Mother of Giants

Unsolved problem:

Does the Mother of Giants quasihalt? If so, how many steps does it take to quasihalt?

The Mother of Giants is a collection of adjacent Turing machines, some of which are Cryptids in the 5-state Beeping Busy Beaver problem that probviously quasihalt. They must all be proven to halt or not if we want to solve BBB(5).

The TMs are all the "children" of 1RB1LE_0LC0LB_0LD1LC_1RD1RA_---0LA where children means all the TMs created by filling in the undefined E0 transition.

Overview of the children

An overview of the 8 most interesting children.

Machine	Status
1RB1LE_0LC0LB_0LD1LC_1RD1RA_1RC0LA (bbch) and 1RB1LE_0LC0LB_0LD1LC_1RD1RA_1LC0LA (bbch)	Probviously quasihalting Cryptids
1RB1LE_0LC0LB_0LD1LC_1RD1RA_1LA0LA (bbch)	Probviously quasihalting Cryptid
1RB1LE_0LC0LB_0LD1LC_1RD1RA_0LC0LA (bbch)	Probviously quasihalting Cryptid
1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA (bbch) (current champion) and 1RB1LE_0LC0LB_0LD1LC_1RD1RA_0LD0LA (bbch)	Quasihalt after $>10^{14\,006}$ steps.
1RB1LE_0LC0LB_0LD1LC_1RD1RA_1RD0LA (bbch)	Quasihalts after about $10^{12\,978}$ steps.
1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RD0LA (bbch)	Quasihalts after about $10^{4\,079}$ steps.

1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RD0LA

Quasihalts after about $10^{4\,079}$ steps. Analysis by Shawn Ligocki:

let
* G(n, m) = 0^inf <E 0^n 1^m 0^inf
then
* G(4k+1, m+2) -> G(7k+ 8, m+1)
* G(4k+2, m+2) -> G(7k+ 6, m+1)
* G(4k+3, m+2) -> G(7k+14, m)
* G(4k+4, m+2) -> G(7k+12, m)
* G(n, 1) -> G(1, n+3)
* G(n, 0) -> Spin Out
and for completeness (although we can never reach G(0, m) by applying these rules):
* G(0, m+2) -> G(7, m)
and finally, at step 4 we are in G(1, 1).

As before, iterating these relations leads to a sort of "2-stage" Collatz-like behavior:
* G(1, 1)
* G(1, 4)
** G(8, 3)
** G(19, 1)
* G(1, 22)
** G(8, 21)
** G(19, 19)
** G(42, 17)
** G(76, 16)
** G(138, 14)
** G(244, 13)
** G(432, 11)
** G(761, 9)
** G(1338, 8)
** G(2344, 7)
** G(4107, 5)
** G(7196, 3)
** G(12598, 1)
* G(1, 12601)
** G(8, 12600)
** G(19, 12598)
...
** G(1876213840710521598391379605525487806521622108810778432527722867392307436766159159315654305525834552560172561652032118063458627370309987185176524224005388698700446967268349729419384342385261686600728253800107702390205303703109911013679543376784994065131479319486732208008605302317371513424001996538578227420511014587912085639154043098929108973505716971259211164084851171147584492195641828442403195419357046301200977722765015982127629619366127227507539382553155352425623517192820793588314301840633848042122283284527197514334793445174332133304252240350530614310123013246222482691057522680607329801073658994807307108563977792602674937915145127717601763644562450026084468482011253492787037376937107425102889344711991496275353115166560777221320784075353693443765935139868061950972649919070516499077698769361018135243624369764825602272332642411663431603264357025025605181511270774475664657506683051463739351416893336641567178700404415824985611548629898352736718482963400988142981278524634899806152389037871669098549259903261356156435822856442997107359268004720245053216395652317114834047445010107196444627059271727548915696497640596592913994488623239627564677236759609661035594688933526335837056500047460869283130575904150442715133699885285716657627307991036082003072952095275483832305837708309180971389464320818551264951669392266140549796563778468321806984538951296296552524443477377012382914243026356951988052699080778947387243967930857325467739175240274180541636551276654811833607927610176824006302410755537315645723335878886137585589884378754363497710819886157044977428594424751379562193743624891055837449756993817095474523342307530343858042369299814007843193782820421402610428397007107529266963233646286254595510683332294534735691214261351756352969840115522216379692717569241523917906031879093475856928361266209321993314289669996574460021380115709710519270157850694498748462253621222466800698577194648206832226152375452456786050873778773714111576650409478390819793529657854689649591879416169292957147115001498762307255727596635383777744252057, 0)
* Spin Out

See https://www.sligocki.com/2022/04/03/mother-of-giants.html for details.