4 dif­fer­ent data sets were tested with the sim­u­la­tion of and the res­ults were ob­tained, they were:

1. A pub­lic­a­tion at with time in­ter­val 1.
   Max­imum cu­mu­lat­ive res­ul­ted at if black hole is im­ple­men­ted at (45.1%; Graph 1a).
   Max­imum cu­mu­lat­ive res­ul­ted at if black hole is im­ple­men­ted at (35.8%; Graph 1b).

Graph 1a: a pub­lic­a­tion of , with cu­mu­lat­ive plot­ted against time of im­ple­ment­ing black hole in lin­ear scale

Graph 1b: a pub­lic­a­tion of , with cu­mu­lat­ive plot­ted against time of im­ple­ment­ing black hole in lin­ear scale ()

2. A pub­lic­a­tion at (i.e., a pub­lic­a­tion with 1 hour in real time) with time in­ter­val 0.01.
   Max­imum cu­mu­lat­ive res­ul­ted at if black hole is im­ple­men­ted at (51.8%; Graph 2a).
   Max­imum cu­mu­lat­ive res­ul­ted at t = 900 if black hole is im­ple­men­ted at t = 480.40 (53.3%; Graph 2b).

Graph 2a: a pub­lic­a­tion of , with cu­mu­lat­ive plot­ted against time of im­ple­ment­ing black hole in lin­ear scale

Graph 2b: a pub­lic­a­tion of , with cu­mu­lat­ive plot­ted against time of im­ple­ment­ing black hole in lin­ear scale ()

3. A pub­lic­a­tion at (i.e., a pub­lic­a­tion with 2 hours in real time) with time in­ter­val 0.01.
   Max­imum cu­mu­lat­ive res­ul­ted at if black hole is im­ple­men­ted at (53.2%; Graph 3a).
   Max­imum cu­mu­lat­ive res­ul­ted at if black hole is im­ple­men­ted at (36.2%; Graph 3b).

Graph 3a: a pub­lic­a­tion of , with cu­mu­lat­ive plot­ted against time of im­ple­ment­ing black hole in lin­ear scale

Graph 3b: a pub­lic­a­tion of , with cu­mu­lat­ive plot­ted against time of im­ple­ment­ing black hole in lin­ear scale ()

4. A pub­lic­a­tion at (i.e., a pub­lic­a­tion with 100 minutes in real time) with time in­ter­val 0.01.
   Max­imum cu­mu­lat­ive res­ul­ted at if black hole is im­ple­men­ted at (50.8%; Graph 4a).
   Max­imum cu­mu­lat­ive res­ul­ted at if black hole is im­ple­men­ted at (43.5%; Graph 4b).

Graph 34: a pub­lic­a­tion of , with cu­mu­lat­ive plot­ted against time of im­ple­ment­ing black hole in lin­ear scale

Graph 4b: a pub­lic­a­tion of , with cu­mu­lat­ive plot­ted against time of im­ple­ment­ing black hole in lin­ear scale ()

All 4 data sets showed sim­ilar res­ults that max­imum cu­mu­lat­ive was ob­tained at the end of the pub­lic­a­tion if black hole is im­ple­men­ted at 50% of the pub­lic­a­tion, while they showed dis­tinct res­ults that max­imum cu­mu­lat­ive was ob­tained at the end of the pub­lic­a­tion if black hole is im­ple­men­ted at dif­fer­ent time (de­pend on and ). No con­clu­sion can be drawn for pro­gres­sion.

As has no ef­fect on after black hole is im­ple­men­ted, it can be con­cluded that has no al­ter­a­tion on the res­ult on the time of im­ple­ment­ing black hole for max­imum cu­mu­lat­ive to be ob­tained based on the ma­jor as­sump­tion of con­stant ef­fect of , , , , and throughout the pub­lic­a­tion.

