The Gran Turismo Association, frequently abbreviated GT Association, is presumably the sanctioning body for all races and events in the Gran Turismo series.
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The GT Association is the in-game fictional company that seemingly sanctions the races held in Simulation/GT Mode. It may be based on the real-life company of the same name that supports the Autobacs Super GT Series. It's logo can be seen in the intro FMV for Gran Turismo 4, on the side of the Ford GT LM Race Car Spec II's livery, on the "My Home" main screen in Gran Turismo 5, and during loading sequences in Gran Turismo 6. An alternate logo can be found on the start/finish gantry on various tracks, such as Tokyo R246, where it has appeared since Gran Turismo 3: A-Spec.
However, aside from its logo appearing in several areas across multiple games, no other information is provided about the GT Association, leaving its purpose and motive unknown.