User:HelpMe/Analyses

register machine

Small Minsky busy beaver values

	1-state	2-state	3-state	4-state	5-state	6-state	7-state	8-state	9-state	10-state	11-state	12-state
1-register	MBB(1, 1) = 1	MBB(2, 1) = 3	MBB(3, 1) = 5	MBB(4, 1) = 8	MBB(5, 1) = 12	MBB(6, 1) = 16	MBB(7, 1) ≥ 21	MBB(8, 1) ≥ 27	MBB(9, 1) ≥ 33	MBB(10, 1) ≥ 40	MBB(11, 1) ≥ 48	MBB(12, 1) ≥ 56
2-register				MBB(4, 2) = 10	MBB(5, 2) = 24	MBB(6, 2) = 49	MBB(7, 2) = 231	MBB(8, 2) ≥ 2,682	MBB(9, 2) ≥ 124,686	MBB(10, 2) ≥ 13,146,374	MBB(11, 2) ≥ 14,289,323,033,648
3-register								MBB(8, 3) ≥ 3,394	MBB(9, 3) > 4.493 × 1074	MBB(10, 3) > 10↑↑4.03	MBB(11, 3) > 10↑↑17.74
...
∞-register	MBB(1) = 1	MBB(2) = 3	MBB(3) = 5	MBB(4) = 10	MBB(5) = 24	MBB(6) = 49	MBB(7) = 231	MBB(8) ≥ 3,394	MBB(9) > 4.493 × 1074	MBB(10) > 10↑↑4.03	MBB(11) > 10↑↑17.74	MBB(12) > 10↑↑17.74

In the above table, cells are highlighted in orange when there are known Cryptids (mathematically-hard machines) in that class, and cells are highlighted in tan when the existence of a Cryptid is given by using a known one with fewer states or registers.

It is known that MBB(n, 1) is computable (it is suspected that $\mathrm{MBB}(n,1)=\lfloor \frac{(n+1{)}^2}{3}\rfloor$), and that MBB(m, n) for n > 1, as a function of m, grows uncomputably fast

table of champions

Trivial champions (MBB(n, 1))

[0+(+1)_]^n_0-0*

Let A(x) = (n + 1):[x]
A(x) --(x)--> A(0)
A(0) --(1)--> *:[0]

A:[0]
--(n)--> A(n)
--(n)--> A(0)
--(1)--> *:[0]
MBB(n + 1, 1) ≥ 2n + 1

MBB(1, 1) ≥ 1
MBB(2, 1) ≥ 3
MBB(3, 1) ≥ 5

notes for later when i have enough brainpower to do analyses

[0+(# + 1)]^(2n + 2)_0-(# + 1)*_[0-(# + 1)(4n + 5 - #)]^n_0-(n)(n + 1) (probably not correct)
 4: 0+B_0+C_0-D*_0-BC | 8
 7: 0+B_0+C_0+D_0+E_0-F*_0-GE_0-CD | 21
10: 0+B_0+C_0+D_0+E_0+F_0+G_0-H*_0-IG_0-JF_0-DE | 40
13: 0+B_0+C_0+D_0+E_0+F_0+G_0+H_0+I_0-J*_0-KI_0-LH_0-MG_0-EF | 65

 5: 0+B_0+C_0+D_0-E*_0-DB | 12
 8: 0+B_0+C_0+D_0+E_0+F_0-G*_0-HC_0-FB | 27
11: 0+B_0+C_0+D_0+E_0+F_0+G_0+H_0-I*_0-JH_0-KG_0-EF | 48

 6: 0+B_0+C_0+D_0+E_0-F*_0-DE | 16
 9: 0+B_0+C_0+D_0+E_0+F_0+G_0-H*_0-IG_0-EF | 33
12: 0+B_0+C_0+D_0+E_0+F_0+G_0+H_0+I_0-J*_0-KI_0-LH_0-FG | 56

i can't be bothered to do [0+(+1)_]^m_[0-(if (# = m + n - 1) {m} else {+1})(if ((1 - #) % n = 0) {*} else {(1 - #) % n})]^n. you do it.

