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(This is the community metapuzzle, discussed in this meta thread. It requires answers from nine other puzzles linked below.)

You find a mysterious device sitting on the ground. It's labelled in a language you don't recognize and doesn't seem to be like anything you've ever seen before. Picking it up, you notice no form of input or output except the button that was face down... which you now seem to have released by lifting it off the ground. The device starts to make a whirring noise. As it vibrates, you suddenly find yourself transported to all sorts of locations, time periods, and situations with seemingly nothing in common...

As the whirring slows down, you hear a lot of yelling... and notice several spears pointed directly at you.

Welp, this is bad.

You're gonna have to find a way to calm these people down, and fast. You desperately try to think of things to show them to convince them that you're worth keeping around, at least out of curiosity. Mathematics? No, they don't have algebraic notation, so you'd have to somehow convey a concept without words. Maybe there's something on your phone? But there aren't any internet connections in ancient Rome (or at least, you think that's where you are). While fumbling around in a desperate panic, you notice that the back of the device has come loose, and inside there appears to be some sort of diagram. Maybe this was placed in there intentionally? Could this give you the clue you need?

enter image description here

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Metapuzzle Answers

Gordon K: Silicon
Beastly Gerbil: Bigfoot
Hugh Myers: Ditto
Rand Al'Thor: Simplistic
Alconja: Violin
GentlePurpleRain: Incognito
Julian Rosen: Idiotic
'Q': Digit
Dcyfj: Windowshopping

I noticed:

The only vowels are I and O, and an I starts every such string, treat these as binary.

Plugging these into the using this decoding blatantly stolen from ffao

Results in

Gate A: BeastlyGerbil + Gordon K = 4 + 6 = 10
Gate B: Alconja ^ Hugh Myers = 5 ^ 2 = 25
Gate C: Hugh Myers * GentlePurpleRain = 2 * 10 = 20
Gate D: Julian Rosen - Rand Al'Thor = 13 - 7 = 6
Gate E: Dcyfj / 'Q' = 9 / 3 = 3

And the next step:

Gate F: D - Ground = 6 - 0 = 6
Gate G: B * C = 25 * 20 = 500
Gate H: A ^ E = 10 ^ 3 = 1000
Gate I: D / D = 6 / 6 = 1
Gate J: E + D = 3 + 6 = 9

And now, to get letters

We're in Rome (maybe), and these look like Roman Numerals so...
VIVID MIDI MIX

So I pull out my phone

Open the music app, and play this sick video game music jam that was apparently downloaded onto it by whoever drew this diagram. The centurions are impressed by this wondrous sound, and don't even stab me a little bit.

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