Freaky Fraction Spitting

Take a pile of nine pebbles or nine coins or nine cucumbers (but you can't use zucchini!) and split them into two piles. Then split each of those piles into two piles. And keep doing this until you have nine piles each with one object.

What could be mathematically astounding about such a process? A scary, tremendous amount of astonishing mathematical stuff! But here, in this video, I present a particularly freaky invariant to do with fractions.

Can you figure out what is going on?