Batter-Runner Steps Over First

 
The batter grounds to the shortstop who throws to first base. The first baseman catches the throw as the batter-runner's foot is passing over the far end of first base -- but the foot clearly never touches the base. In other words, the batter-runner's foot had passed over (but never touched) the entire base before the ball arrived.
 
Is the runner signaled out for not touching the base or is the runner signaled safe for beating the ball to the base? What does the field umpire signal and verbalize?
 
Call
There is no exact rule book reference that I could find to address this situation. However, the interpretation (from Jaska/Roder and from the PBUC 2013 manual) is:
 
If a runner goes by a base within a body's length, he has either touched or passed the base.
 
In either case, he has acquired the base:
- If he touched the base, he is not vulnerable to an appeal.
- If he passed the base without touching it, he is considered to have touched it until there is a proper defensive appeal.
 
The defense has the responsibility to recognize a failure to touch the base.  The appeal must be obvious - unmistakably indicated by voice or action.

In the case below, the proper mechanic is to signal and verbalize "safe", indicating that the runner beat the play. If the defense appeals by tagging the runner or the base before the batter-runner returns to first, then he is declared out.