MBB(4)

0+B_1+C_0-BD_1-C*
discovered by Bard, December 2025

Let A(x) = C:[0, x]
A(x + 1) --(2)--> A(x)
A(0) --(2)--> *:[0, 0]

A:[0, 0]
--(4)--> A(2)
--(2)--> A(1)
--(2)--> A(0)
--(2)--> *:[0, 0]
MBB(4, 2) ≥ 10

MBB(5)

0-DB_0+C_1-ED_1+A_1-B*
discovered by Bard, December 2025

Let A(x) = D:[x, 0]
A(2k) --(7k + 5)--> *:[k + 1, 0]
A(2k + 1) --(7k + 9)--> A(k + 2)

A:[0, 0]
--(3)--> A(1)
--(9)--> A(2)
--(12)--> *[2, 0]
MBB(5, 2) ≥ 24

MBB(6)

0+B_1-FC_1+D_0-CE_0+A_1-A*
discovered by Bard, December 2025

Let A(x) = C:[x, 0]
A(2k) --(7k + 6)--> *:[k + 2, 0]
A(2k + 1) --(7k + 10)--> A(k + 3)

A:[0, 0]
--(2)--> A(1)
--(10)--> A(3)
--(17)--> A(4)
--(20)--> *:[4, 0]
MBB(6, 2) ≥ 49

MBB(7)

0+B_0+C_0+D_1-GE_1+F_0-EC_1-A*
discovered by Rick Griffiths, October 2003

Let A(x) = C:[0, x]
A(2k) --(11k + 6)--> A(3k + 2)
A(2k + 1) --(5k + 3)--> *:[3k + 1, 0]

A:[0, 0]
--(12)--> A(4)
--(28)--> A(8)
--(50)--> A(14)
--(83)--> A(23)
--(58)--> *:[34, 0]
MBB(7, 2) ≥ 231

MBB(8)

0+B_1-CC_0+D_1-HE_1+F_1+G_0-EC_1-A*
discovered by elgan65536, September 2026

Let A(x) = E:[x, 0]
A(3k) --(19k + 10)--> A(4k + 3)
A(3k + 1) --(19k + 14)--> *:[4k + 3, 0]
A(3k + 2) --(19k + 21)--> A(4k + 5)

A:[0, 0, 0]
--(4)--> A(2)
--(21)--> A(5)
--(40)--> A(9)
--(67)--> A(15)
--(105)--> A(23)
--(154)--> A(33)
--(219)--> A(47)
--(306)--> A(65)
--(420)--> A(89)
--(572)--> A(121)
--(774)--> *:[163, 0]
MBB(8, 2) ≥ 2.682

0+B_0+C_1-GD_1+E_0-DF_2-HG_2+A_2-D*
discovered by -d, June 2026

Let A(x) = G:[0, x, 0]
A(2k) --(20k + 12)--> *:[0, 5k + 3, 0]
A(2k + 1) --(20k + 23)--> A(5k + 6)

A:[0, 0]
--(10)--> A(3)
--(43)--> A(11)
--(123)--> A(31)
--(323)--> A(81)
--(823)--> A(206)
--(2.072)--> *:[0, 518, 0]
MBB(8, 3) ≥ 3.394

MBB(9)

0+B_0+C_0+D_1-IE_1+F_1+G_1+H_0-EC_1-A*
discovered by mineralli, September 2026

let uh A(n) = E:[n, 0 0]
A(2n) —(23n + 12)—> :[9n + 4, 0, 0]
A(2n + 1) —(23n + 25)—> A(9n + 10)

A:[0, 0, 0]
—(4)—> A(3)
—(48)—> A(19)
—(232)—> A(91)
—(1.060)—> A(415)
—(4.786)—> A(1.873)
—(21.553)—> A(8.434)
—(97.003)—>:[37.957, 0, 0]
MBB(9, 2) ≥ 124.686

