Astra finds 17-state Beeping Busy Beaver equivalent to Collatz conjecture
Prompted by a question about Collatz, Astra found a 17-state Beeping Busy Beaver machine that quasihalts exactly when Collatz is false, though no formal proof exists yet.
A 17-state Beeping Busy Beaver problem equivalent to the Collatz conjecture. This post got me wondering if there was an equation for Collatz. If unbounded trajectories are hard for Turing Machines to distinguish from collectively all the halting trajectories, maybe it'd be easier with polynomials because you can represent any individual input and can condition on the variable. Apparently not quite, but an OpenAI paper[1] implies there's a Diophantine equation with infinitely many solutions if Collatz is true and finitely many if Collatz is false. So then I thought about Beeping Busy Beavers[2], and Astra found this 17-state machine that quasihalts exactly when Collatz is false: 1RL0RB_1LC1RB_0LK0LD_1RE1LD_0RF0RB_0LG0LG_1LH1LG_0RA0RI_1RJ1RI_1RF1RF_0RA1LK_0LM0LM_0LN1LM_1RO1LN_0RA0RP_0RQ1RP_1RL1RQ "Initial state: A. Blank symbol: 0. Beep state: O. A beep occurs on every exit from O, regardless of the symbol read. There are 17 working states, A through Q, and no halt transitions. The beep state is included in the count." No formal proof, but I had my bots write their notes here. [3] [1] https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Single-fold-Diophantine-representations-September-24-2026/paper.pdf [2] https://wiki.bbchallenge.org/wiki/Beeping_Busy_Beaver [3] https://github.com/Mira-Cult/math/blob/main/turing-machines/collatz-beeping17/README.md
This post quotes another post: x.com/ereliuer_eteer/status/2113795429385347111