An Elementary Compactness Criterion for Twin-Prime Cofactors in Consecutive Semiprime Chains

We establish an unconditional compactness criterion connecting consecutive semiprime chains with the existence of twin-prime cofactors. Let \(S_1<S_2<S_3<\cdots\) denote the increasing sequence of semiprimes, including squares of primes. Fix a prime \(p\ge2\) and an integer \(m\ge2\). Suppose that \(m\) consecutive semiprimes \(S_k<S_{k+1}<\cdots<S_{k+m-1}\) satisfy \(\gcd(S_{k+i},S_{k+j})=p\) for every pair of distinct indices \(0\le i<j<m\), and assume that their prime cofactors \(S_{k+i}/p\) are all greater than \(3\). We prove that the compactness inequality \(\boxed{\frac{S_{k+m-1}-S_k}{p}<4(m-1)+2\left\lfloor\frac{m-1}{2}\right\rfloor}\) necessarily implies the existence of an index \(j\) such that \(\boxed{S_{k+j+1}-S_{k+j}=2p.}\) Consequently, the corresponding prime cofactors form a twin-prime pair. The proof uses the fact that the prime cofactors of consecutive semiprimes sharing a prescribed prime factor must themselves be consecutive primes. In the absence of twin primes, every successive cofactor gap is at least \(4\). Furthermore, two consecutive gaps cannot both equal \(4\), since three primes greater than \(3\) cannot have the form \(q,q+4,q+8\). These restrictions yield the stated lower bound on the normalized span of any twin-free chain. For the special case \(m=3\), the criterion reduces to \(\boxed{S_{k+2}-S_k\le8p,}\) which guarantees that at least one adjacent pair has twin-prime cofactors. We also establish a conditional infinitude criterion: if infinitely many distinct prescribed-GCD semiprime chains of any fixed length \(m\) satisfy the compactness inequality, then infinitely many twin-prime pairs exist. The result provides an elementary connection between the arithmetic structure of consecutive semiprimes, prime-gap restrictions, and the twin-prime conjecture. It complements existing constructions of arbitrarily long prescribed-GCD semiprime chains derived from the Maynard--Tao theorem and the Chinese remainder theorem. The compactness criterion and its logical consequences are proved unconditionally. However, the existence of infinitely many semiprime chains satisfying the required compactness inequality remains unproved. Therefore, this work does not establish the twin-prime conjecture. The contribution is presented as an elementary sufficient criterion and a possible direction for further investigation, without asserting independent novelty.