A Symmetric Classification of Prime Numbers: Correlational, Identity, and Inversion Symmetry

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.