After unlocking the Eudaimonia Machine stage, enter it and make your way to the Directer (the only enemy on the stage). If this is your first or second time entering the stage (i.e. when you can get both this and the Gracia's Mirror), it'll be non-hostile. On the first attempt, take the option "Too hard". This will unlock the relic, which adds Endless Mode, and the achievement. On the second attempt, the Directer will give you the relic outright.