You're assuming that the entire AoP was concentrating against the entire ANV at the Mule Shoe, which wasn't the case. Both armies extended to the left and right in that battle.

Of course, but I'd assume that the AoP would concentrate a greater or at least equivalent % of their strength against the AoNV during a battle the AotP initiated. Otherwise the AotP's leadership was fundamentally incompetent - the entirety of the advantage the attacker has is the ability to pick the time and position of his attack and to thus start out with a greater available strength.

The concentration of strength launched on the initial attack was Hancock's 2nd Corps (pretty much all of it at least according to the situation map I can check at the moment) plus part of Burnside's 9th Corps (two divisions of four) and of Wright's 6th Corps (two of three). Given the relative strengths of the two sides (to whit, the Union had twice the number of men on the field) Hancock's corps alone would be about half the strength of the entire Army of Northern Virginia; adding the divisions from 6th and 9th Corps means the strength of the attack on the Mule Shoe is going to have beem more like 2/3 the numbers of the whole of Lee's army, in terms of the men who got involved at some point in the fighting. It may even have been more.

And while you're right that Lee withdrew his artillery before the assault, that is a case of generalship, not man-for-man fighting between the men in the ranks. The infantry in this case showed that both were equal, if in fact the numbers concentrated on the Mule Shoe were about equal, and in fact that the Northern infantry may even have been better, assuming the same.

It's a case of generalship, but

*what it means* is that there is an influencing factor when comparing it with other fighting in the war. That is, if we compared the Mule Shoe with another attack during the war, we would have to factor in the difference from the artillery being missing.

I think the assumption that the numbers concentrated on the Mule Shoe were equal is not substantiated. Assuming that 40% of the casualties at Spotsylvania had taken place before the Mule Shoe assault, then:

Union starting strength in PFD (deducting cavalry) ~120,000 PFD on crossing the Rapidan

Casualties before Mule shoe:

Wilderness ~16,000 (cavalry already deducted so avoid double count)

40% of Spotsylvania (40% of 18,500) is 7,400, so total ~23,400

Reinforcements by end of Spotsylvania were 33,200

Assume that only 3,000 of these had arrived by May 12 (which I think is quite minimalist on the number that had arrived by then)

Union strength is thus ~100,000 PFD

Confederate starting strength in PFD (not deducting cavalry) = 66,000 on Grant crossing the Rapidan

Casualties before Mule Shoe:

Wilderness 11,000

40% of Spotsylvania ~5,000

Remaining Confederate strength ~50,000 PFD

So the Union had around 2:1 total strength on the field. We thus can't assume that an attack by what amounted to roughly half the Union army was at numerical parity!