In[]:=
TestPlot2[init_,max_,incr_:2^20]:=ListLinePlot[Length[Last[#]]&/@NestList[PostTagSystemFinalState[#,incr]&,init,max],PlotTheme"Minimal"]
In[]:=
FindHeuristicHaltingTimePlot2[init_,max_,delta_:1,min_:3]:=Catch[Module[{new=Null,old},Do[old=new;new=PostTagSystemFinalState[init,2^n];If[old=!=Null&&Abs[Length[Last[old]]-Length[Last[new]]]≤delta,Throw[n]],{n,min,max}];Echo[{init,TestPlot2[init,500,8Round[2^max/(8×500)]]}];TooBig[init]]]
In[]:=
FindFinalParameters[init_,t_]:=With[{final=PostTagSystemFinalState[init,t]},GeneratePostTagSystemHistory[init,t,final]]
In[]:=
With[{n=21},With[{res=Flatten[ResourceFunction["ParallelMapMonitored"][Table[FindHeuristicHaltingTimePlot2[{p,IntegerDigits[#,2,n]},32,2,5],{p,0,2}]&,Range[0,2^n-1]]]},{KeySort[Counts[res/.TooBig[_]TooBig]],Cases[res,TooBig[x_]x]}]]
(kernel 108)
>> {1,{0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,1,1,1,1,1,0}},

(kernel 130)
>> {0,{1,0,1,0,1,0,1,0,0,1,1,0,1,0,1,1,1,1,1,1,0}},

(kernel 117)
>> {0,{1,0,1,0,1,0,1,1,1,1,0,1,0,1,1,1,1,0,0,1,0}},

(kernel 65)
>> {0,{1,0,1,1,0,0,1,1,1,1,0,1,0,1,1,1,1,1,1,0,1}},

(kernel 40)
>> {2,{1,0,1,1,0,1,1,0,0,1,1,1,1,0,1,0,1,1,1,1,1}},

(kernel 118)
>> {0,{1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,1,1,1,1,0}},

Out[]=
{64165044,7918727,8635392,9178090,1076576,11121114,1258472,1356684,1430318,1519614,1615193,172927,182611,193779,203570,211268,22550,2321,2445,26788,27395,30260,3112,TooBig6,{{1,{0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,1,1,1,1,1,0}},{0,{1,0,1,0,1,0,1,0,0,1,1,0,1,0,1,1,1,1,1,1,0}},{0,{1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,1,1,1,1,0}},{0,{1,0,1,0,1,0,1,1,1,1,0,1,0,1,1,1,1,0,0,1,0}},{0,{1,0,1,1,0,0,1,1,1,1,0,1,0,1,1,1,1,1,1,0,1}},{2,{1,0,1,1,0,1,1,0,0,1,1,1,1,0,1,0,1,1,1,1,1}}}}
In[]:=
With[{n=10},With[{res=Flatten[ResourceFunction["ParallelMapMonitored"][Table[FindHeuristicHaltingTimePlot2[{p,IntegerDigits[#,2,n]},32,2,5],{p,0,2}]&,Range[0,2^n-1]]]},{KeySort[Counts[res/.TooBig[_]TooBig]],Cases[res,TooBig[x_]x]}]]
Out[]=
{62511,7372,889,919,1032,1248,161,{}}
In[]:=
With[{n=10},With[{res=Flatten[ResourceFunction["ParallelMapMonitored"][Table[FindHeuristicHaltingTimePlot2[{p,IntegerDigits[#,2,n]},15,2,5],{p,0,2}]&,Range[0,2^n-1]]]},{KeySort[Counts[res/.TooBig[_]TooBig]],Cases[res,TooBig[x_]x]}]]
(kernel 94)
>> {2,{0,0,1,1,1,1,1,1,0,1}},

Out[]=
{62511,7372,889,919,1032,1248,TooBig1,{{2,{0,0,1,1,1,1,1,1,0,1}}}}
In[]:=
With[{n=11},With[{res=Flatten[ResourceFunction["ParallelMapMonitored"][Table[FindHeuristicHaltingTimePlot2[{p,IntegerDigits[#,2,n]},32,2,5],{p,0,2}]&,Range[0,2^n-1]]]},{KeySort[Counts[res/.TooBig[_]TooBig]],Cases[res,TooBig[x_]x]}]]
Out[]=
{64792,7864,8243,976,1057,115,1293,1311,163,{}}
In[]:=
With[{n=11},With[{res=Flatten[ResourceFunction["ParallelMapMonitored"][Table[FindHeuristicHaltingTimePlot2[{p,IntegerDigits[#,2,n]},15,2,5],{p,0,2}]&,Range[0,2^n-1]]]},{KeySort[Counts[res/.TooBig[_]TooBig]],Cases[res,TooBig[x_]x]}]]
(kernel 31)
>> {1,{0,1,0,0,1,1,1,1,0,1,0}},

