Redemption is a very loaded word, because of its meaning for Christianity, where Christ the Redeemer is considered to have paid for the souls of sinners with his own life, there by redeeming - buying them back - from sin. Whereas when I talk about redemption, I mean the secondary meaning of the word, which is “the act of making something better” and by this definition I think Jaime IS on a redemption arc.

For me, redemption is a journey for which there is no endpoint. There’s no balance sheet, no moment in a redemption arc (or shouldn’t be) when someone says “OK, you did this and this and this and now you’ve paid for these bad things you did and you’re redeemed.” There’s never going to be a moment where we can say “Oh, Jaime did this, and that makes up for Bran”; if that were the case, then he already build up 500,000 souls worth of good karma when he saved King’s Landing and he can still kill 499,999 people and be on the plus side of the ledger, which is obviously a ridiculous assertion. It never works that way. (I prefer what the Talmud says: “Whoever destroys a soul, it is considered as if he destroyed an entire world. And whoever saves a life, it is considered as if he saved an entire world.” In other words, each action - good or bad - is complete in itself, and is not a tick in a cosmic ledger of good and evil.)

Redemption is also not about restitution; Jaime can’t restore Bran’s legs any more than he can grow a new hand for himself. He can never “make it up” to the person he actually injured. He can only do his best to do right by others he encounters, NOT to do those evil things for love again. For me, redemption lies less in adding up the measure of “good things vs. bad things” ("what’s one boy against a kingdom?" as Stannis asked when the answer is always going to be "one boy's life" as Davos answered) and more in recognizing what you have done that is evil, changing what you can change and trying to be a good person even if no one around you believes you can be. It’s holding your ground even when your sister mocks you and your father disowns you and doing what you KNOW is right. And so for me, by that measure, Jaime IS on a redemption arc.