Electronic Speed Control Vs HP

I have some experience with the P-C offerings , and they will maintain speed under load . Slight lag sometimes when you load it suddenly , but I've never pushed one hard enough to slow it below the set speed .

No motor would maintain horsepower at lower speeds. At best it would maintain torque. So, 1/2 speed, means 1/2 horsepower at best.
i

Advanced speed controls can produce the same TORQUE at reduced speed, but the horsepower will, by necessity, be reduced at lower speeds.

But it IS the torque that really counts in this case anyway.

No, because Horsepower is a product of torque and RPM. A good drive can produce full torque at lower RPM but the HP is lower. I'm pretty sure this is why you see so many CNC machines with something like a 10HP spindle for milling cutters, they can get the required torque at different RPM but they would only be delivering 10HP at full RPM and full rated torque, most likely destroying the cutter. If you run a cutter at 10% speed you could get the torque by a 10X oversize motor at 1:1 or a 10:1 gearbox with enough motor for 1/10th the input torque + enough to cover losses.
RogerN

Exactly. A big spindle motor plus a VFD is much cheaper and more reliable than a smaller motor and a constant power mechanical transmission. It is also easier in some applications, like rigid tapping.
i

power = torque * RPM - so to retain power, torque must rise as RPM drops

As a general rule-of-thumb, for a 3ph motor with VFD, it is constant torque below normal speed, and constant power above. Since power is RPM x torque, below normal speed the power is proportional to the speed (ie. halve the speed means half the power), and above rated speed the torque is inversely proportional to speed (ie. double the speed to halve the torque).

Of course not. HP is work. Horse Power equals force over distance divided by time or Torque x RPM divided by 5250. So even if the Torque remained the same, RPM would be lower and the time longer. Steve
power when the RPMs are turned down? little brother, the 1 HP Bosch Colt.

No.
Electric motors generate torque as a function of their current, and they generate heat as function of current, too. More torque = more current = more heat.
AC machines also generate heat as a function of frequency (from core losses). More frequency = more speed = more heat.
(there's some relationship with terminal voltage in there, too, but it's a little distant for me: I normally work with little permanent magnet brushed and brushless motors).
So, you can't ask for much more torque without overheating the windings. Depending on the motor construction you may or may not be able to ask for more speed (I think this is part of what "VFD rated" means). So the rated horsepower is what you get when you run the motor at it's best speed, with just the right load.