1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF

1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF (bbch) is a halting BB(7) TM which runs for over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 \uparrow^{11} 2 \uparrow^{11} 3} steps. It was discovered by Pavel Kropitz on 10 May 2025 (Discord link) and analyzed by Shawn Ligocki (here) on 13 May 2025.

Analysis by Shawn Ligocki

Consider general configurations matching the regex:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0^\infty \; 11 \; (1 \; (01)^*)^* \; 0011100 \; \textrm{A>} \; 0^\infty}

Low level rules

              01 1 01^n 0011100 A> 00   -->                 1 01^n+2 0011100 A>
           01^3 11 01^n 0011100 A> 0^6  -->            1 01^n+5 1 01 0011100 A>
01^3 (1 01)^k+1 11 01^n 0011100 A> 0^6  -->  1 01^n+6 (1 01)^k 11 01 0011100 A>
 011 (1 01)^k   11 01^n 0011100 A> 0^2  -->  1 Z> 111 01^n+1 00 101^k+2

Mid level rules

Let

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B(a; b, c, ..., z) = 0^\infty \; 111 \; (01)^{3z+1} \; 1 \; \cdots \; 1 (01)^{3c+1} \; 1 \; (01)^{3b+1} \; 1 \; (01)^0 \; 1 \; (01)^{3a+1} \; 0011100 \; \textrm{A>} \; 0^\infty }

and let B(a; [x]*k, y, ...) = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B(a; \underbrace{x, \cdots, x}_k, y, ...)} (In other words, [x]*k represents k repeats of the value x in a config).

then

B(a; b+1, ...) -> B(2a+4; b, ...)
B(a; [0]*k, 0, n+1, ...) -> B(0; [0]*k, a+2, n, ...)
B(a; [0]*k) -> Halt(3a + 2k + 9)

Start at step 8178: B(2, [1]*12)

High level rule

Let

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{l} g_0(x) & = & 2x + 4 \\ g_{k+1}(x) & = & g_k^{x+2}(0) \\ \end{array}}

then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B(a; \underbrace{0, \cdots, 0}_k, n, ...) \to B(g_k^n(a); \underbrace{0, \cdots, 0}_k, 0, ...) }

Bound

Let

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{l} a_0 & = & 2 \\ a_{k+1} & = & g_k(a_k) \\ \end{array} }

then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B(a_0; \underbrace{1, \cdots, 1}_k) \to B(a_k, \underbrace{0, \cdots, 0}_k) \to \text{Halt}(3 a_k + 2 k + 9) } and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textrm{Start} \to B(a_0; \underbrace{1, \cdots, 1}_{12}) }

and so this TM halts with a sigma score of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma = 3 a_{12} + 33 }

Note that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_k(x) = (2 \uparrow^k (x+4)) - 4} and so for $k\geq 2$,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{k+1} + 4 > 2 \uparrow^k 2 \uparrow^k 3 }

and so this TM halts with sigma score Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma > 2 \uparrow^{11} 2 \uparrow^{11} 3} .

This bound is pretty tight: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma < 2 \uparrow^{11} 2 \uparrow^{11} 4 = 2 \uparrow^{12} 4} .