The Convergence Threshold in Exclusion Set Sieves

In this short note, we present a mathematical observation concerning a variant of the Sieve of Eratosthenes, referred to as an Exclusion Set Sieve. In particular, we investigate a sieve system where all integers below a chosen starting number N are excluded, and N is treated as the first candidate. We observe that the set of surviving E-primes eventually coincides exactly with the standard set of prime numbers. We provide a complete characterization of the surviving composite numbers, state the main threshold result with a formal proof, and relate the sieve to the ideal {n >= N} of the multiplicative monoid of positive integers and to the atoms of an associated monoid. Specifically, we prove that the exact threshold for this coincidence is p^2, where p is the largest standard prime strictly smaller than N. We also show that the number of surviving composites is asymptotic to pi(N)^2 / 2. Version 3 Note: This version significantly expands the paper by adding Theorem 2, which provides an asymptotic formula for the number of surviving composites C(N). It also corrects formatting issues from the previous version and adds a new reference (Hardy & Wright) to support the analytical bounds used in the proof. The paper is now 4 pages.