.. How do I open the cylinder puzzle inside the fireplace safe?

First you need to find the missing piece. It'll be a wheel, a disc about four inches across and a dark grey in color. You won't run across it for a while, so you can ignore the cylinder till then.

After you get into the spy toys display in the Cathedral hub --

How do I get into the spy toys display?

-- you'll find the wheel in there. Pop it in, and it's time to start on the actual puzzle.

Do you remember where you've seen symbols like these? Two places, actually: the archives, and "Mom's Case Notes." Read to the end of this hint to see how to arrange the symbols.

First, notice that the wheels can be moved around. That must mean they need to be in a specific order, but what order?

The rows of symbols you got from the archives look like they spell something, but they don't. Rather, each chunk shows you a piece of the correctly arrayed symbols.

It's as if Kate took a picture of the completed puzzle, then cut it into pieces and mixed them around. All you have to do now is put them back in order.

For Amateurs, this isn't actually that hard, given that each symbol appears only once. See how one chunk ends with a percent sign, and another begins with a percent sign? That means the second chunk goes right after the first one. You can probably solve the entire puzzle quite easily --

-- especially when you realize that the chunk that starts with a big black arrow is the first one in the array.

For Masters, it's a little harder because the chunks have empty spaces in the middle of them. Still, the same rules apply -- look for matching symbols in two chunks and put them together.

For example, one chunk has the wiggly equals sign that means "approximately" at the start, followed by two spaces. Another piece has that "approximately" in the middle, followed by a space and an asterisk. You now know that the asterisk goes two spaces after the "approximately."

But if you're about to tear your hair out, here's the way to arrange the symbols:

Ah, but you're not done yet. See how the wheels are in five groups with arrows above them? That means they can be turned in groups.

Go back to the "Mom's Case Notes" first page and look at the bottom. See anything that could tell you how to turn the wheels?

What you want is the notation that says, "x8 x3 x6 x4 x7." That's telling you to turn the first group 8 times, the second one 3 times, and so on.

The message you get is cryptic: "Music themes reveal ace." It'll be a long time before you know what that means (sigh).

Endgame: After You Call the Scientist