The 4 above-men­tioned data sets were re­peated with the sim­u­la­tion of (i.e., Level 3), (i.e., Level 4), (i.e., Level 5), and (i.e., Level 6). The res­ults were ob­tained and were

1. A pub­lic­a­tion at with time in­ter­val 1 (Graph 5).

Graph 5: a pub­lic­a­tion of , with cu­mu­lat­ive plot­ted against time of im­ple­ment­ing black hole in log scale

2. A pub­lic­a­tion at (i.e., a pub­lic­a­tion with 1 hour in real time) with time in­ter­val 0.01 (Graph 6).

Graph 6: a pub­lic­a­tion of , with cu­mu­lat­ive plot­ted against time of im­ple­ment­ing black hole in log scale

3. A pub­lic­a­tion at (i.e., a pub­lic­a­tion with 2 hours in real time) with time in­ter­val 0.01 (Graph 7).

Graph 7: a pub­lic­a­tion of , with cu­mu­lat­ive plot­ted against time of im­ple­ment­ing black hole in log scale

4. A pub­lic­a­tion at (i.e., a pub­lic­a­tion with 100 minutes in real time) with time in­ter­val 0.01 (Graph 8).

Graph 8: a pub­lic­a­tion of , with cu­mu­lat­ive plot­ted against time of im­ple­ment­ing black hole in log scale

The res­ult can be sum­mar­ized with the fol­low­ing tables:

The above in­vest­ig­a­tions il­lus­trate the fact that im­ple­ment­ing black hole at dif­fer­ent times does af­fect the cu­mu­lat­ive ob­tained at the end of the pub­lic­a­tion, thus af­fect­ing the time re­quired for pub­lic­a­tion and ef­fi­ciency of gain­ing for growth. Sim­u­la­tion across dif­fer­ent data sets also demon­strates the con­sist­ency of im­ple­ment­ing black hole at 50% of the pub­lic­a­tion for op­tim­iz­a­tion, and the hy­po­thesis that the dur­a­tion of pub­lic­a­tion seems to be in­de­pend­ent to the ab­so­lute time of im­ple­ment­ing the black hole (only the re­l­at­ive dur­a­tion does).

Im­ple­ment­ing black hole at dif­fer­ent times does not con­sist­ently af­fect the cu­mu­lat­ive ob­tained at the end of the pub­lic­a­tion. Cal­cu­la­tions across dif­fer­ent data sets demon­strate fluc­tu­at­ing res­ult on the time im­ple­ment­ing the black hole. Graphs 5–8 re­veal a fact that the value of is a ma­jor factor af­fect­ing the time for im­ple­ment­ing black hole to ob­tain max­imum cu­mu­lat­ive , which is dif­fer­ent from that by ma­nip­u­lat­ing cu­mu­lat­ive rho.

The above sim­u­la­tions took on a ma­jor as­sump­tion of a pub­lic­a­tion that had the same levels of , , , , and throughout, which al­lowed the ma­nip­u­la­tion of cu­mu­lat­ive in a single in­de­pend­ent vari­able set­ting and hence val­id­ated the fair com­par­ison among in­de­pend­ent vari­ables. However, such as­sump­tion was prac­tic­ally im­possible dur­ing the ac­tual situ­ation. As the ef­fect of vari­ables on the cu­mu­lat­ive is com­plex and highly de­pend­ent on the act­ive­ness of player, it was also the­or­et­ic­ally chal­len­ging to sim­u­late the ex­act ef­fects on all avail­able data I had in my ex­cel. To eval­u­ate the gen­eral/​rough ef­fects of , , , , , and , I will ex­plore them in the view of the equa­tion of and in-game and in turn eval­u­ate the ef­fect of such on the graphs, hence provide a more re­fined hy­po­thesis.

1. The cost of pur­chas­ing , , , , , and has no ef­fect on the graphs, as the graphs plot the cu­mu­lat­ive , not the a player have at a spe­cific .

2. The ef­fect of , , and will shift the graphs of cu­mu­lat­ive up­ward and right­ward at a non-lin­ear scale, as dir­ectly de­pends on , , and .

3. and has no ef­fect on the graphs of cu­mu­lat­ive , as they have no re­la­tion­ship on .

4. The ef­fect of , , and will shift the graphs of cu­mu­lat­ive up­ward and right­ward at a non-lin­ear scale, as dir­ectly de­pends on , , and .

