Isolated Clusters of Consecutive Semiprimes with Prescribed Prime GCD

We derive an unconditional isolation theorem for arbitrarily long clusters of consecutive semiprimes with a prescribed prime greatest common divisor. Let \(\Omega(n)\) count prime factors with multiplicity, and let \(S_1<S_2<\cdots\) enumerate all integers with \(\Omega(n)=2\), including squares of primes. For every fixed prime \(p\ge2\), integer \(m\ge2\), and real number \(R>0\), we prove the existence of infinitely many blocks \(S_k<\cdots<S_{k+r-1}\), with \(r\ge m\), such that \(\gcd(S_{k+i},S_{k+j})=p\) for all distinct indices in the block and \(S_{k+r-1}-S_k\le pH_m\), where \(H_m\) depends only on \(m\). Moreover, the gaps from this cluster to the immediately preceding and following semiprimes both exceed \(R\): \(S_k-S_{k-1}>R\) and \(S_{k+r}-S_{k+r-1}>R\). The prime cofactors \(S_k/p,\ldots,S_{k+r-1}/p\) are consecutive primes in the ordinary prime sequence. The proof uses the Maynard--Tao theorem for admissible families of linear forms and a Chinese remainder theorem construction that makes all noncandidate integers in a bounded central interval and two arbitrarily long exterior buffers have at least three prime factors. We also establish an analogous statement for any prescribed positive integer greatest common divisor \(g\) in the sequence of integers having exactly \(\Omega(g)+1\) prime factors. As a finite illustration, \(1378011=3\cdot459337\), \(1378023=3\cdot459341\), and \(1378029=3\cdot459343\) are consecutive semiprimes with pairwise gcd \(3\), total span \(18\), and exterior semiprime gaps of \(13\) on both sides. The result is an unconditional consequence of established sieve and congruence methods. It does not imply the twin-prime conjecture, and independent novelty is not claimed.