K-CORRECTION 2026: Exact Multiplicity Identities, Moment Bounds, Rough Complements, and the Remaining Barrier to Strong Goldbach

This paper develops an exact counting framework for the strong Goldbach problem. For an even integer E≥6, let R(E) count odd primes p for which E−p is also prime. The K-correction identity R(E)=N(E)−D(E)+K(E) follows from an exact pointwise identity for the number of distinct small prime divisors of E−p. We develop the associated multiplicity histogram, factorial moments, finite inclusion–exclusion identities, and a weighted three-moment lower bound, while explicitly auditing the information these quantities do and do not provide. A second decomposition uses rough complementary integers: for E^(1/3)<z<√(E−3), every rough complement is prime or semiprime, giving the exact identity R(E)=S(E,z)−B(E,z)+C(E,z). A Liouville-sign identity supplies an equivalent signed formulation. We isolate the remaining analytic task as a uniform separation between rough prime survivors and semiprime contamination. The manuscript ends with a three-step conditional closure scheme: prove a positive lower bound for S, prove a strictly smaller upper bound for B (with the small-complement correction retained), and combine these estimates. The first two uniform estimates are not established here. Accordingly, this is a rigorous partial-results manuscript, not a proof of the strong Goldbach conjecture.