Sheaf Primality via Primality Testing Framework

This paper proposes a novel primality testing framework that reinterprets the notion of primality as a global geometric object over the arithmetic scheme Spec(ℤ). By integrating exponential approximation, modular congruence, p-adic valuation, and elliptic curve regularity, we construct a multilayered filter structure formalized as a sheaf over Spec(ℤ). The resulting object, called the Primality Sheaf, admits a global section if and only if a given natural number is prime. We prove this equivalence and formulate each filtering layer as a local sheaf section, ensuring compatibility via gluing conditions. This approach offers a categorical and geometric reformulation of classical number theory, connecting primality to modern tools in algebraic geometry and sheaf theory.