The Black Omen was destroyed in 12000 BC because of all we said.
As for Lavos, I've just checked something in the game, and it's a little strange... When you clear the Black Omen in 12000 BC, Lavos is shown erupting on the ocean, then the party fights him in his blue wavy pocket dimension. But if you lose the battle at this point, the Day of Lavos bad ending is shown! Lavos is shown erupted (already on the surface, see attachment) on the 1999 AD world map, and the Arris Director sequence is shown.
The fact which is disturbing me is that if you lose in that scenario, Lavos is shown already erupted on the 1999 AD ground... With this BIG hole on the ground, Chronopolis and even the people of 1999 AD should have records and evidences of his existence.
The only explanation I can see (besides the "gameplay" explanation), is that maybe Lavos erupts on the ground only if you lose to him. If you don't lose to him, he would still be in 1999 AD because of the bucket Gate, but apparently under the ground or still totally inside his pocket dimension. The disturbing thing is that you don't see him erupt from underground, you see him already emerged on that island with a big magmatic hole.
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