As above, so below is older than The Kybalion and better attested than anything else in it. The line descends from the Emerald Tablet, a short Hermetic text that survives first in Arabic, embedded in early compilations of secrets of creation, and reaches Latin Europe by the twelfth century. Its opening states, in the usual rendering, that what is below is like that which is above. The Kybalion tightens this into a slogan and supplies the reverse direction, so that the reading runs both ways.
This is the load-bearing principle for a reference work like this one. Every table of correspondences the atlas holds, planet to metal, letter to path, colour to sephirah, rests on the assumption that a pattern found at one level of the world can be read off another. Nothing in the sources argues for that assumption. They take it as given and get on with the tabulating.
Two cautions belong with it. Correspondence is a rule for reading rather than a rule for inferring: a shared attribution between two things is a claim somebody made, not evidence that they behave alike. And the tables disagree with each other, sometimes flatly, which is why this atlas records the tradition a correspondence belongs to instead of presenting any one system as the correspondence. A principle that licenses reading across levels does not license collapsing the traditions that do the reading.
Sources The Kybalion: The Definitive EditionPhilip Deslippe (ed.) with Tabula Smaragdina: Ein Beitrag zur Geschichte der hermetischen LiteraturJulius Ruska