... with a few modifications.
The Cal. 1403 in the Dead Second Tourbillon had the 'pointer' removed from the tourbillon cage (see images), and a new tourbillon bridge (cock is the correct term). In addition there were a few other changes from the original Cal. 1498. You know of the change from brass to 18K RG for the movement, but the entire gear train is different to incorporate the seconds display, and the power reserve indicator is not only in a different position, but is a different design, incorporating the differential system designed for the Chronometer Souveraine. The second 'dots' around the tourbillon have also been removed and there were changes to the mainspring barrel.
Andrew
The 'original' Tourbillon Souveraine (Cal. 1498) with seconds indication built into the one-minute tourbillon

The 'Dead Seconds' Tourbillon Souveraine (Cal. 1403)