The Number That Marks the Edge of Computation

A Turing machine with just five rules — read a square, write a zero or a one, step left or right, switch state — starts on a blank tape and runs for exactly 47,176,870 steps before it halts, leaving 4,098 ones behind. No five-rule machine that halts can ever run longer, a record proven optimal in 2024. That number is the busy beaver for five states, and it sits at the start of one of the strangest functions in mathematics. Every machine and every number in this video is real: the champions are simulated on an actual tape, step by step. From there the busy beaver function outgrows every formula you could ever write down, which is why no algorithm can compute it. BB(6) is already larger than anything physical, and BB(745) is a definite whole number that the standard axioms of mathematics provably cannot pin down. A handful of simple rules is enough to reach the edge of what can be known. Chapters: 0:00 A five-rule machine 0:54 The rules of the game 2:25 The champions climb 3:39 Watch one run 4:42 Why not just compute the next 6:03 Faster than every formula 6:56 Six rules, beyond the universe 8:06 The number math can't reach 9:14 Coda Music by Vincent Rubinetti Download the music on Bandcamp: https://vincerubinetti.bandcamp.com/a...