0+B_0+C_1-HD_0-EG_0-FI_1+D_2+H_0+A_2-B*
discovered by sheep, September 2026

Let A(x, y) = D:[x, 0, y]
A(2k, y) –(7k + 6)–> A(3k + 3, y + 1)
A(2k + 1, y + 1) –(7k + 5)–> A(3k + 1, y)
A(2k + 1, 0) –(3k + 3)–> *:[0, k, 0]

A:[0, 0, 0]
--(3)--> A(2, 0)
--(13)--> A(6, 1)
--(27)--> A(12, 2)
...
--(2.753.550.047.052.785.432.549)--> A(1.180.092.877.308.336.613.950, 31)
--(4.130.325.070.579.178.148.831)--> A(1.770.139.315.962.504.920.928, 32)
--(6.195.487.605.868.767.223.254)--> A(2.655.208.973.943.757.381.395, 33)
...
--(123.361.440.505.394.191.838.159.894.807.429.323.394.336.791.980.165.278.392.551.786.831.696.080.997)--> A(52.869.188.788.026.082.216.354.240.631.755.424.311.858.625.134.356.547.882.522.194.356.441.177.569, 0)
--(79.303.783.182.039.123.324.531.360.947.633.136.467.787.937.701.534.821.823.783.291.534.661.766.355)--> *[0, 26.434.594.394.013.041.108.177.120.315.877.712.155.929.312.567.178.273.941.261.097.178.220.588.784, 0]
MBB(9, 3) ≥ holy fuck 449.388.104.698.221.698.839.011.045.369.921.106.650.798.313.642.030.657.001.438.652.029.750.007.257

MBB(10)

0+B_0+C_0+D_1-EF_0+A_1+G_1+H_0-IE_0-J*_0-FD
discovered by mineralli, October 2026

Let A(x) = F:[x, 0]
A(3k) --(15k + 18)--> A(8k + 12)
A(3k + 1) --(5k + 4)--> *:[0, 2k + 2]
A(3k + 3) --(15k + 16)--> A(8k + 8)

A:[0, 0] --(4)--> A(3)
--(4)--> A(3)
--(33)--> A(20)
--(106)--> A(56)
--(286)--> A(152)
--(766)--> A(408)
--(2.058)--> A(1.100)
--(5.506)--> A(2.936)
--(14.686)--> A(7.832)
--(39.166)--> A(20.888)
--(104.446)--> A(55.704)
--(278.538)--> A(148.556)
--(742.786)--> A(396.152)
--(1.980.766)--> A(1.056.408)
--(5.282.058)--> A(2.817.100)
--(4.695.169)--> *:[0, 1.878.068]
MBB(10, 2) ≥ 13.146.374

0+B_1-AC_2-ED_1+E_1+F_0-DG_2-JH_2+I_1-HE_2-A*
discovered by sheep, October 2026

Let A(x) = E:[0, 0, x]
A(3k) --(28 × 2^k - 5k - 21)--> A(4 × 2^k - 2)
A(3k + 1) --(20 × 2^k - 5k - 16)--> *:[0, 4 × 2^k - 3, 0]
A(3k + 2) --(56 × 2^k - 5k - 23)--> A(8 × 2^k - 1)

Let t3 = 2^5864062014807
Let t4 = 2^((t3 + 4) / 3)

A:[0, 0, 0]
--(20)--> A(5)
--(84)--> A(15)
--(850)--> A(126)
--(123145302310681)--> A(17592186044414)
--(7 × t3 - 29320310074043)--> A(t3 - 1)
--(5 × t4 - 5 / 3 × t3 - 38 / 3)--> *:[0, t4 - 3, 0]
MBB(10) ≥ 5 × t4 + 16 / 3 × t3 + 281474976712738 / 3 > 10^^4.03

MBB(11)

0+B_0+C_0+D_1-AE_1+F_0-GC_0-HF_0-I*_0-JB_1+K_1+E
discovered by mineralli, September 2026
(+16 steps by sheep, September 2026)

Let A(x) = E:[x, 0, 0]
A(4k) --(19k + 8)--> A(9k + 4)
A(4k + 1) --(19k + 10)--> A(9k + 4)
A(4k + 2) --(7k + 4)--> *:[0, 3k + 1, 0]
A(4k + 3) --(19k + 12)--> A(9k + 5)

