Consecutive Semiprimes with Prescribed Prime GCD and Bounded Cofactor Gaps

We establish a bounded-gap transfer theorem for consecutive semiprimes with a prescribed prime greatest common divisor, conditional on a residue-uniform version of the 186-gap prime-tuple sieve result. Let $S_1 < S_2 < S_3 < \cdots$ denote the increasing sequence of semiprimes, including squares of primes. Assuming the required residue-uniform two-prime sieve estimate for admissible 40-element tuples of diameter 186, we prove that for every fixed prime $p \geq 5$, there exist infinitely many indices $k$ satisfying $$\gcd(S_k,S_{k+1}) = p$$ and $$2 \leq \frac{S_{k+1}-S_k}{p} \leq 186.$$ Moreover, the corresponding cofactors $S_k/p$ and $S_{k+1}/p$ are consecutive primes. The proof combines a bounded-prime-gap sieve for admissible 40-element tuples with a Chinese remainder theorem construction that forces unwanted intermediate integers to have at least three distinct prime factors, thereby excluding them from the semiprime sequence. The argument transfers the assumed residue-uniform prime-tuple estimate to globally consecutive semiprimes with a prescribed prime greatest common divisor. As a consequence, under the same sieve hypothesis, for every fixed prime $p \geq 5$, there exists an even integer $$d_p \in \{2,4,6,\ldots,186\}$$ such that infinitely many consecutive semiprimes satisfy $$\gcd(S_k,S_{k+1}) = p,\qquad S_{k+1}-S_k = p d_p.$$ This provides a bounded-gap relaxation of the conjecture concerning first semiprime gcd occurrences and twin-prime cofactors, reducing the previously considered cofactor-gap bound of 600 to 186 under the stated sieve hypothesis. The result establishes a conditional infinite-family theorem for every prescribed prime gcd, without assuming the twin-prime conjecture. It does not establish that the first occurrence has twin-prime cofactors or that infinitely many twin primes exist. The contribution is a transfer argument connecting bounded prime gaps, admissible prime tuples, and the arithmetic structure of consecutive semiprimes. The 186 bound depends on the stated residue-uniform sieve input associated with recent 2026 preprint research and is not claimed here as an independently established unconditional theorem.