There are three references to NX-01 being "dilithium-powered".
First, "Cold Front":
Tucker: "Oh. Well, I guess that covers the basics. Any questions?"
Annoying alien: "How do you regulate positron flow in your dilithium matrix?"
Tucker: "Good one. I'll bring up the schematic of the reactor assembly and you can see for yourself. If you'll just come with me. We use a series of magnetic constrictors to align the positron stream."
We could argue here that Tucker sidesteps the question (while secretly thinking "What the hell is dilithium?"): the ship has no dilithium matrix at all, but
instead uses magnetic constrictors to align the positron stream. That's in keeping with the TNG backstage technobabble where dilithium indeed performs a focusing function for antimatter streams.
Second, "Two Days and Two Nights":
Risan official: "If you add a few days to your stay, you could give everyone a chance to enjoy themselves. The additional charge would be minimal."
Archer: "It's tempting, but we need to keep some dilithium to run our engines."
This is less ambiguous. Dilithium is said to be engine-related, and this is no ploy because Archer is soon shown handing over some actual dilithium (or at least a glowing jar that's suggested to contain dilithium) to the official as payment.
However, this dialogue and scene was ultimately edited out of the episode, and dilithium is not mentioned in any of the actually aired scenes.
Third, "Bound":
Kelby: "The injectors feed into the dilithium chamber."
D'Nesh the Orion: "That's where the matter and antimatter mix."
Kelby: "That's right."
D'Nesh: "The crystals let you control the reaction."
Kelby: "That's right."
That's directly from the TNG Tech Manual, no ambiguity left.
Whether Kelby is talking about a radical upgrade or the systems that were installed prior to "Broken Bow" is open to speculation. And we never actually see the dilithium. Is it big, natural crystals, like in TNG and some TOS episodes describing emergencies and nonregular use? Is it ping-pong paddle shapes, possibly containing matrices of microcrystals, as seen in TOS "Alternative Factor" supposedly describing regular dilithium? Is it something else altogether?
As for the Tyme Barrier, it could be
a) a set value for speed, a former record that has been broken
b) a physical barrier in space that stopped older ships from utilizing a shortcut route
c) a structure on Earth, completely unrelated to starship speed; Tyler only mentions it to the castaways because they will be mightily surprised otherwise when they return to Earth, fully expecting to see the Tyme Barrier still standing
Timo Saloniemi