ANNOUNCEMENT FOR NEW LARGE NUMBER LIST!
Σ(5) is proven to be 4098 as of June 2024!
S(643) is the smallest known value of the maximum shifts function that is undecidable in ZFC set theory as long as it is consistent. In other words, S(643) is the smallest known number undecidable in ZFC generated by maximum shifts function in the sense that the equality 1+⋯+1⏟m=S(643)is unprovable in ZFC set theory for any meta-theoretic natural number m as long as it is consistent.
The first such TM constructed had 7,918 states. Stefan O'Rear soon improved the bound to 1,919 states, and later to 748 states.
In 2023, Johannes Riebel slightly improved the bound to 745 states.
The bound was significantly improved to 643 and then 636 states in 2024 by Rohan Ridenour.