A:[0, 0, 0]
--(4)--> A(3)
--(12)--> A(5)
--(29)--> A(13)
--(67)--> A(31)
--(145)--> A(68)
--(331)--> A(157)
--(751)--> A(355)
--(1.684)--> A(797)
--(3.791)--> A(1.795)
--(8.524)--> A(4.037)
--(19.181)--> A(9.085)
--(43.159)--> A(20.443)
--(97.102)--> A(45.995)
--(218.474)--> A(103.487)
--(491.561)--> A(232.844)
--(1.106.017)--> A(523.903)
--(2.488.537)--> A(1.178.780)
--(5.599.213)--> A(2.652.259)
--(12.598.228)--> A(5.967.581)
--(28.346.015)--> A(13.427.059)
--(63.778.528)--> A(30.210.881)
--(143.501.690)--> A(67.974.484)
--(322.878.807)--> A(152.942.593)
--(726.477.322)--> A(344.120.836)
--(1.634.573.979)--> A(774.271.885)
--(3.677.791.459)--> A(1.742.111.743)
--(8.275.030.777)--> A(3.919.751.420)
--(18.618.819.253)--> A(8.819.440.699)
--(41.892.343.318)--> A(19.843.741.571)
--(94.257.772.460)--> A(44.648.418.533)
--(212.079.988.037)--> A(100.458.941.701)
--(477.179.973.085)--> A(226.032.618.829)
--(1.073.654.939.443)--> A(508.573.392.367)
--(2.415.723.613.741)--> A(1.144.290.132.824)
--(5.435.378.130.922)--> A(2.574.652.798.858)
--(4.505.642.398.002)--> *[0, 1.930.989.599.143, 0]
MBB(11, 2) ≥ 14.289.323.033.648

0-KB_1-IC_1+D_1+E_2-CF_0-AG_0+H_1-GI_2+J_2+B_0-B*
discovered by Ortwin Gasper, March 2002
(+11 steps by sheep, October 2026)

Let A(x) = I:[x, 0, 0]
A(3k) --> A((20 × 4^k + 1)/3) if k ≥ 1
A(3k + 1) --> A((80 × 4^k + 1)/3) if k ≥ 1
A(3k + 2) --> *:[0, (20 × 4^k - 2)/3, 0] if k ≥ 1

A:[0, 0, 0] --(140)--> A(27)
--> A((20 × 4^9 + 1) / 3) = A(1.747.627)
--> A((80 × 4^582.542 + 1) / 3) ≡ A(4.897.384) mod 3^15
--> A((80 × 4^1.632.461 + 1) / 3) ≡ A(2.637.523) mod 3^14
--> A((80 × 4^879.174 + 1) / 3) ≡ A(813.591) mod 3^13
--> A((20 × 4^271.197 + 1) / 3) ≡ A(72.385) mod 3^12
--> A((80 × 4^24.128 + 1) / 3) ≡ A(84.526) mod 3^11
--> A((80 × 4^28.175 + 1) / 3) ≡ A(35.479) mod 3^10
--> A((80 × 4^11.826 + 1) / 3) ≡ A(14.202) mod 3^9
--> A((20 × 4^4.734 + 1) / 3) ≡ A(268) mod 3^8
--> A((80 × 4^89 + 1) / 3) ≡ A(2.077) mod 3^7
--> A((80 × 4^692 + 1) / 3) ≡ A(394) mod 3^6
--> A((80 × 4^131 + 1) / 3) ≡ A(235) mod 3^5
--> A((80 × 4^78 + 1) / 3) ≡ A(21) mod 3^4
--> A((20 × 4^7 + 1) / 3) ≡ A(12) mod 3^3
--> A((20 × 4^4 + 1) / 3) ≡ A(6) mod 3^2
--> A((20 × 4^2 + 1) / 3) ≡ A(2) mod 3^1
--> *:[0, (20 × 4^0 - 2) / 3, 0] ≡ *:[0, 0, 0] mod 3^0
MBB(11, 3) > 10^^17'74