All this data is potentially out of date, and should be taken with a truckload of salt
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'The Alexylva Paradox is paranational corporate entity founded in YC114 by capsuleer Saede Riordan. It acts as a defense force and trade organisation for the Origin Colonial Authority, the government of Origin. It also acts as a governing entity for capsuleers operating within the Origin system.
Its known space operations were initially based in The Citadel, where the corporation established a sprawling research complex in the Suroken system. Shunning publicity, the Paradox corporation had early links to the Angel Cartel and was flagged as a mid- to high-threat entity by CONCORD – but no evidence of illegal activity was ever tied to the corporations' flagship laboratories.
As the Suroken laboratory complex grew, so did Paradox's ambitions. Funnelling money laundered through the Jita markets into shadowy projects involving artificial intelligence, cybernetic enhancements, and advanced planetary terraforming, Riordan and her allies soon set their sights on territory unrestricted by the laws and politics of Empyrean space. Since January of YC115 they began progressively moving their operations out of Suroken into Anoikis, shuttling huge amounts of materials out of Known Space and settling into a system they have dubbed Origin.
Since the governmental reform Paradox is responsible for the majority of spacebound assets such as orbital customs facilities, starbases and various other related infrastructure. The corporation is also responsible for the system’s safety and security from hostile capsuleer corporations. The wider Colonial Authority interacts with Paradox primarily in an advisory manner, consulting with the capsuleers in their decision making, leaving most of the decisions up to the Corporation’s dedicated leadership of Coordinators and Sub-Coordinators. A growing fleet of battleships and strategic cruisers guards the corporation's valued secrecy well, hunting and dispatching any uninvited visitors.
Riordan's stated corporate goals sway heavily towards transhumanist political dissent and revolutionary thought, and has stated desires to bring down the governments of the cluster and replace them. She is an outspoken activist, holding many controversial views on drone rights, Sansha's Nation, and the governments of the Big4. The corporation holds neutral standings towards Sansha's Nation and some believe that ties with the Angel Cartel still remain.
Recent leaks  have revealed Project Susurrus, which seeks to create benevolent artificial superintelligence. Several strong AIs with a greater then human intelligence and learning capacity, are currently hosted aboard OCR servers on Promethean Station, a secret AI research station located on Perihelion.