The Four Human Errors

The Four Human Errors were originally Lost Ones whose Pure Concepts corrupted their minds by erasing their identities and memories while increasing their powers to colossal levels capable of destroying the world. However, it was revealed by Pandæmonium that the Human Errors caused this destruction in an attempt to return to their homeland. After the four Human Errors were contained or eliminated, the executioners became tasked with killing the the Lost Ones to prevent future Human Errors from occurring.

Pandaemonium
The Pure Concept of Chaos, Pandaemonium appears as a young girl whose sole purpose is to sow chaos into the world. She was ultimately contained in the eternal fog, however she was partially released by Akari's attempt to escape Pandaemonium's monster in the fog by using her Pure Concept of Time.

Mechanical Society
The Pure Concept of Vessel, this ability resulted in Lost Ones to gradually lose their sanity with each use of their Pure Concept by eroding their memories at the cost of developing the powers.