Nemausus/Menausus, the third layer of Arcadia, didn't fall into Mechanus — it became Daanvi.Think about it — the description of Daanvi sounds just like the description of Arcadia. Arcadia is a lawful and slightly-good plane, while Daanvi is a straight-up lawful plane. And how did Arcadia lose its third layer? That layer became too lawful.
The Lady of Pain is Haruhi Suzamiya.Because it is the LAW!