1RB2LC1RC 2LC---2RB 2LA0LB0RA - Revision history

Polygon: Even further simulation found

2026-04-19T16:28:05Z

Even further simulation found

← Older revision		Revision as of 16:28, 19 April 2026
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	After active exploration on the #bb3x3 channel by LegionMammal and dyuan, LegionMammal found (and dyuan confirmed) a configuration A(1,c) (defined [https://discord.com/channels/960643023006490684/1259770474897080380/1259968221218607145 here]) which halts and for which a huge "wall" of previous A(1, c') values all reach it. This gives strong evidence that the TM probviosly halts since jumping over this wall is very "unlikely".		After active exploration on the #bb3x3 channel by LegionMammal and dyuan, LegionMammal found (and dyuan confirmed) a configuration A(1,c) (defined [https://discord.com/channels/960643023006490684/1259770474897080380/1259968221218607145 here]) which halts and for which a huge "wall" of previous A(1, c') values all reach it. This gives strong evidence that the TM probviosly halts since jumping over this wall is very "unlikely".

	The TM has been simulated for ~~1052884886930621967473049580 (~1e27)~~<ref>https://discord.com/channels/960643023006490684/1259770474897080380/1324065684677988505</ref> steps, reaching A(1,~~713176529181758~~(~~~7e14~~)).<ref>https://discord.com/channels/960643023006490684/~~960643023530762341~~/~~1275459473112170569~~</ref>		The TM has been simulated for over 10<sup>27</sup><ref>https://discord.com/channels/960643023006490684/1259770474897080380/1324065684677988505</ref> steps, reaching A(1, 1059491543272883(~1e15)).<ref>https://discord.com/channels/960643023006490684/1259770474897080380/1390819028137803829</ref>

	== dyuan01's Rules ==		== dyuan01's Rules ==

Polygon: Added simulation progress

2026-04-19T16:25:17Z

Added simulation progress

← Older revision		Revision as of 16:25, 19 April 2026
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	This is holdout #758 on Justin's 3x3 mugshots. After about 1.8 million steps and up to some relabelling, it is equivalent to holdout #153: {{TM\|1RB0LB0RC_2LC2LA1RA_1RA1LC---}}. And if you start in state C it is a [[permutation]] of the same machine. Together, they simulate a complex set of [[Collatz-like]] rules with two decreasing parameters.		This is holdout #758 on Justin's 3x3 mugshots. After about 1.8 million steps and up to some relabelling, it is equivalent to holdout #153: {{TM\|1RB0LB0RC_2LC2LA1RA_1RA1LC---}}. And if you start in state C it is a [[permutation]] of the same machine. Together, they simulate a complex set of [[Collatz-like]] rules with two decreasing parameters.

	After active exploration on the #bb3x3 channel by LegionMammal and dyuan, LegionMammal found (and dyuan confirmed) a configuration A(1,c) (defined [https://discord.com/channels/960643023006490684/1259770474897080380/1259968221218607145 here]) which halts and for which a huge "wall" of previous A(1, c') values all reach it. This gives strong evidence that the TM probviosly halts since jumping over this wall is very "unlikely".		After active exploration on the #bb3x3 channel by LegionMammal and dyuan, LegionMammal found (and dyuan confirmed) a configuration A(1,c) (defined [https://discord.com/channels/960643023006490684/1259770474897080380/1259968221218607145 here]) which halts and for which a huge "wall" of previous A(1, c') values all reach it. This gives strong evidence that the TM probviosly halts since jumping over this wall is very "unlikely".

			The TM has been simulated for 1052884886930621967473049580 (~1e27)<ref>https://discord.com/channels/960643023006490684/1259770474897080380/1324065684677988505</ref> steps, reaching A(1,713176529181758(~7e14)).<ref>https://discord.com/channels/960643023006490684/960643023530762341/1275459473112170569</ref>

	== dyuan01's Rules ==		== dyuan01's Rules ==
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	And as n gets way bigger, these ranges of repeat will increase exponentially.		And as n gets way bigger, these ranges of repeat will increase exponentially.

