On a Conditional Approach to the Twin Prime Conjecture via Sifted Composite Gaps

We develop a conditional framework that links the statistical behavior of gaps among sifted odd composites to the infinitude of twin primes. Central to our approach is Conjecture 1 (Uniform Gap Sparsity), which asserts that short adjacent gaps \( \leq 8 \) in the sifted composite sequence occur only with asymptotic zero density as the sieve level grows. Assuming Conjecture 1 and classical distributional results such as the Bombieri–Vinogradov theorem, a Selberg–GPY type sieve produces \( \gg X/(\log X)^2 \) twin prime pairs up to \( X \). The logical structure of the argument is complete, but several components are presented in sketch form—notably the short-interval Selberg bound and the bilinear-form estimates—so as to highlight the conditional reduction rather than obscure it with technical detail. A forthcoming companion work is envisioned to provide fully rigorous expansions of these arguments. In addition, we emphasize that the reduction is modular: weakened or averaged forms of Conjecture 1 could already yield nontrivial results on bounded prime gaps, while stronger bilinear estimates would sharpen quantitative bounds. Thus, even if Conjecture \ref{con:A} in its full uniformity is too strong, natural weakened variants may still suffice to establish conditional progress toward the twin prime conjecture.