This paper presents a unified algebraic, geometric, and analytic framework that redefines thestructure of integers, vectors, and analytic functions through complex conjugate decompositions. Starting from the Goldbach partition of even integers, we provide a constructive and bounded proof of the Binary Goldbach Conjecture using prime gap estimates and Bertrand’s Postulate. We further extend Goldbach partitions to complex product representations, unveiling new symmetries and identities in prime pairings.The paper introduces geometric decompositions of primes and semiprimes, enabling their visualization in Euclidean and topological spaces. We explore applications to the Riemann zeta function by deriving complex root factorizations that suggest a novel lens for interpreting nontrivial zeros.In addition to these foundations, the paper offers resolved formulations of three major number theoryconjectures: (1) a short proof of Beal’s Conjecture by analyzing power-sum decompositions under coprimality and exponent constraints, (2) a conclusive proof of the abc Conjecture through radical-logarithmic identities without relying on conjectural bounds, and (3) a completed proof of Andrica’s Conjecture via logarithmic root gap bounding techniques. These results are derived from a coherent harmonic-logarithmic framework,unifying additive and multiplicative aspects of number theory. Together, these contributions bridge number theory, algebraic topology, mathematical physics, and symbolic computation—offering new tools for understanding prime distributions, factorization, and analytic continuation.