Isolated Arithmetic Progressions of Semiprimes with Prescribed Prime GCD

We establish an unconditional existence theorem for arbitrarily long arithmetic progressions of individually isolated semiprimes with a prescribed prime greatest common divisor. Let \(\Omega(n)\) denote the number of prime factors of \(n\), counted with multiplicity. For every fixed prime \(p\ge2\), progression length \(m\ge2\), and isolation radius \(R\ge1\), we prove that there exist infinitely many arithmetic progressions \(A_0<A_1<\cdots<A_{m-1}\) of semiprimes satisfying \(\Omega(A_i)=2\), \(\gcd(A_i,A_j)=p\) for all \(i\ne j\), and \(A_{i+1}-A_i=D\) for a positive common difference \(D\). Furthermore, every selected semiprime is surrounded by a semiprime-free neighborhood of the prescribed radius: \(\boxed{\Omega(A_i+t)\ge3\quad\text{whenever }1\le|t|\le R.}\) The proof combines the positive-relative-density form of the Green--Tao theorem, the prime number theorem in arithmetic progressions, and a Chinese remainder theorem construction that forces all specified neighboring integers to have at least three prime factors. We also extend the method to any prescribed integer greatest common divisor \(g\ge1\) in the sequence \(\{n:\Omega(n)=\Omega(g)+1\}\), and show that the common difference may be constrained to be divisible by any prescribed positive integer. As an explicit illustration, \(309=3\cdot103\), \(489=3\cdot163\), and \(669=3\cdot223\) form a three-term arithmetic progression of semiprimes with pairwise greatest common divisor \(3\), and each has no other semiprime at distance \(1\). The selected terms need not be consecutive in the complete semiprime sequence, and their common difference is not asserted to be uniformly bounded. The theorem is a derived consequence of established mathematics. It does not prove the twin-prime conjecture, and independent novelty is not claimed.