Fractran - Revision history

C7X: Name of cryptid /* Visualizing Fractran Programs' Space-Time Diagrams */

2026-04-12T19:36:00Z

Name of cryptid Visualizing Fractran Programs' Space-Time Diagrams

← Older revision		Revision as of 19:36, 12 April 2026
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	\|[[File:Fractran_22_Cryptid.webp\|alt=The space-time diagram of ~~the BBf(22) Cryptid~~\|460x460px]]		\|[[File:Fractran_22_Cryptid.webp\|alt=The space-time diagram of Fenrir\|460x460px]]
	The space-time diagram of ~~the BBf(22) Cryptid~~		The space-time diagram of Fenrir
	\|[[File:Hydra.webp\|alt=The space-time diagram of Hydra.\|460x460px]]		\|[[File:Hydra.webp\|alt=The space-time diagram of Hydra.\|460x460px]]
	The space-time diagram of Hydra.		The space-time diagram of Hydra.

Int-y1: style

2026-04-12T09:09:13Z

style

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	== Definition ==		== Definition ==
	A ~~fractran~~ program is a list of rational numbers <math>[q_0, q_1, ~~...~~ q_{k-1}]</math>called rules and a ~~fractran~~ state is an integer <math>s \in \mathbb{Z}</math>. The numerator and denominator of any rational number fraction do not share any prime factors (they are in reduced form). We say that a rule <math>q_i</math> applies to state <math>s</math> if <math>s \cdot q_i \in \mathbb{Z}</math>. If no rule applies, we say that the computation has halted otherwise we apply the first applicable rule at each step. In that case we say <math>s \to t</math> and <math>t = s \cdot q_i</math> and <math>i = \min \{ i : s \cdot q_i \in \mathbb{Z} \}</math>. As with [[Turing machines]], we will write <math>s \xrightarrow{N} t</math> if <math>s \to s_1 \to \cdots \to s_{N-1} \to t </math> (s goes to t after N steps) and <math>s \to^* t</math> or <math>s \to^+ t</math> if <math>s \xrightarrow{N} t</math> for some N≥0 or N≥1 (respectively). We say that a program has runtime N (or halts in N steps) starting in state s if <math>s \xrightarrow{N} t</math> and computation halts on t.		A Fractran program is a list of rational numbers <math>[q_0, q_1, \dots, q_{k-1}]</math> called rules and a Fractran state is an integer <math>s \in \mathbb{Z}</math>. The numerator and denominator of any rational number fraction do not share any prime factors (they are in reduced form). We say that a rule <math>q_i</math> applies to state <math>s</math> if <math>s \cdot q_i \in \mathbb{Z}</math>. If no rule applies, we say that the computation has halted otherwise we apply the first applicable rule at each step. In that case we say <math>s \to t</math> and <math>t = s \cdot q_i</math> and <math>i = \min \{ i : s \cdot q_i \in \mathbb{Z} \}</math>. As with [[Turing machines]], we will write <math>s \xrightarrow{N} t</math> if <math>s \to s_1 \to \cdots \to s_{N-1} \to t</math> (s goes to t after N steps) and <math>s \to^* t</math> or <math>s \to^+ t</math> if <math>s \xrightarrow{N} t</math> for some N≥0 or N≥1 (respectively). We say that a program has runtime N (or halts in N steps) starting in state s if <math>s \xrightarrow{N} t</math> and computation halts on t.

