Pi in the Mandelbrot set everywhere

The numerical phenomenon of $\pi$ appearing at parameters $c=1\mathrm{/}4$, $c=-3\mathrm{/}4$ and $c=-5\mathrm{/}4$ in the Mandelbrot set $\mathcal{M}$ has been known for over 30 years. In 2001, the first proof was provided by Aaron Klebanoff for the parameter $c=1\mathrm{/}4$. Very recently in 2023, an even sharper result for $c=1\mathrm{/}4$ was proved using holomorphic dynamics by Paul Siewert. This new proof also provided a conceptual understanding of the phenomenon. In this paper, we give, for the first time, a proof of the known phenomenon for the parameters $c=-3\mathrm{/}4$ and $c=-5\mathrm{/}4$, which is also conceptual, and we provide a generalization of the phenomenon and the proof for all bifurcation points of the Mandelbrot set.