...uh
Is White a primary pigment?

Anyway, to answer your question. The color white can be a pigment but it is never a primary color. Contrary to your assertion, the color white is easily produced by Earthly means, titanium dioxide being a popular white with artists.

There's color, then there's pigment, then there's hue. The pigment white cannot be created with other pigments because the pigment white is the lack of all color. The hue white is created by a combination of light of all colors. The color white can be either a pigment or a hue, one being a lack of color, the other being all colors. But you already know this.
Seeing as the color, cannot be produced by paint, dyes, food coloring or other Earthly methods.
I know it can be produced from a light prism so don't troll.