	Let <math>\Omega(n)</math> be the total number of prime factors of a positive integer n. In other words, <math>\Omega(2^{a_0} 3^{a_1} \cdots p_n^{a_n}) = \sum_{k=0}^n a_n</math>. Then given a rule <math>\frac{a}{b} </math> we say that <math>\text{size} \left( \frac{a}{b} \right) = \Omega(a) + \Omega(b) </math>. And the size of a ~~fractran~~ program <math>[q_0, q_1, ~~...~~ q_{k-1}]</math> is <math>k + \sum_{i=0}^{k-1} \text{size}(q_i) </math>.		Let <math>\Omega(n)</math> be the total number of prime factors of a positive integer n. In other words, <math>\Omega(2^{a_0} 3^{a_1} \cdots p_n^{a_n}) = \sum_{k=0}^n a_n</math>. Then given a rule <math>\frac{a}{b}</math> we say that <math>\text{size} \left( \frac{a}{b} \right) = \Omega(a) + \Omega(b)</math>. And the size of a Fractran program <math>[q_0, q_1, \dots, q_{k-1}]</math> is <math>k + \sum_{i=0}^{k-1} \text{size}(q_i)</math>.

	BB_fractran(n) or BBf(n) is the maximum runtime starting in state 2 for all halting ~~fractran~~ programs of size n. It is a non-computable function akin to the [[Busy Beaver Functions]] since Fractran is Turing Complete.		BB_fractran(n) or BBf(n) is the maximum runtime starting in state 2 for all halting Fractran programs of size n. It is a non-computable function akin to the [[Busy Beaver Functions]] since Fractran is Turing Complete.

	== Vector Representation ==		== Vector Representation ==
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	Let the vector representation (for a sufficiently large n) for a state <math>a = 2^{a_0} 3^{a_1} \cdots p_{n-1}^{a_{n-1}}</math> be <math>v(a) = [ a_0, a_1, \dots, a_{n-1} ] \in \mathbb{N}^n</math> and the vector representation for a rule <math>\frac{a}{b}</math> be <math>v \left( \frac{a}{b} \right) = v(a) - v(b) \in \mathbb{Z}^n</math> (Note that this is just an extension of the original definition extended to allow negative <math>a_i</math>).		Let the vector representation (for a sufficiently large n) for a state <math>a = 2^{a_0} 3^{a_1} \cdots p_{n-1}^{a_{n-1}}</math> be <math>v(a) = [ a_0, a_1, \dots, a_{n-1} ] \in \mathbb{N}^n</math> and the vector representation for a rule <math>\frac{a}{b}</math> be <math>v \left( \frac{a}{b} \right) = v(a) - v(b) \in \mathbb{Z}^n</math> (Note that this is just an extension of the original definition extended to allow negative <math>a_i</math>).

	Now, rule q applies to state s iff <math>v(s) + v(q) \in \mathbb{N}^n</math> (all components of the vector are ≥0) and if <math>s \to t</math> then <math>v(t) = v(s) + v(q)</math>. So the ~~fractran~~ multiplication model is completely equivalent to the vector adding model. For presentation, we will represent a ~~fractran~~ program with a matrix where each row is the vector representation for a rule.		Now, rule q applies to state s iff <math>v(s) + v(q) \in \mathbb{N}^n</math> (all components of the vector are ≥0) and if <math>s \to t</math> then <math>v(t) = v(s) + v(q)</math>. So the Fractran multiplication model is completely equivalent to the vector adding model. For presentation, we will represent a Fractran program with a matrix where each row is the vector representation for a rule.

	For example, the BBf(15) champion (<code>[1/45, 4/5, 3/2, 25/3]</code>) in vector representation would be:		For example, the BBf(15) champion (<code>[1/45, 4/5, 3/2, 25/3]</code>) in vector representation would be:
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	\end{bmatrix}</math>		\end{bmatrix}</math>

	In this representation, it becomes much easier to reason about ~~fractran~~ programs and describe general rules. It is also very easy to calculate the size of a rule or program in vector representation. It is the sum of absolute values of all elements in the matrix + number of rules (number of rows).		In this representation, it becomes much easier to reason about Fractran programs and describe general rules. It is also very easy to calculate the size of a rule or program in vector representation. It is the sum of absolute values of all elements in the matrix + number of rules (number of rows).

