Arkhe/Arche is a mechanic specific to Fontaine involving two opposite forces: Pneuma and Ousia. When the two come in contact, they annihilate each other like particles and antiparticles. Certain Fontaine characters are Pneuma-aligned, while others are Ousia-aligned.
Using Arkhe attacks can debuff certain enemies. This is similar to some abilities in Star Rail. Arkhe was invented by Alain. Contacting Arkhium crystal can cause certain negative effects on people, as shown in the Geode Mine and Meropide.
From the story of Rene and Jakob, we learn that abyssal energy has priority over elements. Arkhe appears to be a mechanic separate from the elements. Is there any relationship between arkhe, the abyss realm, and human realm (of Celestia)?
The 4.0-4.2 AQ seemed more focused on the prophecy, the masquerade of Focalors, elemental authority and dragons. It was to emphasize the greater conflict between Celestia/Heavenly Principles and the elemental dragons and how the archons are like pawns of Celestia, with the gnosis sealing away a part of the dragons' elemental authority.
However, we still do not know much about arkhe. Fontaine still has some mysteries.