Prime-Free Interval Constraints for Consecutive Semiprimes: A Partial Result on the Twin-Cofactor Conjecture

This deposit contains two related research notes on consecutive semiprimes whose greatest common divisor is a prime \(p\geq5\). The work is motivated by computational observations suggesting that the first occurrence of each such common prime factor has twin-prime cofactors. The first note establishes that if \(pq 2\left(\frac{r-q}{2}\right)^{40/21}p^{19/21}.\]Consequently, for sufficiently large \(p\), the additional cofactor-size condition\[r\leq2^{61/21}p^{19/21}\]forces \(r-q=2\). The companion note formulates and critically examines the proposed assertion that infinitely many distinct primes \(p\) admit consecutive semiprimes satisfying this size condition. We prove that this assertion would imply infinitely many twin-prime pairs: the cofactors must differ by two, while the lower bound \(r>2p^{19/21}\) ensures that the resulting twin pairs are unbounded. However, the companion note also establishes substantial obstructions to this particular proposal. Writing \(x=pr/2\), every sufficiently large qualifying configuration would produce a prime-free interval \((x-p,x)\) with\[p\geq\frac12x^{21/40}.\]Any uniform prime-interval bound of length \(o(x^{21/40})\) would therefore exclude the proposed infinite family. In particular, the prime-gap bounds implied by the Riemann Hypothesis show that RH implies the bounded-cofactor conjecture is false. The note also discusses Runbo Li's preprint, arXiv:2308.04458v8, which states a short-interval result with exponent \(0.52\). Taking that stated theorem as valid would likewise rule out the bounded-cofactor conjecture. The preprint's proof and numerical calculations are not independently verified in these notes. These obstructions concern the additional cofactor-size restriction; they do not by themselves resolve the original first-occurrence twin-cofactor conjecture. Without imposing that size restriction, we also prove that consecutive semiprime pairs with twin-prime cofactors for infinitely many distinct common prime factors would imply infinitely many twin primes. The notes distinguish established deductions, conditional implications, and unproved existence assertions. They do not claim a proof of the twin-prime conjecture, a resolution of the original first-occurrence conjecture, or a disproof of the Riemann Hypothesis.