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Legacy gene-modification (otherwise known as Rare Skills), typically stemming either from Dawn State or Alhazredian origin. In short, Rare Skills are genetically encoded and passed down abilities (and a combination of genetics and the linker core traits that are passed from mother to child during pregnancy) created in these earlier epochs and still surviving in various bloodlines to the modern day. Of course, without proper maintenance these abilities will naturally degrade as they dilute by generation (the Alhazredian ones moreso than the Dawn States', given the timeframes involved), but they can still occasionally flare back up by in a genetic freak of nature, such as Caro.
Compared to Linker Cores and Mage Ranks, Rare Skills are the place were lineage really does matter, though it can just as equally be a double-edged sword (see Mei). And yes, a real problem in the Warring States era was the tendency for various fucked-in-the-head engineered lines to launch coups and take over.
Summons would be a good example of passed down Alhazredian 'Rare Skills', though they're typically not called as such.
For the closest equivalent in modern times, see the Combat Cyborg Incident.
Carim (Data analysis)
Carim is rather like a superpowered Uno in some ways, though a bit more specialised and of Belkan origin. She basically has this vast, cold encoded machinery in her mind and Linker Core that takes in all the information she's exposed to and analyses it, finding and matching patterns with inhuman scope. On a mundane level, she just seems to know things, gets hunches - that boy likes that girl, and is probably going to ask her out in the next week, it would be a good idea to stock up on supplies now because they're going to run out soon, her boss doesn't like the new secretary she's been assigned, but if she puts a word in the right ear here and here, she can get the poor girl reassigned to someone else and make everyone happy. It gives her an air of calm unflappability which is at least in part largely responsible for why she holds such a high position at such a young age. She doesn't even realise what's going on with that bit, the low-level stuff just comes to her as things she knows, and she's learnt to trust her hunches because they are almost always right. Pattern-finding on that level is easy for something like her.
And then we get the bigger stuff. See, her geneline is really meant to be plugged into a dedicated data feed in the war room of the Saint's Cradle, with about twenty of them taking in vast reams of data from an entire warfront and finding patterns in it, looking at seemingly uncorrelated troop movements and fortifications and famines and crime sprees and so on to predict major offensives that the enemy is preparing for weeks or months in advance. Then they forward that information to the savant-like strategic genius of a Saint King, who decides on a course of action which they can then make predictions on and give probabilities and success percentages and so on for, and that kind of thing. The level of warfront control this kind of thing gave Ancient Belka was one of the things that made it such an overwhelming military power, and it let them basically just walk over any of the smaller kingdoms that weren't other Dawn States like Galea or Ossiria or whatever. A small five-world fiefdom vs that isn't a war, it's a farce.
Except she's not doing that. Because she doesn't know how her Rare Skill actually works. She's taking in the limited information that she picks up from her position in the Church and from news and general knowledge, and then whenever she wants a "prophecy", she's basically yanking mental levers (since she doesn't have the training to assimilate her mind with the encoded machinery to properly get chains of reasoning and probabilities and such) until she hits one that makes it spew out whatever it's got in a sort of "automatic report" mode. There's actually something of the Blindsight Vampire in her; her consciousness is basically this teeny tiny interaction module bolted onto this vast inhuman pattern analysing system that gives her a fever whenever she gives a prophecy in Ancient Belkan, which she doesn't even speak properly and so has problems translating.
As a result, the savant-adjunct part of her can take reports from the TSAB and the known fact that Olivie's last battle was on Midchilda along with the fact that the Shroud went missing years ago, and then add in military movements and Regius' positions and military conflicts and the fact that Nanoha was injured by something which is of late-Belkan manufacture... and put these all together to get a high probability that the Saint's Cradle has been discovered by someone who is almost certainly attempting to reactivate it. But Carim herself isn't using it right, she's not giving it enough data, yanking on levers until it gives it in a fairly vague "automatic report" mode and then viewing it as a prophecy rather than as a statistical forecast. So it's not nearly as useful as it could be if she were properly trained and knew how to use it.
Possibly Alhazredian, possibly not... more likely one of the early Dawn States created his lineage. But really, if you look at him... he can turn into a small, relatively inconspicuous animal which can sit around someone's shoulders and defend them from any attacks that come at them, he's terrifyingly good with binds and barriers to shield himself and others or hold and harass enemies, and he can cast at a freakishly high level near-instantly and without a Device. He shows all the clear signs of being from a bodyguard lineage for VIPs.
From a Berserker Lineage, an old Belkan attempt to create shock-troops which may have granted her additional abilities where it not for generational degradation. In Mei, the only thing she's gotten out of it is a reduced fear response, which whilst helpful to a degree in a combat situation is not terribly adaptive to an ordinary civilian life or, indeed, self-preservation.
Mei has been very lucky with how her life has turned out, because she got a stable nurturing environment with the care provided by a non-carrier and with a non-carrier older sister. Even, despite that, she was an incredibly hard-to-handle child, in trouble all the time, got in fights, saw psychiatrists, and was in trouble with the police before the age of 11.
You can't judge a proper purebred by Mei, who's a mongrel degraded lineage withered away to not much left. She's actually expressing the reduced fear more strongly than her father, because of normal variation. Certainly, at least some lineages were made with various control methods, like the capacity to turn it off and on, or various hard-coded spells which could be triggered for control purposes, and the like. One of the issues, of course, was the fact that often the reduced fear responses were set by default to "always on" so if someone tried to copy them they'd have psychopaths on their hands without the non-genetic elements to moderate it. There are other berserker lineages already mentioned, who are the ones who really berserk and so the fear-shutdown really kicks in when they go into full fighting mental mode, and at least when everything working properly they can turn it off.
Basically, you can't judge what was intended by Mei, because she's just a girl who got a bad hand in the genetic draw. She just has something wrong in her head because she has a similar combat stress response to Zest without having got that way by being a soldier for decades, and it's made her life hard; if she actually gets diagnosed properly, there are things that can help (including medication which may help if she has a mostly-intact control system, which is basically a hack to give her control of it) if they can correctly identify which lineage she comes from and see if there are any methods on book.