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 $\operatorname {BB} (5)\geq 47176870{\phantom {}}$ and $\Sigma (5)\geq 4098$ at the time.

Analysis

Let $g(x):=0^{\infty }\;{\textrm {<A}}\,1^{x}\;0^{\infty }$. 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}}

$${\begin{aligned}g(x)&\to {\frac {5x+18}{3}}&&{\text{if }}x\equiv 0{\pmod {3}}\\g(x)&\to {\frac {5x+22}{3}}&&{\text{if }}x\equiv 1{\pmod {3}}\\g(x)&\to {\text{HALT}}&&{\text{if }}x\equiv 2{\pmod {3}}\end{aligned}}$$