User:MrSolis/Playground

The 5-state busy beaver (BB(5)) winner is 1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA (bbch). Discovered by Heiner Marxen and Jürgen Buntrock in 1989, this machine proved that $\mathrm{BB}(5)\ge 47176870\phantom{}$ and $\mathrm{\Sigma }(5)\ge 4098$ at the time.

Analysis

Let $g(x):=0^{\mathrm{\infty }}\text{<mrow data-mjx-texclass="ORD"><mo><</mo>A</mrow>}{1}^x{0}^{\mathrm{\infty }}$. Then, Failed to parse (unknown function "\c"): {\displaystyle \begin{align} g(3x)& \xrightarrow{5x^2+19x+15}&g(5x+6),\\ g(3x+1)&\xrightarrow{5x^2+25x+27}&g(5x+9),\c\ g(3x+2)&\xrightarrow{6x+12}&0^\infty\;1\;\textrm{Z>}\;01\;001^{x+1}\;1\;0^\infty. \end{array}} $$\begin{array}{rlrl}g(x)& \to \frac{5x+18}{3}& & \text{if }x\equiv 0(\mathrm{mod}3)\\ g(x)& \to \frac{5x+22}{3}& & \text{if }x\equiv 1(\mathrm{mod}3)\\ g(x)& \to \text{HALT}& & \text{if }x\equiv 2(\mathrm{mod}3)\end{array}$$