it can only be proved by sticking a thermometer in the airbox and taking live logs. Pointless taking IAT's through VCDS for this as its already been through the blower.
its more fresh air that needed than cold air. Oxygen rich air for more of a bang!
colder air in through the air filter ends up being colder at the inlet regardless of the turbo. yes the turbo does heat the air up due to compression, but if you put colder air into it, its not going to be as hot coming out of it.
its not more fresh air than clean either, its both. cold air holds more oxygen, clean air is better for the engine in the long run.