User:RobinCodes/Work on BB Domains

Progress

As of May 2026,

Domains with active filtering going on (decider-wise): BB(2,6), BB(7), BB(4,3) - Andrew Ducharme. BB(3,4) - XnoobSpeakable

Work on BB Domains

	2-state	3-state	4-state	5-state	6-state	7-state	8-state	9-state
2-symbol	BB(2) = 6	BB(3) = 21	BB(4) = 107	BB(5) = 47,176,870 2 Jul 2024	1101 Holdouts list (+1 informal) (+15 Rocq-holdouts) 16+6 Cryptids BB(6)	17,823,260 No Handwork Andrew Ducharme, Terry Ligocki Phase 2, Stage 6 BB(7)	Expected no. of holdouts: 147 billion ($1.47\cdot 10^{11}$) BB(8)
3-symbol	BB(2,3) = 38	4 Holdouts list 1 Cryptid, Exhausted BB(3,3)	5,127,263 No Handwork Terry Ligocki, Andrew Ducharme Phase 2, Stage 3 BB(4,3)
4-symbol	BB(2,4) = 3,932,964	11,362,197 No Handwork XnoobSpeakable, Lúkos, Andrew Ducharme Phase 3 Stage 2 BB(3,4)	Expected no. of holdouts: 1.53 quadrillion ($1.53\cdot 10^{15}$) BB(4,4)
5-symbol	60 Holdouts list (5 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	412,086 No Handwork Andrew Ducharme, Terry Ligocki Phase 2 Stage 4 BB(2,6)
7-symbol	Enumeration started! 37% Done Expected no. of holdouts: 3 billion ($3\cdot 10^9$) Current: ~1 billion BB(2,7)
8-symbol	Expected no. of holdouts: 5.51 trillion ($5.51\cdot 10^{12}$) BB(2,8)

Futher domains with some information:

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

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

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

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

Enumeration

The enumeration process for BB(2,7) has started, and 10% 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: 6 total

Domain	2-state	3-state	4-state	5-state	6-state
2-symbol	0	0	0	0	1
3-symbol	0	0	----	---	---
4-symbol	0	---	---	---	---
5-symbol	5	---	---	---	---

BB(6): June 1 2026

1 informal

• mxdys, Pomme, Autumn Pan, vyx^[1][2][3][4]: 1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB---

BB(2,5): June 1 2026

4+1 informal

• Dyuan:
  1. Dekaheptoid, unverified: 1RB3RB1LB---2RB_2LA1RA4LB2LA2RA (source), equivalently (1.2) 1RB3RA2LB1LB1RB_2LA2RA4LA1LA---
• Legion:
  1. 1RB2RA3LA4LA2RB_2LA---1LA1RA3RA (1)
  2. 1RB3LA4LA1LA2RA_2LA4RB---0RA0LA (2)
  3. 1RB3LA4LA2RB1LA_2LA4RB---3RA3LA (3)

BB(3,3): June 1 2026

None