	=== Relationship to VAS / Petri Nets ===		=== Relationship to VAS / Petri Nets ===
	Using vector representation, ~~fractran~~ programs are a deterministic version of [[wikipedia:Vector_addition_system\|Vector Addition Systems (VAS)]] (and, equivalently, [[wikipedia:Petri_net\|Petri Nets]]). VAS are identical to ~~fractran~~ programs in vector representation except that the rules are unordered and non-deterministic, they are used to model distributed systems where precise order of rule execution cannot be predicted. Interestingly, many problems about VAS are actually decidable, but their runtimes are extremely slow. Notably, the reachability problem (given states A and B are there a sequence of rules so that <math>A \to^* B</math>) is "Ackermann-complete" meaning that the optimal algorithm has worst-case runtime akin to the famously fast-growing Ackermann function.<ref>Czerwiński, Wojciech; Orlikowski, Łukasz (2021). ''Reachability in Vector Addition Systems is Ackermann-complete''. 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS). https://arxiv.org/abs/2104.13866.</ref>		Using vector representation, Fractran programs are a deterministic version of [[wikipedia:Vector_addition_system\|Vector Addition Systems (VAS)]] (and, equivalently, [[wikipedia:Petri_net\|Petri Nets]]). VAS are identical to Fractran programs in vector representation except that the rules are unordered and non-deterministic, they are used to model distributed systems where precise order of rule execution cannot be predicted. Interestingly, many problems about VAS are actually decidable, but their runtimes are extremely slow. Notably, the reachability problem (given states A and B are there a sequence of rules so that <math>A \to^* B</math>) is "Ackermann-complete" meaning that the optimal algorithm has worst-case runtime akin to the famously fast-growing Ackermann function.<ref>Czerwiński, Wojciech; Orlikowski, Łukasz (2021). ''Reachability in Vector Addition Systems is Ackermann-complete''. 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS). https://arxiv.org/abs/2104.13866.</ref>

	== Visualizing Fractran Programs' Space-Time Diagrams ==		== Visualizing Fractran Programs' Space-Time Diagrams ==
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	== Deciders ==		== Deciders ==
	[[File:Fractran deciders.png\|alt=Fractran deciders\|thumb\|All Fractran deciders summarized and their relations, shared by Daniel Yuan on [https://discord.com/channels/960643023006490684/1438019511155691521/1439001835904958655 14 Nov 2025]]]Many specialized deciders have been invented to prove ~~fractran~~ programs non-halting. See image at right. There are three extra deciders: [https://discord.com/channels/960643023006490684/1438019511155691521/1449775657554022531 Spanning Vectors Masked,] which should be very effective, but implementing it is in-progress, a version of Spanning Vectors Masked - [https://discord.com/channels/960643023006490684/1438019511155691521/1453217977385091092 Masked Linear Invariant] - which is very powerful, and some holdouts were removed by [[User:Sligocki\|Shawn Ligocki]] with [https://lsv.ens-paris-saclay.fr/Software/fast/ FAST] (Fast Acceleration of Symbolic Transition systems), a pre-existing general tool.		[[File:Fractran deciders.png\|alt=Fractran deciders\|thumb\|All Fractran deciders summarized and their relations, shared by Daniel Yuan on [https://discord.com/channels/960643023006490684/1438019511155691521/1439001835904958655 14 Nov 2025]]]Many specialized deciders have been invented to prove Fractran programs non-halting. See image at right. There are three extra deciders: [https://discord.com/channels/960643023006490684/1438019511155691521/1449775657554022531 Spanning Vectors Masked,] which should be very effective, but implementing it is in-progress, a version of Spanning Vectors Masked - [https://discord.com/channels/960643023006490684/1438019511155691521/1453217977385091092 Masked Linear Invariant] - which is very powerful, and some holdouts were removed by [[User:Sligocki\|Shawn Ligocki]] with [https://lsv.ens-paris-saclay.fr/Software/fast/ FAST] (Fast Acceleration of Symbolic Transition systems), a pre-existing general tool.

