The Busy Beaver value S ( n ) is the maximum number of steps that an n-state 2-symbol Turing machine can perform from the all-zero tape before halting. S was historically introduced by Tibor Radó in 1962 as one of the simplest examples of an uncomputable function. We prove that S (5) = 47,176,870 using the Coq proof assistant. The proof enumerates 181,385,789 Turing machines with 5 states and, for each machine, decides whether it halts or not. Our result marks the first determination of a new Busy Beaver value in over 40 years and the first Busy Beaver value ever to be formally verified, attesting to the effectiveness of massively collaborative online research (bbchallenge.org).

We design and implement a novel class of highly precise network instrumentation and apply this tool to perform the first exact packet-timing measurements of a wide-area network ever undertaken, capturing 10 Gigabit Ethernet packets in flight on optical fiber. Through principled design, we improve timing precision by two to six orders of magnitude over existing techniques. Our observations contest several common assumptions about behavior of wide-area networks and the relationship between their input and output traffic flows. Further, we identify and characterize emergent packet chains as a mechanism to explain previously observed anomalous packet loss on receiver endpoints of such networks.