User:RobinCodes/Work on BB Domains

Progress

As of 1 April 2026,

Domains with active filtering going on (decider-wise): BB(2,6), BB(7) - Andrew Ducharme

Work on BB Domains

	2-state	3-state	4-state	5-state	6-state	7-state	8-state
2-symbol	BB(2) = 6	BB(3) = 21	BB(4) = 107	BB(5) = 47,176,870 2 Jul 2024	1161 Holdouts list 16+6 Cryptids BB(6)	18,036,852 No Handwork Andrew Ducharme, Terry Ligocki Phase 2, Stage 6 BB(7)	Expected no. of holdouts: 147 billion ($1.47\cdot 10^{11}$)
3-symbol	BB(2,3) = 38	4 Holdouts list 1 Cryptid, Exhausted BB(3,3)	9,401,447 No Handwork Terry Ligocki Phase 2, Stage 2 BB(4,3)
4-symbol	BB(2,4) = 3,932,964	12,435,284 No Handwork XnoobSpeakable, Lúkos Phase 2, Stage 10C BB(3,4)	Expected no. of holdouts: 1.53 quadrillion ($1.53\cdot 10^{15}$) BB(4,4)
5-symbol	60 Holdouts list (9 informal) 2+4 Cryptids BB(2,5)	Lower bound: (due Racheline 2024)$>f_{\omega }(2{\uparrow }^{15}5)>f_{\omega }^2(15)$ Expected no. of holdouts: 11.3 trillion ($1.13\cdot 10^{13}$) BB(3,5)
6-symbol	545,005 No Handwork Andrew Ducharme, Terry Ligocki Phase 2 Stage 4 BB(2,6)
7-symbol	Enumeration started! 09xxx done (10/100) Expected no. of holdouts: 3 billion ($3\cdot 10^9$) Current: 300 million BB(2,7)
8-symbol

Futher domains with small information:

• BB(8):
• BB(11): Lower bound: (Racheline 2024)

$$f_{\omega }^2(2\uparrow \uparrow 12)$$

• BB(12): Lower bound: (Racheline 2024)

$$f_{\omega }^4(2\uparrow \uparrow \uparrow 4-3)$$

• BB(2,8): Expected no. of holdouts: 5.51 trillion ($5.51\cdot 10^{12}$)

Enumeration

The enumeration process for BB(2,7) has started, and 5% has so far been completed. BB(8) is (tested from a random sample) even harder to enumerate. These processes will take a very long time and much collaborative work to complete. As for BB(3x5), BB(4x4), BB(5x3), enumeration may still be possible, but also incredibly difficult. Any domains after these are probably complete out-of-reach of current technology, just to enumerate. Using crowdsourced enumeration, one more layer may be possible.

Informal Proofs

Unconfirmed, informal proofs of machines for each domain by equivalence class: 11 total

Domain	2-state	3-state	4-state	5-state	6-state
2-symbol	0	0	0	0	2
3-symbol	0	0	----	---	---
4-symbol	0	---	---	---	---
5-symbol	9	---	---	---	---

BB(6): 1 March 2026

1 informal, 1 unverified

• Likely correct: (2)
  • Shawn, star, Pomme, @mxdys: BMO 7 verified
  • mxdys, Pomme, Autumn Pan, vyx^[1][2][3][4]: 1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB---
  • 1RB0RC_0LC0LB_0LD1LC_0LE1LA_0LF---_1RF1RA (halt, Racheline, informal)

BB(3,3): 28 March 2026

None

BB(2,5): 28 March 2026

7 informal, 2 unverified (6 informal, 1 unverified by eqv. class)

• Dyuan:
  1. BMO 3 (2 machines in 1 eqv. class) (source)
  2. Dekaheptoid, unverified: 1RB3RB1LB---2RB_2LA1RA4LB2LA2RA (source), equivalently 1RB3RA2LB1LB1RB_2LA2RA4LA1LA---
  3. BMO 3 variant: 1RB0RA3LA4LA2RA_2LB3LA---4RA3RB (source)
• Legion:
  1. 1RB2RA3LA4LA2RB_2LA---1LA1RA3RA (1)
  2. 1RB3LA4LA1LA2RA_2LA4RB---0RA0LA (2)
  3. 1RB3LA4LA2RB1LA_2LA4RB---3RA3LA (3)
• Racheline: 1RB2LB---4LB0RB_1LA3RB4RB4RA1LB

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