Peel the ginger, then using a mandoline, slice ginger as thinly as possible and set off to the side. In a small saucepan, bring 300 g water, 75 g sugar, and 7g calcium lactate to a boil.
Turn off the heat, and add in the ginger. Cover and let this steep for 15 minutes. Strain through a chinois, discarding the ginger slices.
Fill a squeeze bottle with the ginger infusion, then fill 1 inch hemispheric molds. These will be very fragile, so it is wise to make a couple extra, but we want to save enough ginger infusion to store the spheres later.
Place the mold into the freezer for 2 hours or until frozen solid.
To make the sodium alginate bath, add 1400 g water, 65 g sugar, and 6 g sodium alginate to a medium pot and cook until the sugar and sodium alginate have dissolved.
Into a shallow container, place the spheres evenly in the warm sodium alginate bath. Make sure they do not touch one another or then can stick together.
As the spheres thaw, they form a thin membrane on the outside holding the liquid in the center, which should take five minutes.
Fill two bowls with hot water, then dip the spheres to remove any excess alginate and store in the reserved ginger infusion.
Store these in the fridge, they will keep for around 3 days.