The Prime Projection Operator: An Algebraic Characterization of Primality through Pascal's Triangle

We reinterpret the classical property that defines prime numbers within Pascal's Triangle as a linear projection acting on its combinatorial structure. This approach reveals that the behavior of prime numbers can be expressed through three equivalent perspectives: a new factorial-based test of integrality, a divisibility condition along the rows of Pascal's Triangle, and a congruence relation previously known from Fermat's theorem. Although the computational performance is similar to that of traditional Fermat tests and subject to the same pseudoprime limitations, the proposed framework redefines primality as a manifestation of geometric symmetry within the binomial structure, rather than as a purely numerical property. The study also introduces a counting function to describe how primes appear within Pascal's Triangle and establishes an explicit parallel between this horizontal prime operator and the diagonal operator that generates the Fibonacci sequence. The main original contribution is the formulation of a factorial integrality criterion that detects primality independently of modular arithmetic, offering a unified algebraic and combinatorial interpretation that connects number theory, combinatorics, and polynomial algebra.