I think the real problem lies in saying this fight "proves" that Law's haki mastery was vastly, or even decisively 'higher' than Vergo's. It's easy to see with this fight just from the responses within this thread, that there are many POV's that can reasonably be given for Law's win, and the circumstances of said win, with basically none being able to be "Proven right definitively".
I think Billg7 said it best with:
if the brute force is strong enough in this case law+df+haki>vergo+haki
The bold statement is what seems to be what won Law this fight but with no real definitive, specific "haki advantage;' just a better combination of all abilities at once (and technically by concentrating/ focusing / aiming into a specific limited space like that, his haki could have just been decent with his power + DF ability taking up the slack). Add to that, that before Vergo was enhancing body parts and the bamboo stick; and went to not only haki enhancing his entire body + seemingly growing his upper body and arms a bit(although the last part could be 1 of those size distortions that seem to happen sometimes in OP). When concentrated on body parts, it seems to be concentrated enough but still more so on attacking. Then spread this to include the whole body, and no matter what since you're no longer concentrating it on 1 area it will be weaker, or not so balanced all around. Think the best example would be forced with a marker(Vergo's haki armor) to color in the lines, coloring in a life sized person drawing(Vergo), with 1 of those labyrinth puzzles inside the drawing. Now no 1 would be able to perfectly evenly distribute and color in the drawing with the exact same amount of marker on every spot. That would take too long, and Vergo didn't have that much time as it was 1 strike. Instead going from the 1st, fastest coloring of the drawing would be splotchy with varying levels of marker coloring throughout. Now I'm Not saying Law found 1 spot that was weaker, knew where that was and chose to strike/ slice there( although sadly presenting us with another possibility/ variable seeing as how we do not know exactly how his scanning abilities work).
Instead I'm saying that with Law concentrating his ability into his sword is more like blending the 2 traits of both to enhance + condense both into the sword, or quite possibly in times of need the sword blade's very edge itself (as in technically just 1 line of space, extending as far as whatever part of the room Law wants it to).
! With Room you have absolute control of everything within to order, or rather move things around as you would slowly physically in a real room,but pretty almost instantly; while with a sword you get a bonus in speed in (controlling, targeting, and going thru/ into/ past a target at the same time) strength, (reach not really beating the Rooms 'reach'), cutting ability, and randomness, (as in being able to pick or change targets quickly and randomly before the target can tell). With these qualities combined and condensed into the sword you could get absolute control of what you cut in the room, as well as the increased speed to act, anticipate, and thus allowing possibilities to counter-react in an unexpected way. All this and it makes it harder to know exactly where, when, and from what angle you might be attacking from. And speaking of that randomness ^; if done right that slicing ability could only be applied to, or used/ applied in reinforcing very, specific parts of the sword, (but more specifically parts of the sword's blade)
My Point being, by focusing concentrating his haki and fruit powers to the sword, or the sword's blade's edge he did the opposite of Vergo, and thus basically the same situation Vergo mentioned in his fight with Smoker; except now Vergo was making the same mistake he accused Smoker of making, thus with the result of him losing that key strike just like Smoker lost his to Vergo.
Plus Law is at least a bit of a swordsman, his natural sense of 1 vs 1 weapon holder's timing was bound to be better, than some1 who decides to use a weapon for the hell of it, this 1 time.
Some good points, (that seem to go unmentioned), to point out that also might've had some bearing, or presented even more variables to the situation:
Besides attacking with blackened haki armor, (in all but that final dueling slashes at the end), Vergo when getting hit, unless defending with some clear haki, did not defend with the blackened haki armor. Yet he took the hits and was still fighting while bleeding, like nothing had happened.
This also includes taking blackened haki armor blows from Smoker, + Law's specialty attacks seemingly 'unprotected'.
Also, (unless defending with some clear haki), Law and Smoker both seemed to see Vergo as sometimes so unaffected by their hits, like counter shock, that they weren't sure if he was in fact hit.
Another thing of note is that while Law seemed quite 'alert with the seastone in Smoker's jutte, he did not managed to break it; while Vergo's blackened haki armor enhanced bamboo stick did. Something up till now only Boa was able to do with her DF abilities( not saying it was necessarily was the seastone part of the jutte, or not, just that it's rarely been broken).
CCC:
He never cuts things physically; the sword is just some sort of conduit that lets him focus his ability better.
That's all I'm saying. His sword doesn't make contact with whatever he's attacking, and even if it projects some kind of energy attack, they don't behave like Mihawk's because the target simply separates, like Buggy; it doesn't bleed. I'd agree that the sword is essential (because he always uses it for the cutting moves), and that without it, all he can probably do is move objects around, but not break them apart cleanly. Maybe conduit is the wrong word. But you get the idea. Having it somehow enhances/modifies his ability.He never cuts things physically; the sword is just some sort of conduit that lets him focus his ability better.
Get your point, but has his sword specifically been shown to be cutting the specific parts like the hearts? or was that more specifically his fingers? Technically since still within his room ability, + since he still is a doctor; couldn't he have, consciously or not, put limits on his ability to basically automatically stop any blood spill, thus immediate killings.
Besides if true, this not necessarily "truly cutting things physically" thing; wouldn't that again mean it was his ability, or primarily his ability, not necessarily greater Haki mastery, that beat Vergo.:ninja: