BB(6)

The 6-state, 2-symbol Busy Beaver problem BB(6) is unsolved. With the discovery of Antihydra in 2024, we now know that we must solve a Collatz-like problem in order to solve BB(6).

The current BB(6) champion 1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE (bbch) was discovered by Pavel Kropitz in 2022 proving the lower bound:

$$S(6)>\Sigma (6)>10\uparrow \uparrow 15$$

Techniques

In order to simulate the current BB(6) champion requires accelerated simulation that can handle Collatz Level 2 inductive rules. In other words, it requires a simulator that can prove the rules:

$${\begin{array}{l}C(4k)&\to &Halt({\frac {3^{k+3}-11}{2}})\\C(4k+1)&\to &C({\frac {3^{k+3}-11}{2}})\\C(4k+2)&\to &C({\frac {3^{k+3}-11}{2}})\\C(4k+3)&\to &C({\frac {3^{k+3}+1}{2}})\\\end{array}}$$

and also compute the remainder mod 3 of numbers produced by applying these rules 15 times (which requires some fancy math related to Euler's_totient_function).

Cryptids

Known BB(6) Cryptids:

• Antihydra

Potential Cryptids:

• 1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE (bbch)

Top Halters

The current top 10 BB(6) halters (known by Shawn Ligocki) are

1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE Halt ~10↑↑15.60465
1RB0LA_1LC1LF_0LD0LC_0LE0LB_1RE0RA_1RZ1LD Halt ~10↑↑5.63534
1RB1RE_1LC1LF_1RD0LB_1LE0RC_1RA0LD_1RZ1LC Halt ~10↑↑5.56344
1RB0LE_0RC1RA_0LD1RF_1RE0RB_1LA0LC_0RD1RZ Halt ~10↑↑5.12468
1RB0RF_1LC1LB_0RE0LD_0LC0LB_0RA1RE_0RD1RZ Halt ~10↑↑5.03230
1RB1LA_1LC0RF_1LD1LC_1LE0RE_0RB0LC_1RZ1RA Halt ~10↑↑4.91072
1RB0LE_1LC1RA_1RE0LD_1LC1LF_1LA0RC_1RZ1LC Halt ~10↑↑3.33186
1RB1RF_1LC1RE_0LD1LB_1LA0RA_0RA0RB_1RZ0RD Halt ~10↑↑3.31128
1RB0LF_1LC0RA_1RD0LB_1LE1RC_1RZ1LA_1LA1LE Halt ~10↑↑3.18855
1RB1RZ_0LC0LD_1LD1LC_1RE1LB_1RF1RD_0LD0RA Halt ~10^646456993.24591

The numbers listed are sigma scores. Runtimes are not available, but are presumed to be about $score^{2}$ which is roughly indistinguishable in tetration notation. Fractional tetration notation is described in https://www.sligocki.com/2022/06/25/ext-up-notation.html. For a longer list of halting TMs see https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/6x2.txt. For historical perspective see Pascal Michel's Historical survey of Busy Beavers.

Holdouts

@mxdys's informal holdouts list is down to 5877 TMs as of 4 Aug 2024.