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

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

Do you think I'm a TM specialist?

13

1

The poll was created at 11:04 on August 5, 2014, and so far 14 people voted.
Thanks. Wythagoras (talk) 18:58, August 6, 2014 (UTC)

## Poll 2

What is the smallest value of the Busy Beaver function that beats G, you think?

1

0

1

0

1

2

2

4

1

0

1

0

1

0

0

1

0

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The poll was created at 11:22 on August 5, 2014, and so far 15 people voted.
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

When will $$\Sigma(21) > G$$ be proven?

1

1

2

4

1

1

0

0

The poll was created at 12:33 on August 5, 2014, and so far 10 people voted.
This year hopefully :P. But not now. Wythagoras (talk) 18:58, August 6, 2014 (UTC)

## Poll 4

What is the smallest value of n such that $$\Sigma(n) > G$$ will be proven within 5 years?

1

1

1

1

2

1

1

1

0

1

0

The poll was created at 12:37 on August 5, 2014, and so far 10 people voted.
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)