Code
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Explorations

In[]:=
Module​​{deep=5000,cut=1000,ru,life,evo,data},​​SeedRandom[426778];​​evo=NestListCompoundExpression​​ru=
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RandomRuleMutation
[First[#]],​​life=
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TestLifetime
[ru,cut],​​If[life>=Last[#],{ru,life},#]​​&,{{0,3,2},0},deep;​​evo=Rest[First/@SplitBy[evo,Last]];​​MapCompoundExpression​​data=CellularAutomaton[​​First[#],{{1},0},Last[#]+2],​​data=ArrayPad[#,2]&/@data,​​ArrayPlotdata,ColorRules->
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CustomStyleData
["Colors"],​​ImageSize->{Automatic,26Sqrt[Length[data]+1]},​​Mesh->True,MeshStyle->Opacity[.1]​​​​&,​​evo,evo​​
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,{{{106111661199647248543687855752712667991103904330482569981872649,3,2},1},{{1436806417448229268841782020886939106172156274926505962725203631646476103976426441665552333011538689008536005405439,3,2},2},{{1436847152733480355345275645484811752645185776371467736664070735919373892396647035899988362971423953867456647110022,3,2},3},{{1436857280060308294890441993819175758272740056283600032830736485210955222025270162641072104402142872244646893881679,3,2},5},{{1436857280060308294890441995875964155511132649500814721466220074931398191555526416547834043479150861491417238716221,3,2},6},{{56694206262018521965570612559879930720553908256593181798257380094648468569633352642537223211367684765866169745294390,3,2},10},{{56694206262020074706231149835987511866159485969218933225427844661877868777954456457712502876200451336927115858092733,3,2},12},{{57770621606908582024149739313148722800185717834262746086407495748801985097057874488820759371516182067471999080186039,3,2},45},{{57770621631914327277047164462439365038831250969103259130496056186086157030639136497560403488246565278923685762592680,3,2},54},{{25236503179165107833396023922797634724790104425337549789060588048908397315958709837648668017319178083546300569818856,3,2},82},{{15548752972377252515436531076980007303010714208600208927121786188075056982949535503131257953427736582620378751201779,3,2},131},{{25236491026372914537982205086720512518188433226601610917221509814807298526737639908447953964672654077805972209010753,3,2},161},{{25236491026372914537982205086720512518188433226601610917221509814849044105916932826261907316183669400894842918292834,3,2},166},{{25236491026372914537982205086720512518188433226601610704998187415554547018541221320836571333975860739929702954547536,3,2},175},{{15548752972377252516233372543808051082121123816314026878722049688871768727835041417538541379757890269269179285880078,3,2},209},{{15548752972377252516233372543808051082121123816314026878722049688871768727835041417538541379757886216714026266903811,3,2},227},{{15548734743188962573112645043505315360089502411651619230330797034788794398235329369910658562530355671837450670745382,3,2},266},{{15548734743181340032264087120287538110723045506575884478943972113074692297254283777077976300124988856699839952231568,3,2},723}}
In[]:=
IntegerDigits[15548734743181340032264087120287538110723045506575884478943972113074692297254283777077976300124988856699839952231568,3,3^5]
Out[]=
{0,1,1,2,1,1,0,0,0,0,1,0,0,1,0,1,1,0,0,2,0,2,1,2,0,1,0,0,1,0,0,1,2,2,0,0,1,0,1,0,2,1,2,0,2,0,0,1,0,0,0,1,0,0,0,1,2,0,1,2,0,2,0,0,0,0,1,1,2,0,0,0,1,1,0,1,0,2,0,0,0,1,2,1,1,2,2,0,0,0,2,1,0,2,0,1,2,0,1,1,2,1,0,0,1,2,2,0,2,1,2,1,0,0,0,2,1,1,2,0,1,0,0,2,0,2,2,0,2,0,2,1,0,2,2,2,0,1,1,0,2,0,0,0,1,2,1,0,1,1,0,0,2,0,1,0,0,1,1,0,2,2,0,2,2,0,1,2,0,0,2,2,1,1,0,2,2,0,0,1,0,0,2,2,0,0,1,2,0,1,2,2,2,2,1,2,2,0,0,1,2,1,2,1,1,2,1,1,1,1,0,1,0,2,0,2,0,1,2,2,2,1,1,1,0,0,0,2,2,0,1,0,1,0,2,0,0,0,2,2,0,0,0}
In[]:=
ParallelTable
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TestLifetime
SeedRandom[23423+i];
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RandomRuleMutation
[{15548734743181340032264087120287538110723045506575884478943972113074692297254283777077976300124988856699839952231568,3,2}],1000,{i,200}

Less evolved cases

(not “straining to use all bits”)

All Cases

k=4, r=2

Looping over evolutions:

When evolution has reached further, the results are more fragile

k=3, r=2 case

For larger lifetimes, things are more fragile and there’s no collection of similar cases:

k=4, r=3

[ Does not reach any “better place” ]