On the Density of Prime Imbalances in the Unit Interval

We prove that the set of normalized differences between primes, defined as $S=\{(p-q)\mathrm{/}(p+q):p>q\text{ are primes}\}$, is dense in the open unit interval $(0,1)$. Our proof provides an explicit construction algorithm with quantitative bounds, relying on elementary results from prime number theory including Bertrand’s postulate and explicit bounds on prime gaps in long intervals.