			== References ==

	[[Category:BB(3,3)]][[Category:Potential Cryptids]]		[[Category:BB(3,3)]][[Category:Potential Cryptids]]

Polygon: Moved from Category:Cryptids to Category:Potential Cryptids

2026-03-30T14:18:10Z

Moved from Category:Cryptids to Category:Potential Cryptids

← Older revision		Revision as of 14:18, 30 March 2026
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	And as n gets way bigger, these ranges of repeat will increase exponentially.		And as n gets way bigger, these ranges of repeat will increase exponentially.
	[[Category:BB(3,3)]][[Category:Cryptids]]		[[Category:BB(3,3)]][[Category:Potential Cryptids]]

Sligocki: Remove note on "current" work and move several rulesets that aren't used any more to bottom of page.

2025-11-02T21:10:45Z

Remove note on "current" work and move several rulesets that aren't used any more to bottom of page.

Show changes

Sligocki: /* Large-scale behavior */ Add note about length of stretch.

2025-10-24T15:31:48Z

Large-scale behavior: Add note about length of stretch.

← Older revision		Revision as of 15:31, 24 October 2025
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	If we can find a sufficiently long stretch of consecutive A(1, ''c'') configs that all reach the same halting value, we know that any sequence that reaches that stretch will halt. Conversely, if we find a sufficiently long stretch of A(1, ''c'') configs that skip past a halting value, we know that any sequence that reaches that stretch will skip past the halting value. In the first case, we say that the halting value ''explodes'' in terms of how many A(1, ''c'') configs can reach it; in the second case, we say that the halting value ''dies out''.		If we can find a sufficiently long stretch of consecutive A(1, ''c'') configs that all reach the same halting value, we know that any sequence that reaches that stretch will halt. Conversely, if we find a sufficiently long stretch of A(1, ''c'') configs that skip past a halting value, we know that any sequence that reaches that stretch will skip past the halting value. In the first case, we say that the halting value ''explodes'' in terms of how many A(1, ''c'') configs can reach it; in the second case, we say that the halting value ''dies out''.

	The first and currently only known halting value that does not die out is ''H'' = (3<sup>307</sup>−8282708212)/13959275. It it reachable from a very long stretch of consecutive A(1, ''c'') configs up to ''c'' ≤ ''H''~~−3403055~~. Thus, assuming the forward iteration reaches this very long stretch, it will halt at A(1, ''H''), unless it has already reached some earlier halting value.		The first and currently only known halting value that does not die out is ''H'' = (3<sup>307</sup>−8282708212)/13959275. It it reachable from a very long stretch of consecutive A(1, ''c'') configs up to ''c'' ≤ ''H''−3,403,055. The lower bound of this stretch is not known, but it goes at least to H-8,000,000 ≤ c thus this stretch is well over 4 million long. Thus, assuming the forward iteration reaches this very long stretch, it will halt at A(1, ''H''), unless it has already reached some earlier halting value.

	However, we cannot prove that the forward iteration reaches this long stretch instead of skipping past it. Statistically, the expectation is that Δ''c'' = log<sub>3</sub> ''c'' + O(1), but we cannot place any easy bound on how long the chain can become: in principle, there could be an undiscovered A(1, ''c'') config that goes on for 10<sup>1000</sup> steps or more. So if we want to fully prove that this machine halts at or before A(1, ''H''), we must place an upper bound on Δ''c'' for every step leading up to it. But given the size of ''H'', there is no known practical method to compute such an upper bound.		However, we cannot prove that the forward iteration reaches this long stretch instead of skipping past it. Statistically, the expectation is that Δ''c'' = log<sub>3</sub> ''c'' + O(1), but we cannot place any easy bound on how long the chain can become: in principle, there could be an undiscovered A(1, ''c'') config that goes on for 10<sup>1000</sup> steps or more. So if we want to fully prove that this machine halts at or before A(1, ''H''), we must place an upper bound on Δ''c'' for every step leading up to it. But given the size of ''H'', there is no known practical method to compute such an upper bound.

Sligocki: /* A(a, c) Rules and Timings */ Clarify what T() and U() mean.

2025-10-24T15:25:58Z

A(a, c) Rules and Timings: Clarify what T() and U() mean.

