At the HVAC Hub discord server, I asked this question, and they saying that this is true, but I don't understand how.
I only know that in on/off A/C the compressors always works at 100% speed and turned on/off, and in inverter A/C, the compressor changes its speed.

Of course inverter AC is more efficient than fixed speed compressor units.
Once the room is cooled to the desired temperature the compressor just runs slowly to maintain the temperature instead of running at full speed on and off every few minutes.
This really is pretty basic stuff and should be obvious.
Think of driving a car: you don't keep flooring the accelerator and then suddenly let right off only to repeat this sequence a few seconds later to maintain roughly the same speed. If you imagine the poor fuel comsumption this would give then this corresponds to the similar reduced efficiency of fixed speed AC units in most situations.

This really is pretty basic stuff and should be obvious

And also is wrong

Once steady state is achieved (the condition in which the inverter unit winds down), the air conditioning unit output, averaged over time, equals the heat loss (or gain, if we are in the summer) through the room walls. This heat loss depends linearly and exclusively on the temperature difference between the room (chosen by the user) and outside (determined by the weather), and insulation resistance of the room walls

The air conditioning unit has no effect on the heat loss. In case of an on/off switching unit, the hysteresis beween the on and off temperatures is way less than the temperature difference between indoors and outdoors, so in average it still has no effect on the heat loss

The air conditioning unit then must pump the same quantity of heat/time unit in or out to maintain the temperature difference constant. Heat is energy, heat per time unit is power

Consider 2 compressors with identical COP but different power. One is of the exact power required to counter the heat loss, and the other is 2x the power. The 1x one will work continuously, the 2x one will work with 50% duty cycle. Twice the power for half the time is the exact same energy

So no, thermodynamics absolutely don't confirm this basic stuff



Now let's look at the electrics :

The on/off unit has a simple induction motor. There is no principal limit on the efficiency of such motor. Having to work only in a single set of conditions (same power, same refrigerant pressures, ...) it seems obvious to optimize this motor for working at this exact set of conditions

Limiting the efficiency of such motor is only cost - If using a bigger motor with room for thicker winding, Using lowloss steel in the laminations, etc. The motor can be made to any desired efficiency

The inverter unit has a similar induction motor. It is basically a plain 3 phase motor, though most of them are not for 50/60 Hz but for somewhat higher frequency

Higher frequency allows for smaller cores and less turns in each winding (which means thicker wire can be used in the same motor size), same as in lighting ballasts. However, there are few parts that can pull efficiency down as well :

 - The stator is still made of laminations, which dont behave well above few 100s Hz. (Eddy currents, hysteresis losses, increasing loss in general)

 - The motor must work efficiently in a broad range of speeds and pressures. Is it optimised for the exact ones at which it'll run in your case ?

 - The inverter itself also got losses

Considering that 50/60Hz induction motor efficiency is typically 85%..90%, this gives the theoretical upper limit of energy you might save if the inveter and inverter powered motor would be 100% efficient

In reality, the motor is probably closer to the same 85..90% as the other motor, and the inverter has a few % left behind as well



And at the mechanics :

Running at higher speed (due to higher RPM motor) allows pumping the gas faster, or at the original speed but with smaller pistons

Smaller pistons and all the other mechanical bits have less friction surface, however they work faster for the same output, so the friction loss happens more times

Similar considerations would apply to other compressor types as well (Scroll, vane, etc)

I am not sure how the less surface vs. faster running exactly balance vs each other

Same as with the electrics, the mechanics are the same in principle and only differ in size. Both have losses of the same order of magnitude



At this point, it is no longer about any significant energy savings, but other things :

 - Cost cutting. We have reached the point where a complete inverter circuit cost is cancelled out by the savings on smaller and cheaper motor and mechanics which have to work faster. (And question remains whether the mechanics will last as long as the bigger slower running ones)

 - Blatant shoving of advanced technology everywhere it is not needed

Inverter setup allows the use of more efficient motor designs (lower slip and/or even permanet magnet synchronous) that become harder or even impossible to start on a fixed frequency AC feed, so either can not be used at all without the inverter or need rather complicated starter system that is nearly as complex as the inverter.

Inverter designs do not lock the exact rpm, so there is another "knob" available for the system design allowing better optimization.

And mainly the biggest difference is related to the losses related to the second law of thermodynamics and temperature differences on the two main heat exchangers: The condenser vs ambient air and the evaporator vs the home air.
Inverter system (or any system able to vary rhe compressor power while maintaining its efficiency) reduces the power to maintain the target temperature. That means those two heat exchangers have to be transfering less power, but they remain of the same size, so the same thermal resistance (now simplifying a bit, as in reality the blowers are also throttled down). Less power on the same thermal resistance means condenser condensing the refrigerant at lower temperature so lower pressure. At the same time the evaporator gets warmer, so creating higher pressure vapor. Both of these means there is less pressure differential for the compressor to fight, hence less power demand to run it.

Reduced flow and pressure delta is how the mechanical power is reduced, but how do the mechanical losses behave under those varying conditions ? Are they linear, quadratic, something else ?