When Klaus passed through the 3rd core, he didn't find the Demon Bead there. He wanted to ask the senior about it, but he said nothing and continued through the other cores.
He only pegs it as one of those things that usually happens to the bead. He knew it would appear eventually, and he was right.
The bead was indeed going through a final phase before it fully opened.
Now, however, he was inside the 7th core and was about to approach the gem when he felt the bead appear and start emitting powerful energy from within the third core.
He immediately forgoes the gem and moves to the 3rd core to check what is happening.
When he arrived, he saw the bead surrounded by golden runic circles. They were similar to the ones he had seen when he used the [Falling into Despair] technique.