oggers: I think that I did answer where the increase in efficiency comes from (I certainly was trying to):
"Compression causes a temperature increase of the working fluid (gas).This results in the heat charge (combustion) being introduced at a higher temperature. The result is an increase in thermodynamic efficiency."
CW: I'm not saying that the delta T (not good at typing Greek on my ancient keyboard) of compression is somehow solely the cause of the efficiency increase. Without combustion, it would all be for naught. What I am saying is that delta T of compression results in the heat of combustion (delta Q - from which the work is derived) occurring at a higher temperature. The maths are fairly straightforward and have everything to do with the pressure ratio of the turbine (and its effect on the delta T of compression and the resulting delta T of exhaust, after combustion).
Maybe I'm not expressing it clearly (and perhaps we are even in violent agreement). I will say it simply one more time. Thermodynamic efficiency is a function of Compression Ratio (or a slightly different different function of Pressure Ratio in a turbine). Why? Because of its effect on the temperature changes of the working fluid. What is this effect? The heat of combustion is introduced into the system at a higher temperature.