I bet in book 4 Eragon is gonna find Jeod's old entrance and use it to steal back all the heart of hearts back and the last egg

Then.... He will give it to Roran StrongHammer

What CP has said.

1. CP has said that the rider will be "logic inevitability"
2. CP also mentioned that he hopes it will be a surprise.
At the time, the book we know now as Brisingr was supposed to be the final book of the series. Paolini decided to separate that book in two. Before Brisingr came most people didn’t though that it would be Arya. It was Roran. But after Brisingr was released, people have considered Arya much more likely, not just because of the events that took place in the book itself but Paolini's explanations of some of those certain scenes since then.
3. The rider has to have been in all three books so far.
4. CP has said that there is clues in the books about who the next rider is.
And the only thing that can count as clues for any character is the constant referring Arya with the colour Green. (the colour that the egg has)
5. CP has said that book four continues the adventures of Eragon, Saphira and Arya.
He didn’t just said Eragon and Saphira. He also said Arya. Take it for what you want.


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In the matter of logic inevitability.

Roran:
1. Roran has his family to live for. That would ruin his life with his family, causing him to outlive his wife (his entire world) and his child. Then his possible grandchildren, etc. Depressing much?
2. Then he could not be King, which is probably the leading theory under Roran - no immortal Rulers.
3. So far, his characterization has been developing his physical strength and leadership skills. No diplomatic skills or magical skills. If you were to use the stone argument, it could easily be countered with this - his failure at learning/using magic could be there to show his path will not be a magical one.
4. Is human - very strong, but only for a human. Could not compete with Galby or Murt.
5. On the "Rider blood" argument - blood is no real argument. CP has said that dragon does not choose because of families. He don’t even share that what you then should call rider blood. BromEragon MorzanMurtagh. Roran has no conection there.
6. We have no time to train him. How many books did it take Eragon to reach his current strength? We would need to teach Roran the customs of Riders, how to preform magic and control energy as well as the AL itself, etc. Not time.
7. Wields a hammer, not a sword. It’s a diffrent fighting skill.
8. Due to the above, he is not really "logically inevitable," no?


But let’s take a look at Arya.
1. The dream in Deathwatch. Two people, a man and woman, boarding a ship while two dragon circle above. Now, most think this is Eragon departing the land with the other Rider, their two dragons flying above. This, if true, indicates the Rider is female.
2. Eragon is in love with Arya, Greenie and Saphie will likely be mates. Fits together.
3. She requires less training, as we don't have time to train a new Rider.
4. She is elven and female. The other Riders are human and male. It evens it out a little bit.
5. As an elf, is of superior mental and physical strength - and with one book to go, this is definite advantage.
6. Green. She has green eyes, green magic, etc. CP has said that there was clues in the books about who the next rider will be.
7. CP is trying to put her and eragon on even footing. They have the same strength magically and physically, or close to it, both are Shadeslayers. So making her a dragon rider would be the last part.
8. She has been in all three books.
9. She IS "logically inevitable.

And for those who say that she will be the next queen, take a look at this:

Arya, page 309 of Eldest.
"If I had died in the course of my duty, then a replacement successor would have been chosen from among our various Houses. Even now I would not be required to become queen if I found the prospect distasteful. We do not choose leaders who are unwilling to devote themselves wholeheartedly to their obligation . . . I had many years to perfect those arguments with my mother"

You are right I think it will be Arya but some disagree. Whatdo you think Greenie will be called