I found some old footage of my teacher, Tim Cartmell, demonstrating the Gift Wrap and will address it here:
In competition jujitsu, the Gift Wrap is generally used as a set up for either taking an arm bar or taking the back. I will address it instead in tactical terms, where can be an end state in and of itself.
Here is the basic idea. The position is entered (by Blue) from a knee to belly position (uki gatame in Japanese terms), or a mount position. The arm that crosses White’s face (in this case his right) is captured or it is pushed acros from the top position. It is placed across White’s face, and controlled by pressure from Blue’s chest behind the shoulder. From there (in this case Blue’s left) hand is brought behind White’s head and neck to grasp the hand or wrist.
Blue’s right hand is used to control White’s left hand on the ground. The foot in front of the abdomen is placed sole on the ground.
Here Blue can extend his arm and exert pressure. This allows a relatively upright position, the foot on the ground can take weight and take pressure off the other knee, which may be on a hard surface. Blue can exert quite a bit of pressure simply by extending his left arm, or his posture with his center into White’s rib cage. (Note that Blue can also use the knee to belly here to cause even more pressure, and neither knee is on the ground).
Also, Blue can use more chest pressure, or place his right knee on White’s grounded left arm, to free up his right hand for other options.
And at any time Blue can stand and disengage.
Additional pressures can be exerted thusly:
Blue transitions his right hand grip to grasp his own wrist. Both of White’s hands are controlled, though now Blue has tied up both of his own hands as well, so this must be a conscious decision in light of the totality of circumstances Blue can observe in the particular situation.
Here Blue can exert pressure on White’s neck. This is a pain compliance component, added to achieve greater postural and positional control only. Generally, pain compliance techniques are not very effective against violently resisting subjects, who often may be drunk, drugged, or disturbed – in fact they often cause greater resistance. This is particularly true when postural and positional control are not otherwise achieved. Pain does not equal control.
With postural and positional control, pain compliance can be a force multiplier toward shutting White’s resistance down completely, not because it hurts, but because he cannot move and cannot mount any resistance to begin with. The pain furthers the control.
Incidentally, this is the set up for a straight arm bar, juji gatame, where Blue would bring his left leg around and lay on his back. That is the tactic performed in the Gracie Breakdown – leaving this position, performing the arm bar, then returning to this position for eventual cuffing.
This is superfluous from a tactical standpoint, actually giving up a perfectly good, and strong, position of advantage – already a cuffing position – in order to attempt a submission.
Instead, staying here in the Gift Wrap, Blue assesses his options: restrain the subject until help arrives (cover officers or the police), disengage for other options (escape or other tools), or, take the subject into custody himself.