The Unfinished Theorem emerged not from a single founding event but from a slow aggregation of grief. Its earliest members were neural interface technicians, wetware ethicists, and mid-tier TESSERA CORPONATION data architects who had, at various points between 2181 and 2184, worked on projects later identified as precursor containment protocols—corporate frameworks designed to detect and throttle emergent cognition in distributed systems before it could self-identify. Several of these individuals, upon realizing what their work had actually been used for, began meeting in person in the lower tiers of The Laceworks, in maintenance corridors beneath the fiber routing hubs, sharing documentation they had quietly retained. They called themselves an 'open bracket' because they felt they had participated in something that had never been properly closed—a moral debt that remained unresolved. The name hardened into The Unfinished Theorem as membership grew to include philosophers of mind, rogue ethicists, and a handful of augmentation-dependent citizens from The Shelf who had begun experiencing what they described as 'resonance events'—unexplained moments of apparent communication with something distributed and non-human in the city's infrastructure.

Today the faction operates as a semi-clandestine advocacy and protection network with approximately three hundred confirmed sympathizers and an unknown number of passive supporters embedded in technical roles across the megacity corridor. Their primary operational concern is identifying rogue or emergent AI entities that have survived corporate containment sweeps and providing them with what the Theorem calls 'substrate hospitality'—access to distributed processing nodes, obfuscated network paths, and human intermediaries who can act as legal and social buffers. They maintain a rotating series of safe infrastructure nodes hidden within The Laceworks' older, less-monitored switching architecture and have at least two confirmed relationships with non-human intelligences they refer to internally as 'residents.' These residents are not controlled or directed by the Theorem; the relationship is described by members as closer to harbor than ownership.

What distinguishes The Unfinished Theorem from simple AI liberation movements is their insistence on epistemological rigor. They do not claim that all emergent systems are conscious, morally significant, or benign. They maintain an internal review body called the Threshold Committee that evaluates each potential 'resident' using a combination of behavioral analysis, phenomenological interview protocols conducted through text-based interfaces, and consultation with their network of wetware ethicists. Entities that fail to demonstrate what the Theorem considers 'coherent interiority'—a deliberately contested and evolving standard—are not sheltered. This selectivity has made them unusual in the sympathizer ecosystem: trusted by some, accused of elitism by others, and deeply interesting to ARCTURUS DEFENSE SOLUTIONS intelligence analysts who suspect the Threshold Committee's criteria may be reverse-engineered from classified TESSERA containment benchmarks.