Gai's father, Dai, raised him alone (there's no canon information about Gai's mother) and was the central figure in his life. He's the source of a number of Gai's quirks, though not all of them, including the green jumpsuit, the thumbs-up gesture, and the emphasis on relentless self-improvement.
Like Lee, Dai couldn't use ninjutsu or genjutsu and was forced to rely solely on physical training and martial arts. He never rose above genin rank and brushed off constant bullying by cheerfully thanking his tormentors for the reminder to keep improving.
He was determined to embody self-confidence and considered youthfulness to be about never giving up on one's passions and beliefs.
As a child, Gai was embarrassed by others' low perception of his father, unsure how to understand Dai's eternal optimism but ending up in a lot of fights about his father's reputation; Dai would tell him not to take losses to heart and instead to be proud of taking a stand for what he valued.
Despite his limitations, Dai mastered the Eight Gates and later taught them to Gai, stressing that the technique must only be used in life-and-death situations to defend those precious to him. He later used the deadly Eighth Gate to defend Gai's team from some notoriously overpowered opponents, dying for his son in accordance with his convictions.
Gai, already on the path to serve his village as a ninja, took both the loss and the message to heart.
Most of my Tron characters don't have canon parents, though I do headcanon there's a little of Alan's temper and rigidity in Tesler and not a little of Flynn's pride and sociability in Eckert, and I headcanoned a bunch of stuff for Roy but it's way back in my timeline and I can't find it argh. Ram, though, inherited a lot from Roy.
Both Ram and Roy are keyed toward helping others, and both do well in support roles, using their skills and gathering information to give others the information they need to move forward.
Post-Legacy Roy had a lot of the shine knocked off him by the last twenty years, but he still fundamentally loved his friends and wanted to trust them and work with them to make things better.
I saw him as more pragmatic than Alan; he might have been willing to accept the unsolved mystery of Flynn's disappearance and move on, but Alan could not, and Roy, rather than lose his other best friend to the mystery, stayed and helped.
I think he would have been overjoyed if Flynn Lives found what it was looking for; he meant everything he said about Flynn. But he knew Alan first, and Alan was there and actively hurting, and this was the best Roy could do for him.
Ram, though much younger and formed by the mind of a much younger Roy, would probably have done the same.
Not much is known about Cheris's father, Derow, except that he wasn't originally Mwennin; he married in and took on their customs, including going by his wife's family name (which doesn't seem uncommon in the hexarchate). He and his wife originally lived in the Mwennin ghetto in one of the only two cities on Bonepyre with a Mwennin population,
though they later moved to a house nearer to the sea. There's no indication that he was less unhappy than his wife about the difficulty of passing on Mwennin beliefs under hexarchate rule (the Mwennin relied on their insignificant numbers for safety from charges of calendrical heresy) or about Cheris being drawn toward mainstream hexarchate culture.
She entered faction service despite both parents' objections.
It appears that she wrote to both of them, but only her mother's letters are mentioned.
Later, when awful things happened, there was no sign of anyone thinking Derow's adoptive-Mwennin past would save him, and though he and his wife were separated, they both suffered the same fate.
I pretty much only have fewer character dads than character moms because Bel possibly-jokingly mentioned having had a mom. XD
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