← Older revision		Revision as of 15:25, 24 October 2025
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	[[File:758 chain lengths.png\|thumb\|400px\|right\|alt=A 2D plot of blue and orange points. The values are organized into branching trees of intervals that become progressively smaller until they turn into scattered clouds of points.\|Chain lengths starting at A(1, ''c'') configs. Blue points correspond to ''c'' odd, orange points to ''c'' even. The log scale for ''c'' illustrates the recursive structure.]]		[[File:758 chain lengths.png\|thumb\|400px\|right\|alt=A 2D plot of blue and orange points. The values are organized into branching trees of intervals that become progressively smaller until they turn into scattered clouds of points.\|Chain lengths starting at A(1, ''c'') configs. Blue points correspond to ''c'' odd, orange points to ''c'' even. The log scale for ''c'' illustrates the recursive structure.]]

	Notice how in the sequence of A(1, ''c'') configs, the next ''c'' value is always greater than the previous ''c'' value, but not extraordinarily greater. In fact, the difference Δ''c'' over the course of a chain is roughly proportional to the length of the chain. We can see this if we reparameterize the rules in terms of ''t'' = ''a''+2''c'' and ''d'' = (''c''−1)/2, with the invariant that 0 ≤ ''d'' ≤ (''t''−2)/4 at all times:		Notice how in the sequence of A(1, ''c'') configs, the next ''c'' value is always greater than the previous ''c'' value, but not extraordinarily greater. In fact, the difference Δ''c'' over the course of a chain is roughly proportional to the length of the chain. We can see this if we reparameterize the rules in terms of ''t'' = ''a''+2''c'' and ''d'' = (''c''−1)/2, with the invariant that 0 ≤ ''d'' ≤ (''t''−2)/4 at all times (<code>T(t,d) = A(t-4d-2, 2d+1))</code>:

	{\| class="wikitable" style="text-align:left"		{\| class="wikitable" style="text-align:left"
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	In this form, each step increases ''t'' by a small positive value, so this explains why Δ''c'' between A(1, ''c'') configs depends on the number of steps in the chain.		In this form, each step increases ''t'' by a small positive value, so this explains why Δ''c'' between A(1, ''c'') configs depends on the number of steps in the chain.

	What determines the number of steps in the chain? As the plot shows, the chain lengths follow a sort of fractal pattern depending on the magnitude of ''c''. To derive this pattern, we first start with our T(''t'', ''d'') parameterization, combine steps so that ''t'' is always odd, and let ''u'' = (''t''−1)/2, so that 0 ≤ ''d'' ≤ (''u''−1)/2. Looking at rules (b), (c), and (j), we find:		What determines the number of steps in the chain? As the plot shows, the chain lengths follow a sort of fractal pattern depending on the magnitude of ''c''. To derive this pattern, we first start with our T(''t'', ''d'') parameterization, combine steps so that ''t'' is always odd, and let ''u'' = (''t''−1)/2, so that 0 ≤ ''d'' ≤ (''u''−1)/2. Looking at rules (b), (c), and (j), we find (<code>U(u,d) = T(2u+1, d))</code>:

	{\| class="wikitable" style="text-align:left"		{\| class="wikitable" style="text-align:left"

LegionMammal978: add explanation

2025-10-24T13:31:35Z

add explanation

Show changes

Polygon: Added link to Collatz-like

2025-10-09T18:53:11Z

Added link to Collatz-like

← Older revision		Revision as of 18:53, 9 October 2025
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	{{TM\|1RB2LC1RC_2LC---2RB_2LA0LB0RA\|undecided}} is a [[BB(3,3)]] [[holdout]] which appears to [[probviously]] halt. If it can be proven to halt, it will be the BB(3,3) champion in terms of both steps and tape symbols. However, it could also turn out to be a probviously halting [[Cryptid]].		{{TM\|1RB2LC1RC_2LC---2RB_2LA0LB0RA\|undecided}} is a [[BB(3,3)]] [[holdout]] which appears to [[probviously]] halt. If it can be proven to halt, it will be the BB(3,3) champion in terms of both steps and tape symbols. However, it could also turn out to be a probviously halting [[Cryptid]].