	-d released a new decider on 25 Jan 2026: [https://discord.com/channels/960643023006490684/1438019511155691521/1464873923647639703 Beeping Permutation].		-d released a new decider on 25 Jan 2026: [https://discord.com/channels/960643023006490684/1438019511155691521/1464873923647639703 Beeping Permutation].
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	==== Sequential programs ====		==== Sequential programs ====
	All champions up to BBf(14) have very simple behavior. They are all of the form: <math>\left[ \frac{3^{a_1}}{2}, \frac{5^{a_2}}{3}, ~~...~~ \frac{p_n^{a_k}}{p_{k-1}}, \frac{1}{p_k} \right]</math> or in vector representation (limited to k=4):		All champions up to BBf(14) have very simple behavior. They are all of the form: <math>\left[ \frac{3^{a_1}}{2}, \frac{5^{a_2}}{3}, \dots, \frac{p_n^{a_k}}{p_{k-1}}, \frac{1}{p_k} \right]</math> or in vector representation (limited to k=4):

	<math display="block">\begin{bmatrix}		<math display="block">\begin{bmatrix}
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	<math display="block">C(2) \to C(4) \to C(7) \to C(11) \to C(16) \to C(23) \to C(32) \to C(44) \to C(60) \to \text{halt}</math>		<math display="block">C(2) \to C(4) \to C(7) \to C(11) \to C(16) \to C(23) \to C(32) \to C(44) \to C(60) \to \text{halt}</math>






	==== BBf(21) ====		==== BBf(21) ====

Sligocki: /* Fenrir */ Tweak text to use name more clearly.

2026-04-01T16:51:32Z

Fenrir: Tweak text to use name more clearly.

← Older revision		Revision as of 16:51, 1 April 2026
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	<math display="block">C(2) \to C(4) \to C(7) \to C(11) \to C(16) \to C(23) \to C(32) \to C(44) \to C(60) \to \text{halt}</math>		<math display="block">C(2) \to C(4) \to C(7) \to C(11) \to C(16) \to C(23) \to C(32) \to C(44) \to C(60) \to \text{halt}</math>



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	=== Fenrir ===		=== Fenrir ===
	[[File:Fractran 22 Cryptid.webp\|alt=The space-time diagram of Fenrir.\|thumb\|Partial space-time diagram of Fenrir.]]		[[File:Fractran 22 Cryptid.webp\|alt=The space-time diagram of Fenrir.\|thumb\|Partial space-time diagram of Fenrir.]]
	~~The~~ size 22 [[Cryptids~~\|Cryptid~~]] ~~is a group of 3 holdouts that was~~ discovered by Jason Yuen (@-d) and Claude Opus 4.6. Out of 500 holdouts of size 22, Claude Opus 4.6 used Lean to prove that 497 holdouts were non-halting. The remaining 3 holdouts are ~~this Cryptid~~.<sup>[https://discord.com/channels/960643023006490684/1438019511155691521/1485415054475268179]</sup> Discord user @ZTS439 shared [https://discord.com/channels/960643023006490684/1438019511155691521/1487251919444508723 some analysis] and a [https://discord.com/channels/960643023006490684/1438019511155691521/1487252789158613002 Python program] for it. Its name comes from [[wikipedia:Norse_mythology\|nordic mythology]]; [[wikipedia:Fenrir\|Fenrir]] is the wolf that helps destroy the world during [[wikipedia:Ragnarök\|Ragnarök]].		"Fenrir" is a family of 3 size 22 [[Cryptids]] discovered by Jason Yuen (@-d) and Claude Opus 4.6 on 22 Mar 2026. Out of 500 holdouts of size 22, Claude Opus 4.6 used Lean to prove that 497 holdouts were non-halting. The remaining 3 holdouts are the Fenrir family.<sup>[https://discord.com/channels/960643023006490684/1438019511155691521/1485415054475268179]</sup> Discord user @ZTS439 shared [https://discord.com/channels/960643023006490684/1438019511155691521/1487251919444508723 some analysis] and a [https://discord.com/channels/960643023006490684/1438019511155691521/1487252789158613002 Python program] for it. Its name comes from [[wikipedia:Norse_mythology\|nordic mythology]]; [[wikipedia:Fenrir\|Fenrir]] is the wolf that helps destroy the world during [[wikipedia:Ragnarök\|Ragnarök]].

