\(\Sigma(22) \gg f_{\omega+1}(2 \uparrow^{12} 3) > G\)

0 _ 1 r 21 0 1 1 l 21 1 1 1 l 1 1 _ 1 r 2 2 1 1 r 2 2 _ _ r 3 3 _ _ r 14 3 1 1 r 4 4 1 1 l 5 4 _ _ r 6 5 1 _ l 5 5 _ 1 l 1 6 _ _ r 14 6 1 1 r 7 7 _ _ r 6 7 1 1 l 8 8 1 _ l 9 8 _ _ l 15 9 _ _ l 10 9 1 1 r 11 10 1 1 l 9 10 _ 1 r 1 11 1 _ r 12 11 _ _ l 13 12 _ 1 r 11 12 1 1 l 13 13 1 1 l 13 13 _ 1 l 10 14 _ _ r 18 14 1 _ l 8 15 _ 1 l 16 15 1 _ l 1 16 1 1 l 17 16 _ 1 l 1 17 _ _ l 16 17 1 _ l 16 18 _ _ r halt 18 1 _ l 19 19 _ 1 l 20 19 1 _ l 15 20 1 1 l 19 20 _ 1 l 19 21 _ 1 l 0 21 1 _ l 14

State 1 is state 0 of Deedlit's expandal machine

State x is state x of Deedlit's expandal machine for 2 ≤ x ≤ 17

Then, if the w+1 category is empty, it checks whether there is some in the w+2 category.

Then it changes all empty categories (remember, the tape looks like 11111....11111_1_1_1_1...) to ones for the w+1 category. That is about \(f_{\omega}^{-1}(n)\) of the ones currently on the tape.

State 0 and 21 are used to set the input. 1_11, where the head is on the first one and in state 14.

## Snapshots of the tape

After 6 steps, state 14

1 11 ^

After 13 steps, state 2

111 11 ^

After 16 steps, state 18

111 11 ^

After 17 steps, state 19

111 1 ^

After 23 steps, state 1

1 1111 1 ^

After 59 steps, state 1

1 1 1 11 11 1 ^

After 1359 steps, state 10

11 1 1 1 1 1 1 1 111 111 111 111 111 11 1 1 1 ^

## Bound

\(\Sigma(22) > f_{\omega+1}(2 \uparrow^{12} 3) > G\) (See last tape)

## Poll

- Thanks. Wythagoras (talk) 18:58, August 6, 2014 (UTC)

## Poll 2

- Personally I think 11 or 12 (I voted for 12). I would be very suprised if someone could implent Ackermannian growth in 9 states, in other words, I'd be suprised if Sigma(9) > A(100). Wythagoras (talk) 18:58, August 6, 2014 (UTC)

## History

September 9, 2010: r.e.s. proves that \(\Sigma(64) > G\)

April 8, 2013: Deedlit11 proves that \(\Sigma(25) > G\)

September 27, 2013: Wythagoras proves that \(\Sigma(24) > G\) using Deedlit11's results.

October 6, 2013: Wythagoras proves that \(\Sigma(23) > G\) using Deedlit11's results.

August 5, 2014: Wythagoras proves that \(\Sigma(22) > G\) using Deedlit11's results.

## Poll 3

- This year hopefully :P. But not now. Wythagoras (talk) 18:58, August 6, 2014 (UTC)

## Poll 4

- 5 years is a long, long, long time. 5 years ago the wiki was barely created! Still, I think it is unreasonable to think we'd even come close to the real BB machines, so I think that Sigma(17) > G will be proven in five years, but if n is the number of states to beat G, I think that even Sigma(n+2) > G won't be proven within 15 years. Wythagoras (talk) 18:58, August 6, 2014 (UTC)