A convincing math would be something like this
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On top we see numerator contains Tsuction and Tdischarge. Imagine we push Tsuction lower and lower, it becomes smaller and smaller. A smaller number taken away the Tdischarge means Tdischarge is getting larger. Tdischarge is in the numerator so when it grows, Power requirement grows. From this equation it looks like power requirement is increasing as the temperature on the suction side is decreasing.

https://myengineeringtools.com/Compressors/Tools_Compressor_Power.html

 

Here we go
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This time inlet temp is denominator. As P1 inlet temp decreases, P2 blows up becomes very large. As P1 approaching zero P2 becomes infinity. P2 is numerator and when it gets large, Pis gets large e.g. power requirement grows. So again, we see that reduction of inlet stage increases the power requirement of the compressor.

Infinity is a very large power requirement for such a low temperature. It is difficult to move a fluid when the kinetic energy is zero, they just locked up like diamonds in the 5th dimension.

 

Didn't say compressor efficiency changed, I'm saying that that is 1 of 2 reasons of when required turbine energy goes down.

As inlet temp decreases, there is less temp rise during compression.
This then means, there is less temp drop required in the turbine stage, which then leads to less pressure drop. More energy left in the exhaust.

I don't think I can post all the equations needed for the turbine side, it's nothing nearly as simple as the compressor side. You first have to figure out what the compressor needs, then what the turbine can do based on the conditions.

You only changed altitude on matchbot. Leave altitude alone and set temp for 5°, when I do that I get an expansion ratio of 1.77 and like .3% wastegating required. You have like 80% being wastegated, drop the expansion ratio until the % wastegating is as near 0 as you can get.

Leaving everything else the same, I increased temp to 100° and expansion ratio went n/a, not enough energy.
I had to increase expansion ratio to 2.01 with .36% gate to restore power back to the turbine. That right there shows it takes more energy to compress hot air than cold air.

I run into it all the time with compounds, guys try to compress hot boost. Adding cooling between stages decreases drive pressure, decreases egts, and makes a ton more power.