Glad I'm at least close! This truly is a "Dr. Dr." subject.
So I believe the Assumed Temp N1 computed by the FMC (taking both actual and assumed temp into account) will generate an equivalent thrust rating on takeoff. *But*- you get better "lift" and better other performance with the same amount of thrust (at the higher actual air density) due to the true airspeed effect. Thrust does not equal performance.
Effect of True Airspeed
Pilots who are skeptical about reduced-thrust takeoffs often sense that something very important is being taken away. However, there is absolutely no loss of any necessary performance margins involving field length, screen height,1 climb or obstacle clearance. If the airplane’s weight and power setting satisfied the certification standards at the higher temperature, then they certainly will do so at the lower temperature.
Although the takeoff speeds used by the flight crew are indicated airspeeds, actual performance is determined by true airspeed, which is a function of air density. Because we are operating at an actual temperature that is lower than the assumed maximum, true airspeed likewise will be lower.
Because of this true-airspeed effect, we enjoy a great deal of cushion between what the airplane must do and what it actually is doing. We are, in reality, using less runway and achieving a higher climb gradient, or obstacle-clearance margin, than if the ambient temperature was at the maximum for that same weight. Depending on conditions, the effect can be considerable — on the order of several hundred feet in field length. The benefit increases as the difference between the actual and the assumed temperatures increases.