No he is not rediscovering honour, honour is a horse, a joke, a luxury those with lesser concerns can afford but not him. He threatens to launch a child in a trebuchet because honour means less to him than doing what needs doing. I didn't say Jaime began as an honourable knight acting for the greater good, the greater good and the honourable action are often contradictory, that's the point the character exists to explain. He sides with the greater cause over honour more so than any other character, and will continue to do so, that's what makes him unique and will make him a good Hand for the times.
Jaime becoming the new Tywin shouldn't be debatable, Jaime's arc is the furthest developed of all the characters to the point where GRRM is putting it out there plain. GRRM has Genna explain what Tywin meant to her and the realm, and then tell Jaime he's not like him. The rest of that chapter and future Jaime chapters are then dedicated to Jaime proving Genna wrong as he repeatedly does the Tywin.
You've got this all wrong.
First of all, Honor is not a joke or some luxury not to be taken seriously. Ned Stark getting his head chopped off because he adhered to honor is not a condemnation of honor in favor of the harsh real politique of Tywin Lannister. Rather, the series has shown how Ned's honor carries forward even after his death, his name and word still means something to the various people and nobles in Westeros. Further, his strong affection towards his family continues amongst his descendants who wish to do everything to retain their bonds to each other.
Meanwhile, we see that for all of Tywin's cold and calculating scheming his entire family and legacy fell apart the moment he died. And, while he was running things everyone was biding their time to screw him over because he had no honor and endeared no love from the people he ruled. Jaime is not learning to become more like Tywin, it's just the opposite. Brienne showed Jaime what a true knight looks like, something he no longer believed existed. And yet, she is not even a real knight. This is what begins to reawaken that old spirit within him. It's why he gives Brienne Oathkeeper and tasks her with finding Sansa Stark when he had no reason to do so, why he suddenly finds Cersei so distasteful, why he achieves so much success in the Riverlands. He's no longer a jaded warrior that thinks only through the point of his sword, he's trying to become better.
When he threatens Edmure at Riverrun we know that he's bluffing, but he uses the public perception of the Kingslayer to get Edmure to believe he is willing to make good on the threat. But, he would never have actually done it. Jaime still cynically pretends that honor means "shit" but it's all an act because he still cares very much. You left out the best part of his exchange with the Blackfish:
AFFC said:
I had no hand in Lady Catelyn's death, he might have said, and her daughters were gone before I reached King's Landing. It was on his tongue to speak of Brienne and the sword he'd given her, but the Blackfish was looking at him the way that Eddard Stark had looked at him when he'd found him on the Iron Throne with the Mad King's blood upon his blade. "I came to speak of the living, not the dead. Of those who need not die, but shall . . ."
This inner thought is important as it shows how much Jaime still cares about his character and how is pride can still get the better of him. He could easily clear this up with the Blackfish, but he refuses to because he dislikes the way his honor has been questioned, the same way Ned Stark judged him before he could explain himself. So, he keeps silent.
The whole chapter ends with his aunt telling him that it was Tyrion, not Jaime himself, that was Tywin's true son. Jaime can never be the kind of individual that Tywin was, but he can become his own person. The most important thing about Jaime's arc is the dream he has under the weirwood tree that includes Brienne:
AFFC said:
"Sister, why has Father brought us here?"
"Us? This is your place, Brother. This is your darkness." Her torch was the only light in the cavern. Her torch was the only light in the world. She turned to go.
"Stay with me," Jaime pleaded. "Don't leave me here alone." But they were leaving. "Don't leave me in the dark!" Something terrible lived down here. "Give me a sword, at least."
"I gave you a sword," Lord Tywin said.
It was at his feet. Jaime groped under the water until his hand closed upon the hilt. Nothing can hurt me so long as I have a sword. As he raised the sword a finger of pale flame flickered at the point and crept up along the edge, stopping a hand's breath from the hilt. The fire took on the color of the steel itself so it burned with a silvery-blue light, and the gloom pulled back. Crouching, listening, Jaime moved in a circle, ready for anything that might come out of the darkness. The water flowed into his boots, ankle deep and bitterly cold. Beware the water, he told himself. There may be creatures living in it, hidden deeps . . .
