The Filtrate Theory and the Primes of Alpha Filtrate

This paper introduces Filtrate Theory, a number-theoretic framework that approaches prime generation through the algebraic construction of prime-dense integer regions rather than the isolated testing of randomly selected candidates. The study develops the Alpha Filtrate, a quadratic family defined by Uk=5k2−5k+1, derived from the algebraic properties of a set generated by ratios of geometric progressions. The methodology integrates algebraic derivation, analytic investigation of sequence density, multiplicative-order analysis, and computational evaluation to examine the structural behavior of this quadratic family under the base-2 Fermat primality test. The mathematical development establishes several structural properties of the underlying set and derives the Alpha Filtrate from these properties. Computational experiments indicate that, over the tested range, composite terms of the Alpha Filtrate exhibit strong empirical resistance to base-2 Fermat pseudoprime occurrence while maintaining a substantial density of prime values. The study further introduces the Filtrate Resilience Coefficient (FRC), a metric for quantifying the pseudoprime-rejection capability of quadratic families. These findings support Filtrate Theory as a framework for identifying integer families with reduced composite interference during prime candidate generation. Although additional theoretical work is required to establish complete pseudoprime exclusion, the proposed framework provides a foundation for further investigation into deterministic prime-candidate generation and its potential applications in computational number theory and cryptography.