise from admissible topology under projection

The paper does not resolve all open problems.

What it does establish is:

• a complete classification of admissible topological invariants (canonical model)
• a structural explanation for key empirical facts
• a clear separation between established results and open questions

1. Global extension of π2 = 0
   Prove that the triviality of π2 holds on the full configuration space

2. Derivation of maximal chirality
   Show that admissibility enforces strict V−A selection

3. Spectral identification of weak sectors
   Connect SU(2) structure to Weil-block representations

4. Topological–spectral interface
   Understand how admissible invariants map onto spectral hierarchy

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

If you reference this work, please cite:

J. Beau, Topological Invariants of Admissible Configurations: Charge Quantisation, Absence of Magnetic Monopoles, and Projective Chirality in Cosmochrony, Zenodo, 2026.

Portions of the conceptual organisation, structural clarification, and editorial refinement benefited from iterative interactions with large language models used as analytical assistants. All theoretical results and interpretations remain the sole responsibility of the author.

This repository is intended as a research reference.

Critical feedback, independent verification, and extensions of the topological results to the full configuration space are welcome.

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