	This is holdout #758 on Justin's 3x3 mugshots. After about 1.8 million steps and up to some relabelling, it is equivalent to holdout #153: {{TM\|1RB0LB0RC_2LC2LA1RA_1RA1LC---}}. And if you start in state C it is a [[permutation]] of the same machine. Together, they simulate a complex set of Collatz-like rules with two decreasing parameters.		This is holdout #758 on Justin's 3x3 mugshots. After about 1.8 million steps and up to some relabelling, it is equivalent to holdout #153: {{TM\|1RB0LB0RC_2LC2LA1RA_1RA1LC---}}. And if you start in state C it is a [[permutation]] of the same machine. Together, they simulate a complex set of [[Collatz-like]] rules with two decreasing parameters.

	After active exploration on the #bb3x3 channel by LegionMammal and dyuan, LegionMammal found (and dyuan confirmed) a configuration A(1,c) (defined [https://discord.com/channels/960643023006490684/1259770474897080380/1259968221218607145 here]) which halts and for which a huge "wall" of previous A(1, c') values all reach it. This gives strong evidence that the TM probviosly halts since jumping over this wall is very "unlikely".		After active exploration on the #bb3x3 channel by LegionMammal and dyuan, LegionMammal found (and dyuan confirmed) a configuration A(1,c) (defined [https://discord.com/channels/960643023006490684/1259770474897080380/1259968221218607145 here]) which halts and for which a huge "wall" of previous A(1, c') values all reach it. This gives strong evidence that the TM probviosly halts since jumping over this wall is very "unlikely".

Polygon: Added Category:BB(3,3)

2025-09-27T16:18:17Z

Added Category:BB(3,3)

← Older revision		Revision as of 16:18, 27 September 2025
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	# dyuan and I race to find a halting <code>c</code> value that 'explodes' (passes the reachability analysis), first within the 'trivial' halter family to no avail, then within all halter families (writing a program to enumerate halter families is extremely fiddly, at this point I treat some of my own code as a black box not to be touched)		# dyuan and I race to find a halting <code>c</code> value that 'explodes' (passes the reachability analysis), first within the 'trivial' halter family to no avail, then within all halter families (writing a program to enumerate halter families is extremely fiddly, at this point I treat some of my own code as a black box not to be touched)
	# By dint of a somewhat more performant program, I find (and dyuan confirms) the <math>c = \frac{3^{307}-8\,282\,708\,212}{13\,959\,275}</math> we all know and love, probviously sealing the machine's fate		# By dint of a somewhat more performant program, I find (and dyuan confirms) the <math>c = \frac{3^{307}-8\,282\,708\,212}{13\,959\,275}</math> we all know and love, probviously sealing the machine's fate
	[[Category:Cryptids]]		[[Category:BB(3,3)]][[Category:Cryptids]]

Sligocki: Revert arbitrary name. Please stop naming other people's machines.

2025-08-11T05:58:01Z

Revert arbitrary name. Please stop naming other people's machines.

← Older revision		Revision as of 05:58, 11 August 2025
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	After active exploration on the #bb3x3 channel by LegionMammal and dyuan, LegionMammal found (and dyuan confirmed) a configuration A(1,c) (defined [https://discord.com/channels/960643023006490684/1259770474897080380/1259968221218607145 here]) which halts and for which a huge "wall" of previous A(1, c') values all reach it. This gives strong evidence that the TM probviosly halts since jumping over this wall is very "unlikely".		After active exploration on the #bb3x3 channel by LegionMammal and dyuan, LegionMammal found (and dyuan confirmed) a configuration A(1,c) (defined [https://discord.com/channels/960643023006490684/1259770474897080380/1259968221218607145 here]) which halts and for which a huge "wall" of previous A(1, c') values all reach it. This gives strong evidence that the TM probviosly halts since jumping over this wall is very "unlikely".

	The user [[User:Xl643\|Xl643]] Calls this Turing machine '''Smallfoot'''. However, [[User:isokate\|Katelyn Doucette]] disagrees with this name because this Turing machine has no connection to [[Bigfoot]].

	NOTE: As of 16 Jul 2024 there is a lot more active work on the #bb3x3 channel with LegionMammal and dyuan not reflected here.		NOTE: As of 16 Jul 2024 there is a lot more active work on the #bb3x3 channel with LegionMammal and dyuan not reflected here.