Relaxed Almost-Prime Production in Sieve Sequences

The square-safe Sieve Sequence certifies that every survivor p in [Q,Q^2) is prime. Requiring p+2 to survive every same filter would ask for a twin-prime pair. This article studies a deliberately weaker target: requiring p+2 to avoid primes below a smaller threshold, which implies that p+2 has at most two prime factors for all sufficiently large Q. The article proves exact algebraic results for the relaxed weight, including its one-divisor local factor and boundary remainder, a shifted-divisor prime-progression discrepancy, and a bilinear character decomposition. These results do not prove positivity; they identify the remaining averaged prime-progression and locally adapted bilinear estimates.