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Giovanni is Red's father.. Pokemon Origins made this SO CLEAR..
Giovanni looks like hes seen a ghost, hes frozen and says "the way he throws it, its just like.." at the way Red moves, throws his pokeball. he goes on a full minute reflection on how Red's turned out so much like he was.
Giovanni's proud when Red actually defeats him, when he was giving it his all.
Giovanni smiles and tries to hand Red his badge, but Red rejected Giovanni. Giovanni sighs, smiles and declares the DISBANDONMENT of Team Rocket just so Red would accept the badge.. If that's not a father doing that for atonement for his son, I dont know what is..
Giovanni then speaks to Red the way a father would his son, telling him to continue to aim for greater heights.
He watches him ride away into the sunset, and Giovanni reflects upon himself and says "Move forward, take the patch that I did not, YOU will find the way, i will also move forward" this is the sort of life changing event that comes only from a father finding his son after all these years, seeing his son has taken in his own footsteps.
There was already so much speculation Giovanni was Red's father, but this just clears it up.
(picture unrelated)