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RobinCodes: /* Champions */ Fixed Cryptid's name

2026-04-01T09:01:15Z

Champions: Fixed Cryptid's name

← Older revision		Revision as of 09:01, 1 April 2026
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	Known [[Cryptid\|Cryptids]]:		Known [[Cryptid\|Cryptids]]:

	# ~~Size 22 Cryptid~~		# Fenrir
	\|-		\|-
	\|23		\|23
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	<math display="block">C(2) \to C(4) \to C(7) \to C(11) \to C(16) \to C(23) \to C(32) \to C(44) \to C(60) \to \text{halt}</math>		<math display="block">C(2) \to C(4) \to C(7) \to C(11) \to C(16) \to C(23) \to C(32) \to C(44) \to C(60) \to \text{halt}</math>

RobinCodes: /* Cryptids */ Added name; Fenrir

2026-04-01T08:54:25Z

Cryptids: Added name; Fenrir

← Older revision		Revision as of 08:54, 1 April 2026
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	<math display="block">C(2) \to C(4) \to C(7) \to C(11) \to C(16) \to C(23) \to C(32) \to C(44) \to C(60) \to \text{halt}</math>		<math display="block">C(2) \to C(4) \to C(7) \to C(11) \to C(16) \to C(23) \to C(32) \to C(44) \to C(60) \to \text{halt}</math>



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	== Cryptids ==		== Cryptids ==

	=== ~~Size 22 Cryptid~~ ===		=== Fenrir ===
	[[File:Fractran 22 Cryptid.webp\|alt=The space-time diagram of Fenrir.\|thumb\|Partial space-time diagram of Fenrir.]]		[[File:Fractran 22 Cryptid.webp\|alt=The space-time diagram of Fenrir.\|thumb\|Partial space-time diagram of Fenrir.]]
	The size 22 [[Cryptids\|Cryptid]] is a group of 3 holdouts that was discovered by Jason Yuen (@-d) and Claude Opus 4.6. Out of 500 holdouts of size 22, Claude Opus 4.6 used Lean to prove that 497 holdouts were non-halting. The remaining 3 holdouts are this Cryptid.<sup>[https://discord.com/channels/960643023006490684/1438019511155691521/1485415054475268179]</sup> Discord user @ZTS439 shared [https://discord.com/channels/960643023006490684/1438019511155691521/1487251919444508723 some analysis] and a [https://discord.com/channels/960643023006490684/1438019511155691521/1487252789158613002 Python program] for it.		The size 22 [[Cryptids\|Cryptid]] is a group of 3 holdouts that was discovered by Jason Yuen (@-d) and Claude Opus 4.6. Out of 500 holdouts of size 22, Claude Opus 4.6 used Lean to prove that 497 holdouts were non-halting. The remaining 3 holdouts are this Cryptid.<sup>[https://discord.com/channels/960643023006490684/1438019511155691521/1485415054475268179]</sup> Discord user @ZTS439 shared [https://discord.com/channels/960643023006490684/1438019511155691521/1487251919444508723 some analysis] and a [https://discord.com/channels/960643023006490684/1438019511155691521/1487252789158613002 Python program] for it. Its name comes from [[wikipedia:Norse_mythology\|nordic mythology]]; [[wikipedia:Fenrir\|Fenrir]] is the wolf that helps destroy the world during [[wikipedia:Ragnarök\|Ragnarök]].

