Indulgence Weapons (免罪武器, Menzai Buki) are ancient weapons that are linked to EXS abilities.
Indulgence Weapons seem to originate from the Night Blade, although no further information has been given. It is also implied that the weapons (or at least the Insulator) have become lost or missing as Linne had been searching for the Insulator for some time but had no luck in finding it until she met Hyde.
It also appears that the weapons' stronger abilities require a certain amount of power to use as Hyde, who has only been an In-Birth for one month, becomes visibly winded after using a powerful slash while Seth, who has much more experience, is able to use the Eliminator without any apparent effort and is confident that he can use the Insulator's full power himself.
Known Indulgence Weapons
Indulgence of Rending, the Insulator
Indulgence of Sever-Rending, Insulator (断裂の免罪符 [インスレーター], Danretsu no Menzaifu [Insureetaa]) is a katana with a red blade and no guard (making its design almost akin to a cleaver). This weapon is currently in possession of Hyde.
Insulator's power is referred as the Eternity Killer, Loop End (永遠殺 [ループエンド], Eigousatsu [Ruupu Endo]). It is the power to rend apart Existence. This makes it possible to kill In-Births, Voids and even Re-Births.
Indulgence of Seal-Destroying, the Eliminator
Indulgence of Seal-Destroying, Eliminator (封滅の免罪符[エリミネーター], Fuumetsu no Menzaifu [Erimineetaa]) takes the form of a pair of blue daggers shaped like sickles. Originally owned by Kuon, this weapon is currently in possession of Seth.
While Eliminator's power has yet to be fully explained, it is said to possess the opposite power to the Insulator.