Yeah. Theories have to reconcile the facts that

a) Oddly, this came from Rand's own memories, not Lews Therin.

yet

b) the sword is centuries old - not millenia old (not that lacquered scabbard would last this long, anyway).

This is a nagging detail, because it means Lews Therin could have recognized the sword too, but it's Rand who did. First, at least. We don't know that Rand didn't recognized the bladde first, then LTT added his own bit to it that made Rand realise LTT too could have recognized the sword... But it sounds more likely Rand read about it in Tear, about someone whose scabbard had LTT's sigil on it.

Why would Rand find it odd that LTT does not recognize Justice, a sword from over 2000 years after the War of Shadow? Rand has absolutely no idea that the Heroes see themselves in TAR. That's knowledge only Nynaeve and Elayne have.

Sadly, we have little description of Justice. Rand talks of it as a 'great sword', but may not have intended it's large by this so much as emphasizing its legendary status. Rand and his friends grew up laping up stories about Hawkwing and Justice, as is well demonstated by Egwene and Perrin in the stedding episode in EOTW.

Hawkwing holds it one-handed in the scene, so it's not a two-handed sword. Beyond that, all Rand says it that it shone like a mirror. If there is anything very distinctive about it Rand didn't say. It's well to remember that Rand knows his childhood stories, and whatever sword Hawkwing held at Falme Rand would have identified it as Justice, so the fact Rand identified it as Justice doesn't imply the blade seen at Falme had any distinctive feature as such.

The blade Rand has been given appears to be a classic blademaster weapon, very long, with the usual slightly curved blade. If it's an heron-marked blade, it's extremely doubtful it's the blade itself and not the scabbard Rand has recognized.

Rand says:



Rand obviously believes the scabbard with the dragon is also old, otherwise he means there is also a dragon etched on the blade. In any case, there is something, centuries old, which makes this sword & scabbard look as if it's been designed specificially for Rand. It has not been designed specifically for Rand, or at least it's not what he believes. It's not been decorated recently, or if it did, it's a forgery.

But why would Rand finds it odd it's him, not LTT who recognized the blade?

This suggests the blade is a replica of Lews Therin's blade in the War of Shadow. I don't really how LTT could be expected to have recognized a blade (or a scabbard) centuries old unless it's a replica of something older.

The most likely person to have owned a replica of LTT's sword centuries ago, or to have himself a weapon made bearing Lews Therin's sigil is a false dragon. It's barely thinkable anyone else would have. The most likely False Dragon in that area is Guaire Amalasan, who was born and claimed himself in those parts.
There is even a precedent in the series (or Guide) of Amalasan adopting an old symbol - he used as banner the ancient symbol of Aes Sedai (on blue) that the prophecies say the Dragon would conquer under. Logain copied him, incidentally. Taim is also extremely fond of using the dragon sigil. But no one but false dragon (and now Rand) want to be associated with this symbol. No one.

Scholars know very well the red-and-gold serpent with legs is Lews Therin's sigil - that has been established in the series. Moiraine recognized the banner of LTT for what it was on sight, and others have gasped at its mere sight. Why on earth would Hawkwing, of all people, adopt the sigil of the Kinslayer on his scabbard? This symbol is both feared and despised. The only remote way I could see this sword being Justice is if it turns out Justice was a sword Hawkwing took from his archenemy Amalasan when he defeated him, and gave its new name. It still extremely odd he would have kept the scabbard with the dragon on it, in Bonwhin's days no less! Honestly! This sword isn't Justice.

It's just Etzel trying to save his old obession that Tuon and Rand will ally very early and Rand will send the Senchan to save the Tower.

His idea that Rand meant to hand Semirhage down to Tuon has of course not panned out, so now he's conceived this new one, not much better at all IMO, that Tuon would be swayed Rand's side by being given a shiny sword. Surprising he's not found a way to tie the sword to Slayer and Asmodean's murder, yet