Existence of Every Term of OEIS A396673

For every odd prime \(p\), we prove that infinitely many triples of consecutive odd squarefree semiprimes have pairwise greatest common divisor \(p\). These triples can be chosen consecutive in the complete semiprime sequence, with span at most \(pH\), where \(H>0\) is an absolute constant independent of \(p\). Consequently, the first-occurrence sequence OEIS A396673 is well-defined for every index \(n≥2\). The proof combines the established Maynard--Tao theorem with a Chinese remainder theorem construction that excludes unwanted semiprimes. The note includes the congruence construction, a verification of admissibility, and the deduction of global consecutiveness. The result establishes existence and infinite recurrence unconditionally and is presented as a consequence of established methods, without asserting independent novelty.