Exact Normalized Gap Two for Consecutive Almost-Semiprimes with Prescribed Prime GCD

We establish an unconditional theorem concerning exact normalized gaps between consecutive almost-semiprimes with a prescribed prime greatest common divisor. Let \(\mathcal A_3=\{n\ge2:2\le\Omega(n)\le3\}\), where \(\Omega(n)\) denotes the number of prime factors counted with multiplicity, and write its elements in increasing order as \(A_1<A_2<A_3<\cdots\). We prove that for every fixed prime \(p\ge5\), infinitely many indices \(k\) satisfy \(\boxed{\gcd(A_k,A_{k+1})=p,\qquad\frac{A_{k+1}-A_k}{p}=2.}\) Moreover, the smaller element is a semiprime, while the larger has either two or three prime factors. We establish the quantitative lower bound \(N_p(X)\gg_p\frac{X}{(\log X)^2},\) where \(N_p(X)\) counts qualifying pairs whose smaller element is at most \(pX\). The proof combines the Chinese remainder theorem with Lewulis's published theorem on Chen primes in admissible arithmetic progressions. The construction forces every integer strictly between the selected endpoints to have at least four prime factors, thereby establishing consecutiveness in the entire almost-semiprime sequence. We further extend the construction to any prescribed integer greatest common divisor \(g\ge2\). Writing \(r=\Omega(g)\), infinitely many consecutive elements of the ordered set \(\{n\ge2:r+1\le\Omega(n)\le r+2\}\) have greatest common divisor \(g\) and difference exactly \(2g\). The result demonstrates that exact normalized gap two is unconditionally attainable in a minimally enlarged semiprime sequence. It does not establish the infinitude of twin primes or the corresponding result for consecutive ordinary semiprimes. The work is presented as a consequence of established Chen-prime and congruence methods, without asserting independent novelty.