From behind came a great splash. Jaime whirled toward the sound . . . but the faint light revealed only Brienne of Tarth, her hands bound in heavy chains. "I swore to keep you safe," the wench said stubbornly. "I swore an oath." Naked, she raised her hands to Jaime. "Ser. Please. If you would be so good."
The steel links parted like silk. "A sword," Brienne begged, and there it was, scabbard, belt, and all. She buckled it around her thick waist. The light was so dim that Jaime could scarcely see her, though they stood a scant few feet apart. In this light she could almost be a beauty, he thought.In this light she could almost be a knight. Brienne's sword took flame as well, burning silvery blue. The darkness retreated a little more.
"The flames will burn so long as you live," he heard Cersei call. "When they die, so must you."
"Sister!" he shouted. "Stay with me. Stay!" There was no reply but the soft sound of retreating footsteps.
Brienne moved her longsword back and forth, watching the silvery flames shift and shimmer. Beneath her feet, a reflection of the burning blade shone on the surface of the flat black water. She was as tall and strong as he remembered, yet it seemed to Jaime that she had more of a woman's shape now.
"Do they keep a bear down here?" Brienne was moving, slow and wary, sword to hand; step, turn, and listen. Each step made a little splash. "A cave lion? Direwolves? Some bear? Tell me, Jaime. What lives here? What lives in the darkness?"
"Doom." No bear, he knew. No lion. "Only doom."
...
The fires that ran along the blade were guttering out, and Jaime remembered what Cersei had said. No. Terror closed a hand about his throat. Then his sword went dark, and only Brienne's burned, as the ghosts came rushing in.
"No," he said, "no, no, no. Nooooooooo!"
There is a lot of symbolism here but an important note is that Jaime's flame dims when confronted by his ghosts while only Brienne's flame remains. To me this speaks to Jaime's redemption. Jaime's fate is tied with Brienne's fate, his sword flame gutters out because it is his old life as a swordsman. He still clings to his sword even in his dreams, but his sword was responsible for many misdeeds. However, the sword he gave Brienne was given to fulfill an oath and help redeem his honor. Brienne completing her task is tied with Jaime reclaiming his honor and becoming a better person.
His dream is not a call to be more like Tywin, to be more ruthless and callous for the greater good. Instead, it's purely about confronting his perceived past failures as a knight with only Brienne there to defend him, naked. Jaime is not going to become a new Tywin, he's trying to become who he was meant to be, who he was supposed to be before becoming disillusioned.
This is correct. If one takes a look at GRRM's original outline for the planned trilogy, Jaime Lannister was the big bad. Cersei did not even exist as a character. We can see hints of it in the first book. It's Jaime who decides to kill Bran - Cersei tries to stop him. Jaime who hunts down Arya Stark to kill her when she's on the run in the Trident. He's a pretty terrible person in terms of morality. But suddenly when we come to book 3, Jaime is getting his so called 'redemption arc' and Cersei becomes the one dimensionally evil cartoon villain in the books. I did like that the show gave Cersei more complexity and better writing.
Also, just a reminder, that Jaime only abandons Cersei in the last book because he learns about her infidelity. Not because he becomes a better person, or he is humbled or because of Brienne or anything like that. He abandons her because he learns about her affairs and his pride takes a hit. That's all.
This is not true, it was a long time build up. Remember, Jaime refuses to sleep with Cersei in the White Sword tower because he still respects the place so much. She then makes a cruel jab at him. Tyrion's last words linger in his mind, but he doesn't actually believe them at first. Rather, as he slowly watches as Cersei rules he begins to finally see who she truly is and his experience with Brienne has shown him a different kind of woman. It's not pride that he abandons Cersei, it's the wool being pulled from his eyes. She never loved him as he loved her.
