Many people have been bashing the Justice idea.

I still think Rand's new sword is Justice.

On my reread I noticed this quote which I haven't see discussed yet:

"Rand reach down touching the sword he wore at his waist. It was the curved one, with the scabbard of black, painted with the twisting dragon, red and gold. For more reasons than one, it made him think of the last time he had been in Falme.

Based on this quote alone, I think it is pretty obvious that Rand first saw this sword in Falme from one of the Heroes of the Horn.

Now I will add in Rand's other observation about the sword:

"It looked as if it had been designed specifically for Rand-- and yet it was centuries old, unearthed only recently. How odd, that they should find this now, he thought, and make a gift of it to me, completely unaware of what they were holding... He had told no one, not even Min, that he had recognized the weapon. And not, oddly, from Lews Therin's memories-- but Rand's own."

Rand recognizes the sword from his own memories, which makes it even more likely he saw it from one of the Heroes of the Horn. He also mentions that he thinks it is centuries old. This is another reason why several people think the sword is Justice-- because that is the time period of Artur Hawkwing.

The quote below (for those of you who haven't seen it yet) gives even more credence to sword being Justice:

"And men cried unto the Creator, saying: O Light of the Heavens, Light of the World, let the Promised One be born of the mountain, according to the Prophecies, as he was in Ages past and will be in Ages to come,. Let the Prince of the Morning sing to the land that green things will grow and the valleys give forth lambs. Let the arm of the Lord of the Dawn shelter us from the Dark, and the great sword of Justice defend us. Let the Dragon ride again on the winds of time.

Rand's new mysterious sword is almost undoubtedly one of the Aes Sedai Blades made during the Age of Legends. Given the fancy ornamentation of the sword, this was a sword of someone very important during the Age of Legends. Since the sword is a Dragon sword, it is quite possible that this sword was also once used by Lews Therin himself (according to DomA's theory) or by one of his top commanders such as the 100 companions. If it was really Lews Therin's sword how did it end up traveling to Artur Hawkwing? (edit: I changed the part about Lews Therin blowing up his own palace)

This sword does have great significance however. Why else would it have been buried in a special place to be found later?

Min makes this observation about the sword: "Such a strange weapon those scholars had found beneath the submerged statue. The sword felt so old."

This dragon sword is very mysterious. Most of the evidence shows that it is indeed Justice. But why would Hawkwing have the sword buried in Arad Doman under a statue? There is some possibility that the sword was moved after Hawkwing's death. Perhaps an Aes Sedai had a vision or foretelling to bury the sword there.

I accept the fact that this sword is Justice. But where did Hawkwing get this sword in the first place as it is most likely a sword from the Age of Legends? One other oddity is that if the sword was really Justice, neither Tuon nor any of her Seanchan entourage recognized it.

I suspect that this sword has some importance to the future of the story. However, Sanderson has clearly left us puzzled on purpose with this issue.