Andrica's Conjecture — How Tight Are Prime Gaps?

Does $\sqrt{p_{n+1}} - \sqrt{p_n} < 1$ hold for every $n \ge 1$, where $p_n$ is the $n$th prime?

Equivalent form: $p_{n+1} - p_n < 2\sqrt{p_n} + 1$. PNT gives average gaps of order $\ln p_n$, but an average does not control the worst gap. Within explicitly verified ranges, the largest observed difference is $\sqrt{11}-\sqrt7\approx0.6709$ at $n=4$; no theorem proves it is the global maximum over all primes.

Matt Visser (2019) explicitly verified, for $p<2^{64}$, the stronger statement that the difference is below $1/2$ outside six listed exceptions. The largest observed value in that range is $\sqrt{11}-\sqrt7\approx0.6709$ at the exception $p_n=7$. A finite computation proves neither a global maximum nor the conjecture for all primes. RH alone does not directly imply it; stronger gap conjectures such as Cramér or Oppermann would.