Symmetry‑Based Classification of Prime Number Families: A FAIR Dataset

📊 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.