This is similar to the mathstack question, mandelbrot-bulbs-countable. It turns out that every bulb/Cardioid has a hyperbolic center that is the solution of an algebraic equation; and algebraic numbers are countable. The hyperbolic centers are all zeros of the following sequence of equations: $f_1=x$, $f_2=x^2+x$,
The roots of each of the $f_n$ equations are the hyperbolic centers. The main Cardioid of every mini-mandelbrot also has a main hyperbolic center, which represent a subset of these countable algebraic numbers.
For example, consider the roots of $f_4=0$. Two of these zeros are $x=-0.156520166833755\pm1.03224710892283i$, which are the hyperbolic center of the largest period 4 mini-mandelbrot, and its conjugate
$f_4 = x^8 + 4x^7 + 6x^6 + 6x^5 + 5x^4 + 2x^3 + x^2 + x$. Another zero is -1.94079980652948 which is a smaller period 4 mini-mandelbrot. The other zeros are period=4 and period=2 bulbs, as well as the period=1 main Cardioid at x=0.
Another example is $f_3=x^4 + 2x^3 + x^2 + x$. One of the zeros of $f_3=0$ is $-1.75487766624669$, which is the hyperbolic center of the main Cardioid of the largest period 3 mini-mandelbrot.