5. The ef­fect of will shift the graph of cu­mu­lat­ive up­ward at a non-lin­ear scale and cu­mu­lat­ive up­ward at an ex­po­nen­tial scale, as it dir­ectly in­flu­ences in an ex­po­nen­tial man­ner and hence in­dir­ectly.

6. The ef­fect of shortened time buy­ing the vari­ables will shift the graphs of cu­mu­lat­ive and left­ward at a non-lin­ear scale, as it al­lows an earlier growth for cu­mu­lat­ive and in a re­peat­able man­ner.

Over­all, pur­chas­ing , , , , , and have an ef­fect of shift­ing graphs up­ward and slightly right­ward, in­dic­at­ing the im­ple­ment of black hole is pos­sible to be pushed back slightly for op­tim­iz­a­tion.

The above-men­tioned ef­fects were later veri­fied by the sim (with the most op­timal strategy im­ple­men­ted by brute-for­cing dif­fer­ent for im­ple­ment­ing black hole on a pub), which takes into the ac­count of the ef­fects of vari­able pur­chases (i.e., , , , , , and ) and us­age of level chas­ing strategy (Us­ing a ra­tio of ap­prox­im­ately 4x in terms of levels for over ). The dur­a­tion of a pub­lic­a­tion, the time of im­ple­ment­ing black hole and their re­l­at­ive per­cent­age, , has been cal­cu­lated and plot­ted as a graph of re­l­at­ive dur­a­tion against (Graph 9).

Graph 9: Re­l­at­ive time of im­ple­ment­ing black hole plot­ted against dif­fer­ent , with or­ange ho­ri­zontal line as 60% line and yel­low plots as 30 mov­ing av­er­age for re­l­at­ive time

The plot sup­ports the con­sist­ency of im­ple­ment­ing black hole at 60% of the pub­lic­a­tion for achiev­ing a more ideal out­come by op­tim­iz­a­tion of cu­mu­lat­ive . One point worth not­ing is that the re­l­at­ive dur­a­tion for black hole im­ple­ment­a­tion tem­por­ar­ily spiked up upon a and/​or up­grade, in­dic­at­ing a longer pub­lic­a­tion, and hence a later time for im­ple­ment­ing black hole.

Im­ple­ment­ing the black hole at the right time is es­sen­tial for growth since it fixes as well. However, the con­tinu­ity of the pub­lic­a­tion dur­a­tion does not have the same nature of the dis­crete­ness of the solu­tion for , which may lead to sub­op­timal if the hy­po­thesis is strictly fol­lowed.

In re­sponse of this, there are also data from Dis­cord about with par­tic­u­larly higher as a list. One can con­sider se­lect­ively set­ting t with and high as the time of im­ple­ment­ing black hole in­stead of the the­or­et­ical val­ues ob­tained from the hy­po­thesis. From the graphs above, it can be ob­served that se­lect­ing a time for im­ple­ment­ing black hole that slightly de­vi­ates from the hy­po­thes­ized time min­im­ally af­fect the cu­mu­lat­ive ob­tained at the end of the pub­lic­a­tion.

Base on the dis­cus­sion and all avail­able data we have at this mo­ment, it may be reas­on­able to pro­pose a new idle route of pub­lic­a­tion for com­ple­tion of RZ CT after e600 the gen­eral rout­ing will be as fol­low:

1. Take an es­tim­ate on the dur­a­tion of the up­com­ing pub­lic­a­tion wish to be idled. Cal­cu­late the hy­po­thes­ized time for im­ple­ment­ing black hole (i.e., 60% of your pub­lic­a­tion time. Do es­tim­ate a longer time if a and/​or up­grade is close to your pre­vi­ous pub­lic­a­tion point).

2. Set the time for im­ple­ment­ing black hole that is and has a high and is suf­fi­ciently close to the hy­po­thes­ized time cal­cu­lated from pre­vi­ous step.

3. Start play­ing the pub­lic­a­tion with auto­buy all (as miss­ing and pur­chases af­fect the pro­gress heav­ily if it is missed for a sig­ni­fic­ant por­tion of time).

4. When the black hole is im­ple­men­ted, con­tinue to auto­buy un­til the end of pub­lic­a­tion.

5. Re­peat the pro­gress if the next pub­lic­a­tion is also des­ig­nated to be idled.