I am aware of the theorem that $p_{n+1}- p_n \leq n^{0.525}$ which is true for all sufficiently large numbers due to Baker, but if i want to make the implicit "for all sufficiently large numbers" explicit, is it known that $p_{n+1}-p_n \leq c n^{\alpha}$ for all $n \geq 1$ and for small $c$, lets say $c \leq 2$ and $\alpha \leq 0.55$ ?

Any ref that can give me the explicit numbers or a way to construct them would be great.

Thank you, also i posted the question yesterday on MSE