I don’t think Franky is Vegapunk’s kid, but he is related to MADS in both a relative way, but also remember Franky found himself on a ship with all the materials needed to turn himself into a Cyborg. An abandoned ship that had been floating for years.
This was about 10 years prior to present time… MADS split up some time prior to 24 years ago as that’s how old Reiju is and Queen only knew of Judge’s planned modifications, he’d presumably never met any of the children, meaning they were born after the break up. How long after though? We don’t know.
Ages being significant, Franky was 26 when he turned himself into a Cyborg (aboard a long-adrift yet reasonably advanced ship) and was found as a “child” presumably quite young as he was around 8-10 when he met the Roger Pirates (Oden specifically – avoiding the others)… You know what that age range lines up with?
Franky is old enough to have been abandoned around the same time MADS was falling apart. We don’t know how small little Cutty Flam was when Tom found him… but any time prior to him being 8 honestly works. Perhaps he was a kid, born to parents within that organisation but when it all fell apart he was either lost to pirates or his parents went into piracy to avoid the Government or fight them a little. I don’t know.
In any case – I think that ship is a derelict from the days of that scientific organisation, because maybe some of the metalwork and prosthetics and whatnot are explainable… but where does a pirate’s kid get the knowledge to run a body on a carbonated beverage? AND build a complicated hydraulic movement and electrified life-support system capable of sustaining him in the middle of a scrap-boat?
Unless there were either some advanced knowledge there (giving context to how he so easily slipped into going through Vegapunk’s old notes and understanding their potential), or he already knew of those principles and how to make them work (and remember he’d only glimpsed at Pluton’s blueprints at this point, not had multiple years to study them).
I feel like Franky will inherit the will/spark for scientific advancement that MADS had in its early days when it was just Vegapunk, Judge, Queen and Caesar trying to make cool stuff happen before the Government got their hands on them and split them up.
His chosen family, the Straw Hats, will inspire him to do more good than harm with the new findings of MADS.
TLDR; Franky probably isn’t Vegapunk Jr… but he is connected to MADS, either through his lineage, his original enhancements or possibly even both, through that weird Serendipity that happens to Straw Hats.
Theory by AllosaurusJei (https://www.reddit.com/user/AllosaurusJei)