(kernel 142)
>> {0,{1,0,0,1,1,0,0,1,0,1,0}},

(kernel 54)
>> {1,{0,1,1,1,1,0,0,1,0,1,0}},

Out[]=
{64792,7864,8243,976,1057,115,1293,1311,TooBig3,{{1,{0,1,0,0,1,1,1,1,0,1,0}},{1,{0,1,1,1,1,0,0,1,0,1,0}},{0,{1,0,0,1,1,0,0,1,0,1,0}}}}

Systematic version

In[]:=
Last[{},5]
Out[]=
5
THIS IS MESSED UP!!
In[]:=
doall[n_]:=With[{rr=With[{res=Flatten[ResourceFunction["ParallelMapMonitored"][Table[FindHeuristicHaltingTimePlot2[{p,IntegerDigits[#,2,n]},32,2,5],{p,0,2}]&,Range[0,2^n-1]]]},{KeySort[Counts[res/.TooBig[_]TooBig]],Cases[res,TooBig[x_]x]}]},With[{res=Flatten[ResourceFunction["ParallelMapMonitored"][Table[FindHeuristicHaltingTimePlot2[{p,IntegerDigits[#,2,n]},Last[Keys[First[rr]],5]-1,2,5],{p,0,2}]&,Range[0,2^n-1]]]},{KeySort[Counts[res/.TooBig[_]TooBig]],Cases[res,TooBig[x_]x]}]]
In[]:=
doall[8]
In[]:=
Table[doall[n],{n,10,20}]
KernelObject
:Subkernel connected through KernelObject
Name: local
KernelID: 1
 appears dead.
NotebookSave[]
In[]:=
Now
Out[]=
Fri 12 Feb 2021 02:27:27GMT-5
In[]:=
ParallelEvaluate[Needs["PostTagSystem`"]];
In[]:=
With[{n=9},With[{res=Flatten[ResourceFunction["ParallelMapMonitored"][Table[FindHeuristicHaltingTimePlot2[{p,IntegerDigits[#,2,n]},32,2,5],{p,0,2}]&,Range[0,2^n-1]]]},{KeySort[Counts[res/.TooBig[_]TooBig]],Cases[res,TooBig[x_]x]}]]
Out[]=
{61329,7133,822,92,1029,1221,{}}
In[]:=
With[{n=25},With[{res=Flatten[ParallelMap[Table[FindHeuristicHaltingTimePlot2[{p,IntegerDigits[#,2,n]},32,2,5],{p,0,2}]&,Range[0,2^n-1]]]},{KeySort[Counts[res/.TooBig[_]TooBig]],Cases[res,TooBig[x_]x]}]]
(kernel 78)
>> {2,{0,0,0,0,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0,0,1,1,0,1,0}},

(kernel 115)
>> {2,{0,0,0,0,0,1,0,1,0,0,1,0,1,0,0,0,1,0,1,1,1,1,0,1,0}},

(kernel 79)
>> {2,{0,0,0,0,1,1,1,0,1,0,1,1,1,0,1,1,1,0,1,1,1,0,1,0,1}},

(kernel 67)
>> {1,{0,0,0,0,1,1,1,1,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,1}},

(kernel 85)
>> {1,{0,0,0,0,1,1,1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,1,1,1,1}},

(kernel 154)
>> {1,{0,0,0,0,1,0,0,1,1,0,1,0,1,0,0,1,1,1,1,0,1,1,1,1,1}},

(kernel 74)
>> {2,{0,0,0,1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,1}},

(kernel 4)
>> {0,{0,0,0,1,0,0,1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,1,1,1,1}},

(kernel 83)
>> {1,{0,0,0,1,1,1,0,1,0,1,1,1,0,1,1,1,0,1,1,1,0,1,0,1,0}},

(kernel 96)
>> {0,{0,0,0,0,1,0,1,0,1,0,1,1,0,1,0,1,1,1,0,1,1,1,0,1,1}},

(kernel 92)
>> {2,{0,0,0,1,1,0,1,0,1,0,0,1,0,1,1,1,0,1,1,1,0,1,0,1,1}},

(kernel 153)
>> {1,{0,0,0,0,1,0,0,1,1,0,1,0,1,1,1,1,0,0,1,0,1,1,1,1,1}},

(kernel 133)
>> {0,{0,0,0,0,1,1,0,1,0,1,1,1,0,1,1,1,0,1,1,1,0,1,0,1,0}},

(kernel 62)
>> {2,{0,0,1,0,0,0,0,1,1,1,1,0,1,0,1,0,0,1,1,1,1,0,1,1,0}},


Try mostly 1s.... (max of 7 0s)

Testing all 1s....