An Arch is a designation given to a specific individual residing within one of the parallel worlds in the Multiverse. The title was created by Singularity to refer to a person that possesses enough strength and willpower to be able to challenge fate, clashing with his powers of phenomenal affirmation.
The Arch-Eve of a particular world within the Multiverse is primarily a version of Bayonetta, due to her complex history and being a key part of many critical moments regarding the fate of the Trinity of Realities. They are also vital to the stability of the world they inhabit due to their fate-altering powers, with their death quickly leading to the destruction of said world without being present to challenge Singularity. As a result, Singularity promptly makes it his mission to hunt them down. The fate-altering powers of these Arch-Eves are insufficient to resist Singularity's Phenomenal Affirmation, allowing him to systematically kill thousands of these, sometimes seemingly out of nowhere. The only Arch-Eve who is capable of avoiding this fate is the Arch-Eve Origin who possesses fate-altering powers that override those of Singularity's Phenomenal Affirmation, effectively preventing Singularity from landing a certain death on her and making her the only person capable of taking the maniacal AI down on an equal-term fight.
The title of Arch-Eve is not always fixed on a Bayonetta variant, however, and can be assumed by other individuals within a specific world if they have a strong enough will. Jeanne β3 was originally the Arch-Eve of her world until the title was assumed by Bayonetta β3 following her death, and Viola took Bayonetta's place as the Arch-Eve Origin after the latter sacrificed her life to defeat Singularity once and for all.
One of the very first Arch-Eves, the Bayonetta present within Singularity's origin of the Alphaverse, fought against him but was destroyed in the process. However, her consciousness was able to survive and stay within the Alphaverse. As more Arch-Eves were killed, their residual consciousnesses joined with the the Alphaverse Bayonetta, becoming the malevolent force known as Dark Eve.
Arch-Adam
Lukaon, one of the Arch-Adams within the Multiverse.
Though not a term originally defined by Singularity, a world's Arch-Adam within the Multiverse is the counterpart to an Arch-Eve. In every known universe within the World of Chaos, the Arch-Adam is a variant of Luka Redgrave.
According to Luka's own definition, the Arch-Adam of a world is supposed to guide an Arch-Eve to where she needs to be to accomplish her goal. This is a role that Luka fulfilled across the firsttwo games by helping Bayonetta reach Balder and by giving her important information regarding the legends of Aesir.
The Arch-Adam from the Alphaverse was killed by Singularity along with the Arch-Eve, but his consciousness survived. Fuelled by revenge, this Luka called to his other variants within the Multiverse, but only served to transform himself into a beast of hatred known as Dark Adam. That call, one for "the single truth", was a key part of Luka's insatiable desire to discover the truth of things as a journalist.
As more worlds within the Multiverse are destroyed, Luka find himself succumbing more and more to Dark Adam's call as a voice in his head, eventually transforming into the beastly Strider. This faerie power originally belonged to Lukaon, a faerie king Arch-Adam with the power to travel between worlds, but was corrupted by Dark Adam's call when they merged. Lukaon himself was able to find Luka after Bayonetta defeated Dark Adam, using his power to join the hearts of other Arch-Adams from across the Multiverse together and keep such power under control.
Trivia
The terms of Arch-Adam and Arch-Eve are an explicit reference to the creation story of Genesis in the Bible, featuring Adam and Eve as the first humans whose actions shaped the course of humanity. This is further emphasized in how Bayonetta is referred to as the "Witch of Genesis" within her character profile in Bayonetta 3, and the roles of Arch-Adam and Arch-Eve in the Multiverse.
Singularity refers to the playable Bayonetta in Bayonetta 3 as "Arch-Eve Origin" rather than just a normal Arch-Eve. As the game has suggested that she is either the grown up time-displaced Cereza or the Bayonetta from the Main Universe, the addition of "Origin" to her designation could refer to how Bayonetta's interactions with her past self allowed the creation of a parallel world by Cereza traveling back in time with knowledge of the present, essentially being an origin of different realities for other Bayonettas.
Similarly, the game also refers to Luka as "Arch-Adam Origin" after he merges with the hearts of the other Arch-Adams, implying that he is now the only one left within the entire Multiverse.
