A Strengthened Compactness Criterion for Twin-Prime Cofactors in Consecutive Semiprime Chains

Let \(S_1<S_2<\cdots\) enumerate the semiprimes, including prime squares. This note proves that a block of \(m\ge2\) consecutive semiprimes with pairwise greatest common divisor equal to a prime \(p\), and with all prime cofactors greater than \(3\), contains a twin-prime pair whenever\(\frac{S_{k+m-1}-S_k}{p}<6m-8\). The proof is elementary: every prime greater than \(3\) belongs to a unique pair \(\{6a-1,6a+1\}\), and a collection containing no twin pair occupies a distinct such pair for each prime. The resulting criterion strengthens the earlier threshold\(4(m-1)+2\left\lfloor\frac{m-1}{2}\right\rfloor\)for every \(m\ge4\). A quantitative refinement shows that any \(m\) distinct primes greater than \(3\), with total span \(D\), contain at least\(\max\left\{0,\;m-1-\left\lfloor\frac{D+2}{6}\right\rfloor\right\}\)twin-prime pairs. The note includes a verified example where the strengthened criterion applies but the earlier criterion does not, together with an accompanying verification script. For fixed \(p\) and \(m\), infinitely many qualifying compact chains would imply infinitely many twin primes. This implication is proved; the existence of infinitely many such chains remains unproved. The note does not establish the twin-prime conjecture.