DOI: 10.5281/zenodo.18328851
title: Symmetry‑Based Classification of Prime Number Families: A FAIR Dataset

description: 📊  A FAIR Dataset presents a unified collection of prime number families enriched with symmetry‑based invariants. It is designed to support reproducible research and to offer a structural foundation for exploring primes through an alternative coordinate system based on correlational and inversion symmetries.

The dataset includes more than seventy classical and specialized prime families, each annotated with symmetry vectors derived from simple but systematically defined invariants. These vectors induce a discrete geometric structure on the primes, enabling comparisons across families, the identification of recurring patterns, and the development of new classification principles. The framework remains neutral and extensible, inviting researchers to form their own interpretations.

🧠  All data are provided in open, machine‑readable formats, accompanied by Mathematica notebooks used for generation and verification. Adhering to FAIR principles, the dataset supports independent theoretical work, computational experiments, and future extensions.

Beyond a collection of numerical sequences, the dataset offers a structured environment in which primes and prime constellations can be examined through symmetry‑based descriptors that are elementary to compute yet rich in structural implications. By encoding each prime and constellation as a symmetry vector, the dataset enables comparative analysis across families traditionally studied in isolation, opening possibilities for new forms of classification and geometric interpretation.

The construction presented here is intentionally modest in assumptions and open in implications. It does not propose a theory of primes, but provides a set of tools—symmetry invariants, coordinate assignments, and structural vectors—that may serve as a starting point for further inquiry. Mathematics often advances through new ways of looking at familiar objects, and this framework is offered in that spirit: as a lens through which primes may be reorganized or reimagined.

The invariants introduced here do not claim to capture fundamental properties of the primes. Their value lies in the possibilities they open: new comparisons, new questions, and new structural perspectives. The dataset is offered not as a collection of conclusions, but as an invitation to exploration.

🔍  Future researchers may discover connections not suggested here, or reinterpret the invariants in ways that lead to new insights. Such developments are encouraged. If this framework inspires even a single new line of investigation, or offers a useful perspective to someone working in a distant corner of number theory, its purpose will have been fulfilled.

 
description: This publication is part of the research cycle “Linear Infinity → Cyclic Space”, a structured sequence of works exploring how linear asymptotic regimes on the positive half‑line transform into cyclic, stratified, or boundary‑identified geometric structures under metric contraction. The full cycle currently includes:





Part I. — Hybrid Linear–Cyclic Topological Structures for Digital Sequence Encoding and Technosignature Analysis | DOI: 10.5281/zenodo.18473473




Part II. — Informational Geometry of the Positive Half-Line. A World Without Negative Numbers | DOI: 10.5281/zenodo.18474513




Part III. —  A Symmetric Classification of Prime Numbers. Correlational, Identity, and Inversion Symmetry | DOI: 10.5281/zenodo.18520138




Part IV. — Global Affine Time and Metric Uniqueness: A Geometric Characterization of Linear and Cyclic Temporal Structure | DOI: 10.5281/zenodo.18505857


Part V. — Symmetric Signatures of Global Configurations: Topological Rigidity and the Epoch of Convergence in a Case Study of Orion–Giza Correspondence. A Unified Framework for 2D Similarity Invariants, 3D Orientation Geometry, and 4D Precessional Dynamics | DOI:10.5281/zenodo.18651258




Part VI. — Linear Time, Cyclic Geometry. Spectral Phase Structure of Cosmic Emitters | DOI:10.5281/zenodo.18698140


Together, these works form a coherent geometric–informational program tracing the transformation:

linear infinity → asymptotic contraction → metric multiplicity → temporal stratification → cyclic space → spectral phase structure.

Each entry stands independently, yet collectively they outline a unified framework in which linear temporal regimes, cyclic geometries, and spectral invariants emerge as different manifestations of the same underlying structural principles.
description: All project materials, publications and complete machine‑readable metadata are maintained at https://linearcyclic.eu