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RobinCodes: /* BBf(20) */ Forced line-break

2026-04-01T08:51:45Z

BBf(20): Forced line-break

← Older revision		Revision as of 08:51, 1 April 2026
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	\|}		\|}

			<br>
	==== BBf(20) ====		==== BBf(20) ====
	[[File:Screenshot 2026-04-01 104704.png\|alt=Full space-time diagram of the BBf(20) champion.\|left\|507x507px]]		[[File:Screenshot 2026-04-01 104704.png\|alt=Full space-time diagram of the BBf(20) champion.\|left\|507x507px]]
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	<math display="block">C(2) \to C(4) \to C(7) \to C(11) \to C(16) \to C(23) \to C(32) \to C(44) \to C(60) \to \text{halt}</math>		<math display="block">C(2) \to C(4) \to C(7) \to C(11) \to C(16) \to C(23) \to C(32) \to C(44) \to C(60) \to \text{halt}</math>

RobinCodes: /* BBf(20) */ Forced line break

2026-04-01T08:51:16Z

BBf(20): Forced line break

← Older revision		Revision as of 08:51, 1 April 2026
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	<math display="block">C(2) \to C(4) \to C(7) \to C(11) \to C(16) \to C(23) \to C(32) \to C(44) \to C(60) \to \text{halt}</math>		<math display="block">C(2) \to C(4) \to C(7) \to C(11) \to C(16) \to C(23) \to C(32) \to C(44) \to C(60) \to \text{halt}</math>


	==== BBf(21) ====		==== BBf(21) ====

RobinCodes: Added many space-time diagrams

2026-04-01T08:50:52Z

Added many space-time diagrams

← Older revision		Revision as of 08:50, 1 April 2026
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	==== BBf(20) ====		==== BBf(20) ====
			[[File:Screenshot 2026-04-01 104704.png\|alt=Full space-time diagram of the BBf(20) champion.\|left\|507x507px]]
	The BBf(20) champion (running 746 steps):		The BBf(20) champion (running 746 steps):

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	=== Size 22 Cryptid ===		=== Size 22 Cryptid ===
			[[File:Fractran 22 Cryptid.webp\|alt=The space-time diagram of Fenrir.\|thumb\|Partial space-time diagram of Fenrir.]]
	The size 22 [[Cryptids\|Cryptid]] is a group of 3 holdouts that was discovered by Jason Yuen (@-d) and Claude Opus 4.6. Out of 500 holdouts of size 22, Claude Opus 4.6 used Lean to prove that 497 holdouts were non-halting. The remaining 3 holdouts are this Cryptid.<sup>[https://discord.com/channels/960643023006490684/1438019511155691521/1485415054475268179]</sup> Discord user @ZTS439 shared [https://discord.com/channels/960643023006490684/1438019511155691521/1487251919444508723 some analysis] and a [https://discord.com/channels/960643023006490684/1438019511155691521/1487252789158613002 Python program] for it.		The size 22 [[Cryptids\|Cryptid]] is a group of 3 holdouts that was discovered by Jason Yuen (@-d) and Claude Opus 4.6. Out of 500 holdouts of size 22, Claude Opus 4.6 used Lean to prove that 497 holdouts were non-halting. The remaining 3 holdouts are this Cryptid.<sup>[https://discord.com/channels/960643023006490684/1438019511155691521/1485415054475268179]</sup> Discord user @ZTS439 shared [https://discord.com/channels/960643023006490684/1438019511155691521/1487251919444508723 some analysis] and a [https://discord.com/channels/960643023006490684/1438019511155691521/1487252789158613002 Python program] for it.

