He ran the simulations. For 12 satellites with up to 3 Byzantine failures, the input complex wasn't simply connected. It was like a sphere with a wormhole through it. And that meant… impossibility .
: These mathematical structures represent all possible system states. Instead of tracking every interleaving step of a protocol, you view the entire computation as a "frozen" geometric object. distributed computing through combinatorial topology pdf
: Topology was used to prove that "consensus" (all processes agreeing on one value) is impossible in asynchronous systems with even one failure. He ran the simulations
They rewrote the Knot’s protocol. Instead of a single coordinate, each satellite would vote for a region . The protocol used a combinatorial structure called a "chromatic subdivision": each round of communication subdivided the input simplex into smaller, colored simplices, like cutting a triangle into smaller triangles whose corners corresponded to possible local states. And that meant… impossibility