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The supercharger sees only the pressure
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No, the engine see density not pressure.
1. Whenever you include temperature, you are really talking about density.
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Temperature correction does not change the FTH in the pressure altitude scale, only the power at the given pressure altitude.
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IF the engine is equipped with a density controller. If it has a pressure relief valve, then it is dependent upon density altitude.
If it has an absolute pressure valve or a variable pressure controller then it is subject to density altitude effects.