Finite Sieve Identities, Quadratic Minorants and a Non-Covering Formulation of the Binary Goldbach Problem

For a composite integer n ≥ 4 we study the symmetric decompositions 2n = (n − r) + (n + r) through a finite sieve and its Fourier factorisation by the Chinese remainder theorem. We record exact identities: a full-sieve count with restoration of small representations, a spectral identity, and a grouping by conductors. At the cubic level (2n)1/3 every surviving term is a prime or a semiprime, and the number of Goldbach representations is given by an exact identity involving three sums of the Liouville function. On a reserve where the two terms have at most two and three prime factors respectively, we determine the best lower bound obtainable from polynomials of total degree at most two in the numbers of prime factors, and we exhibit two subfamilies with identical second-order moments, one containing no prime pair and one containing a prime pair. We then derive lower bounds for the number of prime pairs by bounding the semiprime complements through their least prime factor, in both orientations, and combine them over intervals; for 2n = 200 000 this certifies at least 222 pairs. All identities and numerical values are checked by an independent program in exact integer and rational arithmetic. No proof of the binary Goldbach conjecture is claimed: the uniform positivity that would be required remains open, and we explain, using established results, why it meets the parity barrier of sieve theory.