Discovering Chaos in Prime Numbers — Computational experiments through the Euler mirror

DOI: https://doi.org/10.5281/zenodo.18258723

This repository is a technical and historical record.

It does not seek persuasion, commercial adoption, or institutional validation. Its sole purpose is to preserve an observable phenomenon, together with the methods and conditions required for its independent reproduction.

All claims documented here are grounded exclusively in operational reproducibility, not interpretative authority.

---

How to access

1. Definitive Version (PDF): Download the complete book as a PDF — Recommended for archival purposes and offline study.
2. Online Reading: Access the HTML version of the book here
3. Laboratory: Run the experiments in the /notebooks directory via Google Colab.

---

Repository structure

• book/ Complete text of the work in Markdown, PDF, EPUB, and HTML formats.

• notebooks/ Fully reproducible computational experiments designed for execution in Google Colab.

• LICENSE.md Terms of use.

• CITATION.cff Academic citation metadata.

---

How to read this repository

1. Read the book in any of the available formats.
2. Execute the notebooks in numerical order.
3. Observe the phenomenon.
4. Draw your own conclusions.

No prior trust, interpretation, or theoretical commitment is required.

Readers interested only in the operational result may proceed directly to the notebooks.

---

Conceptual note — Where quantum chaos resides

This repository functions as a technical custody record.

Its purpose is not persuasion, market positioning, or institutional endorsement, but the rigorous preservation of an observable phenomenon and the precise conditions under which it can be independently reproduced.

The validity of the results depends on execution rather than authority.

---

Central position

This work adopts an operational and verifiable position:

Quantum chaos does not reside exclusively in the system, but in the relation between the system and the mode of observation.

GOE/GUE-type statistics are not automatic ontological properties of isolated mathematical or physical objects. They emerge when observation is aligned with the appropriate structural scale of the phenomenon.

---

Central implication

The widespread belief that GOE/GUE statistics are:

• exclusive to the non-trivial zeros of the Riemann zeta function, or
• inherent only to quantum systems with classically chaotic analogues,

is not structurally necessary.

This repository demonstrates, in a fully operational and reproducible manner, that:

• deterministic operators constructed from prime arithmetic,
• without direct reference to the zeros of the zeta function,
• exhibit GOE-compatible statistics

when observed at the appropriate scale.

When the same systems are observed under incompatible scales, Poisson-type statistics emerge instead.

---

Diagnosis of the recurring error

The recurring error does not lie in:

• the system,
• the arithmetic, or
• a lack of complexity.

It lies in the assumed neutrality of the observational metric, and in the implicit belief that statistical universality is an absolute property of the object rather than a relational phenomenon.

---

Epistemological consequence

GOE statistics should not be interpreted as a mystical seal or an exclusive signature tied to a singular object.

They are a relational signature: they emerge when structural organisation and observational scale enter into coherence.

This repository documents this fact not as conjecture, but as an operationally reproducible phenomenon, verifiable by any reader willing to execute the experiments provided.

Language note

This repository is the English-language counterpart of the Portuguese repository Descobrindo o Caos nos Números Primos, which documents the same phenomenon under a different linguistic and cultural context.

---

This repository exists so that the phenomenon remains accessible, independently of the pace of its interpretation.