Integers are only Primals and Compounds

This study introduces a nuanced framework for classifying integers that transcends the traditional dichotomy of prime and composite numbers. It incorporates: (I) the inclusion of negative integers as primal numbers, (II) the recognition of 1 and -1 as distinct primal numbers, and (III) the classification of 0 as a unique compound number, alongside negative compounds. This approach is rooted in a detailed exploration of prime number sieves across the entire integer spectrum, highlighting the fundamental roles of negative integers, zero, and unity in mathematical constructs. Central to this work is the definition of the complementary pair of divisors and the reconceptualization of integers through their multiplicative relationships, particularly exemplified by the hyperbolic sieve of primes derived from the multiplication table framework. This study argues against the oversimplification inherent in the binary classification of integers into primes and composites based solely on their multiplication or factor properties. Instead, a more comprehensive classification system is proposed, utilizing the concept of divisors to delineate integers with one or two distinct positive divisors as primal numbers, and integers with more than two distinct positive divisors as compound numbers