Quotes / Stable Time Loop

Nagato: Since there's no conclusion to the paradox theory, there's no way to prove there's no paradox.
Kyon: Maybe that's a good enough explanation for you...
Haruhi Suzumiya: Bamboo Rhapsody

1971 Robo: You're going to learn the hell out of hyperdimensional mathematics when you get back. Focus on electrodynamics.
1926 Robo: Why?
1957 Robo: So we'll know how to convert these burnt out powerpacks into a hyper-dimensional bomb to blow up the monster across multiple simultaneous points in space-time.
1926 Robo: How'd you know they burned out?
2009 Robo: Because we just told you, so you'll remember they are, so we can tell you they are. Dah! Study temporal mechanics while you're at it. Short version: when your future self tells you to do something, you do it.
Atomic Robo and the Shadow from Beyond Time

In fact that was even more puzzling, when Harry thought about it. Time had presented him with the finished Prank as a fait accompli, and yet it was, quite clearly, his own handiwork. Concept and execution and writing style. Every last part, even the ones he still didn't understand.

The idea here is that you can travel back in time, but you can't actually change history. So if you go back and try to prevent the assassination of President McKinley, it turns out that you killed him yourself. Or if you go back and try and prevent the Black Plague, it turns out that you are a flea-ridden diseased rat.
Lore Sjöberg, "Alt Text"

I am the Bad Wolf. I create myself. I take the words... I scatter them. Across time and space. My message to lead myself here.
Bad Wolf!Rose Tyler, Doctor Who, "The Parting of the Ways"

Idris: House eats TARDISes.
The Doctor: House what? What do you mean?
Idris: I don't know, something I heard you say.
The Doctor: When?
Idris: In the future.
The Doctor: "House eats TARDISes"?
Idris: Oh! There you go.

The Destiny Trap: You can't change history if you're part of it.
The Eleventh Doctor, Doctor Who, "The Time of the Doctor"

So there's this man. He has a time machine. Up and down history he goes, zip zip zip zip zip, getting into scrapes. Another thing he has is a passion for the works of Ludwig van Beethoven. And one day he thinks, 'What's the point of having a time machine if you don't get to meet your heroes?' So off he goes to 18th-century Germany. But he can't find Beethoven anywhere. No-one's heard of him, not even his family have any idea who the time traveler is talking about. Beethoven literally doesn't exist. (This didn't happen, by the way. I've met Beethoven. Nice chap. Very intense. Loved an arm-wrestle.) No, this is called 'The Bootstrap Paradox'. Google it. The time traveller panics, he can't bear the thought of a world without the music of Beethoven. Luckily he'd brought all his Beethoven sheet music for Ludwig to sign. So he copies out all the concertos and the symphonies... and he gets them published. He becomes Beethoven. And history continues with barely a feather ruffled. But my question is this. Who put those notes and phrases together? Who really composed Beethoven's 5th?
The Twelfth Doctor explains it perfectly, "Before the Flood"

Time Agent 1: Captain, why did you take the Defiant back in time?
Sisko: It was an accident.
Time Agent 1: So you're not contending it was a predestination paradox?
Time Agent 2: A time loop - that you were meant to go back into the past?
Sisko: No.
Time Agent 1: Good.
Time Agent 2: We hate those.
Star Trek: Deep Space Nine, "Trials and Tribble-ations"

"Though we can move through time, time itself is unmovable."

"No one can unmake the past. It's already happened, there's no 'undo'. Similarly, the future already happened. We just haven't reached it yet."

If I ever invent a time machine, I'm going to stay as far away from myself as possible. If I were to have a conversation with myself, wouldn't I have to remember it perfectly, or else my memory of the conversation would be different from the actual conversation and it could result in a paradox that could destroy the universe? Unless, of course, I don't remember the conversation well enough to remember what future me was/is supposed to say. I guess my point is, don't go visit yourself yesterday. Wait a decade or two.
Bob and George (author's commentary)

gallowsCalibrator/Terezi Pyrope, Homestuck

Present!Vanderbeam: Wait! Where did you get this? I was too late to steal it!
Future!Vanderbeam: I used this suit to steal the one I just gave to you.
Present!Vanderbeam: But... then where did you get the one you're wearing?
Future!Vanderbeam: My future self handed it to me two decades ago.

Pierce: "I don't believe you, causality is—
Volair: Pick a number between ten and oh, sixteen.
Pierce: Uh, twelve. Now—
Volair: How old were you when I suddenly decided to paint the kitchen green?
Pierce: Twel— (look of sudden realization). Urk. Uh. muhh.. fuh..
Rhonda: Told ya.

"Don't do anything that affects anything, unless it turns out you were supposed to do it, in which case, for the love of God, don't not do it!"
Professor Farnsworth, Futurama, "Roswell That Ends Well"

"Well, I guess that wraps everything up in a nice, paradox-free bow"
Professor Farnsworth, Futurama, "Bender's Big Score"

Goliath: If I didn't fear the damage you would do to the time stream, I would gladly leave you here.
Xanatos: But you won't, because you didn't. Time travel is funny that way.

"One cannot damage history because history cannot be changed. I went back in time to steal this because history says it disappeared. And history says it disappeared because I went back to steal it. Past, present, future. It's all written in stone, my dear, and nothing you do can ever change it."
Warp, Teen Titans "How Long Is Forever?"