Published February 7, 2026 | Version v1
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A Symmetric Classification of Prime Numbers: Correlational, Identity, and Inversion Symmetry

Authors/Creators

  • 1. Academy of MInistry of Interior

Description

Prime numbers are typically viewed as arithmetically simple yet structurally irregular objects, lacking an intrinsic organizing principle beyond divisibility. This work develops a complementary structural perspective in which every prime number $P>2$ is assigned a well-defined \emph{symmetric signature} determined by three mutually independent invariants.

The first invariant, the \emph{correlational symmetry}, arises from the choice of sign in the expressions $(P\pm1)/4$ and determines which of the two adjacent arithmetic configurations contributes to the definition of a local symmetric quantity $N(P)$. The second invariant, the \emph{identity symmetry}, captures a uniform algebraic relation between the neighboring integers $P-1$ and $P+1$, a relation that holds for all primes and provides a structural baseline for the framework. The third invariant, the \emph{inversion symmetry}, links local and global information by comparing $N(P)$ with the prime index $\mathbb{P}(P)$ through the integer difference

\[
Z(P)=\mathbb{P}(P)-N(P).
\]

Together, these invariants define a classification scheme in which primes are organized not only by numerical order but also by structural properties encoded in their symmetric signatures. The framework is established through three foundational lemmas and a unifying theorem showing that every prime $P>2$ belongs to a unique symmetric class determined by its signature. The construction is entirely elementary and does not aim at predictive or analytic results on prime distribution; rather, it provides a structural coordinate system revealing an additional layer of organization within the prime sequence that complements classical analytic and probabilistic approaches to prime number theory.

A reproducible dataset accompanying this work provides explicit symmetric signatures for large sets of primes, enabling further computational and structural investigations.

Notes

⭐ Highlights

  • Introduces a symmetry‑based structural framework in which every prime number P>2 is assigned a three‑component symmetric signature.
  • Defines the correlational symmetry that selects between (P−1)/4 and (P+1)/4, thereby determining the local symmetric value N(P).
  • Establishes the identity symmetry, showing that the neighboring integers P−1 and P+1 satisfy a universal structural relation independent of the specific prime.
  • Develops the inversion symmetry linking the local value N(P) with the global prime index P(P) through the invariant Z(P)=P(P)−N(P).
  • Demonstrates that the three invariants form a coherent classification scheme, organizing primes into symmetric classes within a structural coordinate system.
  • Provides three foundational lemmas and a unifying theorem integrating the correlational, identity, and inversion symmetries into a single consistent framework.

Series information

This publication is part of the research cycle “Linear Infinity → Cyclic Space → Expansive Chaos”, 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
  • Part VII. — From Topological Models to Telescope Data: Empirical Tests of Hybrid Linear–Cyclic Dynamics in High-Energy Astrophysical Systems | DOI:10.5281/zenodo.20545297

Taken together, these works form a coherent geometric–informational program that traces a continuous transformation of organizational structure:

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

Each component stands independently, yet collectively they outline a unified framework in which linear temporal regimes, cyclic geometries, and spectral invariants emerge as complementary manifestations of the same underlying structural principles. The progression therefore reflects not a sequence of physical phases, but a set of organizational regimes that become observable across mathematical models, geometric constructions, and empirical reconstruction spaces.

All project materials, publications and complete machine‑readable metadata are maintained at https://linearcyclic.eu

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References

  • N. J. A. Sloane, The on-line encyclopedia of integer sequences, a000040, https://oeis. org/A000040, sequence of prime numbers (2024)
  • T. M. Apostol, Introduction to Analytic Number Theory, Springer, 1976
  • G. H. Hardy, E. M. Wright, An Introduction to the Theory of Numbers, 6th Edition, Oxford University Press, 2008
  • D. M. Burton, Elementary Number Theory, 7th Edition, McGraw-Hill, 2010
  • P. J. Cameron, Some notes on symmetry and structure in mathematics, Mathematical Gazette 92 (523) (2008) 1–12