Gearing can be used to multiply torque but I understand that has no effect on power.
As torque is increased by gearing, the maximum rpm of the rear wheels is reduced so despite higher torque the work done (power) is reduced correspondingly.
Gearing multiplies torque and it maintains the same power (as you said it has "no effect") and let's ignore friction.
On your second statement, it is almost correct except the end. The engine when held to a specific RPM continues to make the same power as you shift gears, and the RPM of the rear wheels is reduced exactly the amount that the torque is increased so that the power is THE SAME. The engine is still sitting at the same point on its power curve at the same RPM, remember. The power it is generating didn't get created or destroyed along the way.
I realize this is not the case when you are driving because the car's speed is constant as you shift up and down while the engine's speed changes instead, so imagine we are running the car on a dyno and we set the engine at a certain rpm for testing purposes so that we can illustrate the math.
I've read that a low torque high rpm motor can produce the same amount of peak power as a high torque low rpm motor.
Yes, you can work out the math so that after the transmission, both generate the same torque at the wheels. The high rpm motor might need 4:1 to run at 60 mph and the low rpm motor might need 2:1 at the same 60 mph. But if they make the same power at that speed, the high rpm motor's torque is lower by exactly the amount such that the 4X torque multiplication gives the same wheel torque as the other motor's 2X torque multiplication. We know this because we know they are making the same power. This is why comparing power figures is what counts IMO, because it accounts for the gearing.
Example motors:
High rpm motor makes 100 lbs ft at 8000 rpm and has 4:1 low gearing
Low rpm motor makes 200 lbs ft at 4000 pm and has 2:1 higher gearing
Torque at the wheels:
high rpm motor makes 100 lbs ft * 4 = 400 lbs ft at wheels
low rpm motor makes 200 lbs ft * 2 = 400 lbs ft at wheels
We're going the same speed with both because the wheel rpm is the same:
high rpm motor turning the wheels at 8000 rpm / 4 = 2000 rpm
low rpm motor turning the wheels at 4000 rpm / 2 = 2000 rpm
Horsepower:
High rpm motor is 8000 rpm * 100 lbs ft / 5252 = 152.3 hp
Low rpm motor is 4000 rpm * 200 lbs ft / 5252 = 152.3 hp
The fact that they are both making the same horsepower defines the fact that they are also making the same torque at the wheels, if the gearing is right. If they were making different torque at the wheels, at the same speed, then that means the engines must be making different power.
You will feel the same acceleration because the wheels are being pushed with 400 lbs ft of torque that is generated by 152.3 hp.
Edit: I will throw in my opinion on why the GT-R punches above its weight. From the R32 in 1989, I believe it was one of the early cars to attempt deep integration of computers with the mechanical systems to optimize how the car worked and drove. The Porsche 959 was probably the only other similar car in that era. As the GT-R progressed over the next couple of decades, that active control and its optimization was refined to the current state which some other manufacturers are finally catching up to now. I think other cars that don't perform as well with what should be similar or superior specs are illustrating the effect of many little "holes" in their performance that specs don't reveal. Like optimizing traction at every point in a corner or covering up deficiencies in how the tires lose adhesion. When people say you can stand on the gas and the car launches out of corners, that's what you get when the systems work. The result is that people say the car is too easy to drive, but it's really the triumph of software and hardware working together and reaching that easy to drive goal.
