Suppose every prime $n$ could "explode" once.
An explosion results in $\lfloor \alpha \ln n \rfloor$ particles being
uniformly distributed over the integers in a range $n \pm \lfloor \beta \ln n \rfloor$.
If a particle hits a composite or a previously exploded prime, nothing happens.
If a particle hits a new prime $p$, then $p$ explodes under the same rules.
Here is a little simulation starting at $n=23$,
with $\alpha=10$ and $\beta=5$:

So, on the first step, $n=23$ explodes into $\lfloor 10 \ln 23 \rfloor = 31$
particles, uniformly spreading over $23 \pm 15$. In the run depicted, these particles hit four primes:$11, 13, 29, 31$. Then each of those explodes; and so on. In the last frame, $199$ is hit.

Q1. For which $\alpha$ and $\beta$, if any, will this process almost surely explode an infinite number of primes?

I am hoping the answer is independent of the starting $n$.

Being largely ignorant of number theory, I am wondering if current knowledge of the distribution of the primes is sufficient to answer this question. Thanks for insights or even speculations!

**Q1 Answered**. quid shows that a ballistic range of $\pm \beta \log n$ is not enough
to bridge the prime gaps:
$\pm \beta \log^2 n$ conjecturally suffices, but a range of more than $\pm \beta \sqrt{n}$
is all that current technology can prove.

**Addendum**. Both quid and joro confirm that even assuming RH
(the Riemann Hypothesis),
the range needed for my simulation to *provably* explode all primes is at least $\pm \beta \sqrt n \log n$. May I add this question:

Q2. What is the minimum ballistic $\pm$-range that would suffice to provably (under "current technology") explode (with sufficent $\alpha$) all primeswithout any RH or otherwise conjectural assumptions?