If you could really move heat 300% more efficiently than using a resistive heater to generate heat, couldn’t you just move a bunch of heat to power a steam turbine and get a perpetual motion machine?
It's not entirely clear what you mean. But, you say "if you could really ..." as if ground-source/air-source heat pumps don't do this exact thing; do you think they don't work?
Running a steam turbine by pushing water down a borehole to come into contact with a high-temp (>100°C) source has been done; it's not perpetual motion. Energy pumping the water is << energy output from the steam produced. It's solar energy that's being used, ultimately.
No, first of all perpetual motion machines don't exist, will never exist.
The 300% efficiency means if you had 1 Watt you could use it to power a fan and coolant lines to move heat. Or use 1 Watt to generate heat and then once made to move that heat.
The moving of heat already existing in the air (or in the ground) is more efficient than generating the heat and then moving it.
