A Constructive Exact Counting Framework for Goldbach's Conjecture

Abstract

This paper presents a constructive exact counting framework for computing the number of representations of an even number as the sum of two primes. The framework consists of two components: the First Formula (a recursive generation rule for algebraic deduction) and the Correction Table (a step-by-step global correction rule). The system guarantees that at every step, the corrected theoretical total precisely equals the actual number of surviving pairs after sieving (i.e., Sk≡Rk, not an estimate). Within this framework, all composite cases except the "double-prime case (n=2p)" admit constructive derivations. The double-prime case is precisely reduced to a proposition concerning the boundedness of the correction magnitude. Numerical verification through independent cross-validation confirms that within the range n≤106, the framework prediction holds, and the global maximum relative correction ∣M∣/F=1 occurs only at the boundary point p=5 (n=10). This work is an independent constructive study that does not reference extant sieve literature. The framework originates from the author's original derivation in 1990 and was systematically consolidated in 2026. This paper establishes a constructive exact counting framework that reduces Goldbach's Conjecture to a precise, testable mathematical proposition.