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	=== Frankenstein's Monster ===		=== Frankenstein's Monster ===
			[[File:Frankenstein's Monster.webp\|alt=Partial space-time diagram of Frankenstein's Monster.\|thumb\|Partial space-time diagram of Frankenstein's Monster.]]
	"Frankenstein's Monster" is a size 23 [[Cryptid]]. It was created by tweaking a single instruction in the size 22 champion. This tweak switches it from a unbiased random walk to a biased one and thus makes halting probviously impossible. It is called Frankenstein's Monster since it was found by a combination of exhaustive search and hand design.<sup>[https://discord.com/channels/960643023006490684/1438019511155691521/1449138938215141478]</sup>		"Frankenstein's Monster" is a size 23 [[Cryptid]]. It was created by tweaking a single instruction in the size 22 champion. This tweak switches it from a unbiased random walk to a biased one and thus makes halting probviously impossible. It is called Frankenstein's Monster since it was found by a combination of exhaustive search and hand design.<sup>[https://discord.com/channels/960643023006490684/1438019511155691521/1449138938215141478]</sup>

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	=== Hydra ===		=== Hydra ===
			[[File:Hydra.webp\|alt=Partial space-time diagram of Hydra.\|thumb\|300x300px\|Partial space-time diagram of Hydra.]]
	A size 25 program was produced and golfed by hand to simulate [[Hydra]] rules ([https://discord.com/channels/960643023006490684/1438019511155691521/1449829146040467681 Discord]):		A size 25 program was produced and golfed by hand to simulate [[Hydra]] rules ([https://discord.com/channels/960643023006490684/1438019511155691521/1449829146040467681 Discord]):

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	=== BMO1 ===		=== BMO1 ===
			[[File:Ftran bmo1.png\|alt=Partial space-time diagram of BMO 1.\|thumb\|Partial space-time diagram of BMO 1.]]
	A size 36 program was produced by hand to simulate [[BMO1]] rules ([https://discord.com/channels/960643023006490684/1438019511155691521/1440018895212642424 Discord]):		A size 36 program was produced by hand to simulate [[BMO1]] rules ([https://discord.com/channels/960643023006490684/1438019511155691521/1440018895212642424 Discord]):

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	=== BMO 6 (“Space Needle”) ===		=== BMO 6 (“Space Needle”) ===
			[[File:Space Needle.webp\|alt=Partial space-time diagram of Space Needle.\|thumb\|Partial space-time diagram of Space Needle.]]
	A size 48 program was produced by hand to simulate [https://wiki.bbchallenge.org/wiki/1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD BMO 6] rules ([https://discord.com/channels/960643023006490684/1438019511155691521/1441137371046482071 Discord])		A size 48 program was produced by hand to simulate [https://wiki.bbchallenge.org/wiki/1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD BMO 6] rules ([https://discord.com/channels/960643023006490684/1438019511155691521/1441137371046482071 Discord])

RobinCodes: /* BBf(17) Family */ Added emphasis on the champion values in table

2026-04-01T08:32:19Z

BBf(17) Family: Added emphasis on the champion values in table

← Older revision		Revision as of 08:32, 1 April 2026
Line 356:		Line 356:
	\|51		\|51
	\|-		\|-
	\|17		\|'''''17'''''
	\|2		\|'''''2'''''
	\|3		\|'''''3'''''
	\|107		\|'''''107'''''
	\|-		\|-
	\|18		\|'''''18'''''
	\|2		\|'''''2'''''
	\|4		\|'''''4'''''
	\|211		\|'''''211'''''
	\|-		\|-
	\|19		\|'''''19'''''
	\|2		\|'''''2'''''
	\|5		\|'''''5'''''
	\|370		\|'''''370'''''
	\|-		\|-
	\|20		\|20

RobinCodes: /* BBf(21) */ Added space-time diagram

2026-04-01T08:31:03Z

BBf(21): Added space-time diagram

← Older revision		Revision as of 08:31, 1 April 2026
Line 407:		Line 407:

	==== BBf(21) ====		==== BBf(21) ====
			[[File:Bbf21 champ full.png\|alt=The full space-time diagram of the BBf(21) champion until halting.\|thumb\|The full space-time diagram of the BBf(21) champion until halting.]]


	The BBf(21) champion (running >31M steps):		The BBf(21) champion (running >31M steps):
	<math display="block">\begin{bmatrix}		<math display="block">\begin{bmatrix}