(Engineered) diehards

[Life73]

This produces 13096 loaves in 60 X 60 (parity only needs to be attended to for long streams of lone block laying gliders):
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Code: Select all

x = 60, y = 60, rule = B3/S23
11b6o2b4ob4ob2o25bob2o$13bo7b4obobo2bo10b3o12b3o$17bo6bobo2bo5bo5bo2bo
14bo$12b5o6bo3bobo4bob2obo4bo11b2obo$14b2o21bobo15bob3o$11bob2o24bobo
16b2o$5bo8bo14bo24b3ob2o$5b3o21b3o12b2o9bo$8bo23bo12bo8bo3bo$7b2o3bo
16b2obo9b3o10bo$12bobo14bobo3b2o4bo$11bobo21b2o4b4o$6bobo4bo30bo$7b2o
22b2o6b3o$7bo23b2o6bo2bo$15b2o3b2o2b2o14b2o$15b2o2bo2bo2bo18bo8b2obo$
3b2ob2o11bob2obo18bobo6bo5b2o$3b3o2bo3bo7bo3b2o10bo5bobo8b2ob2obo$2bo
4b2o2bobob2obobob2o6b2o3bobo5bo8b3obob2o$b3ob3o3bobobob2ob2obo6bobo3bo
16bo4b2o$b2obob2o4bo18b2o18bobo2bo2bo$bo49b2obobo$bo2b3obo44bo2bob2o$b
ob2o41bo6bo2bo2bo$21b2o13bo7b3o11b2o$4bo16b2o12bo2bo4bo7b2obo4bo$3bo
31bo2bo4b2o10b2ob2o$31b2o4bo21bo$2ob3o18b2o4bobo17b2obo3bo$2o2bo2bo16b
2o4b2o18bob2o$o6bo6b2o32bobo$2o2bo2bobo5bo11b2o5b2o11bobo2b2o$2bo4bo7b
obo8bobo4bobo12bo4bo$3bo2bobo7b2o3bo4b2o5bo4b2o3b2o7bo$5bo14bobo9b2o4b
2o3b2o7b2o$2bob3o3bo10bo25bob2obo$4bo5bo7bo10b2o9b3o4b2obo2bo$3bo3bo2b
2o5bobo8bobo9b3o7bob2o$12bo5bo4b2o3bo12bo5b2obobo4bobo$12b3o8b2o2b2o4b
2o12bobo2bo5bo$18bo14bo14bo3b2o3bobo$16bo18bo22b2o$12bo4b4o7b2o4b2o20b
o2bo$14bobo11b2o3bo24b2o$14b2o5bo4b2o6bobobob2o12b3o2bo$13b2o3bob2o4b
2o3bo3b2obobo3bo14bo$12bo3bo4bobo7bo5b3o2bo13bo$14bo2bo11b3o5bo2b4o$
18bob2o2b2obo3bobobob2ob2obo$18b4o5bo6b2o3bo$24b3o2b4ob2o3b3obo$24b2o
3b2o7bo3bo$18b2o4bobo2bobo3b2o3bob2o$22b2o9b2ob2o2bo3bo$18b3o4b2obobob
3o2bo2b2o2bo$16bob4ob3o4b2o4bob2obo$14b2o4bo2b2o2b3o2bo3b2o2bo2bo$15bo
2b2ob3obo3bobob3o2b2ob2o$18b2obob3ob8obob2obobo!

I don't know if this is good, but I did have this extended a few hundred generations longer by slightly tweaking the inner contents for fun.

Code: Select all

x = 66, y = 66, rule = B3/S23
15b2o28bo$15b2o5b2o3b2ob2o12b3o$22b2o4bob2o10b3ob2o$28bo14b2ob3o$18b2o
9b3o8b2obo2b3o8b2o5b2o$18b2o11bo8b3o14b2o5b2o$38bo3bo$15b2o21bo2bo18b
2o$7bo7b2o14bo7bobo18b2o$7b3o21b3o6bo5b2o$10bo23bo12bo$9b2o3bo16b2obo
9b3o$14bobo14bobo3b2o4bo$13bobo21b2o4b4o$15bo30bo$10bo22b2o6b3o$11b2o
20b2o6bo2bo$9bobo5b2o3b2o2b2o14b2o$8b2o7b2o2bo2bo2bo18bo$8b2o11bob2ob
o18bobo$7b2o5bo7bo3b2o10bo5bobo$6b3o4bobob2obobob2o6b2o3bobo5bo15b2o$
5bobo5bobobob2ob2obo6bobo3bo14b2o6bobo$2bo3bo7bo18b2o18b2o3bo3bobo$bo
bo53bobo3bo$2b2o54bo$37b2o9bo$23b2o12bobo6b3o$23b2o13b2o5bo17b2o$45b2o
16b2o$33b2o$3b2o21b2o4bobo17b2obo$b6o19b2o4b2o18bob2o$o2b5o8b2o32bobo
$o6b2o8bo11b2o18bobo2b2o$b2o4b2o8bobo8bobo19bo4bo$2b3o2b2o9b2o3bo4b2o
4bo19bo$7b2o13bobo8bobo18b2o$4bo3bo14bo10bo14bob2obo$4b2obo12bo10bo8b
2o3b2o2b2obo2bo$5bobo2b2o7bobo8bobo9b3o7bob2o$11b2o7bo10bo9bo3bo3b2ob
obo4b2o$13bo21b2o5bobo4bobo2bo4bobo$13b2o13b2o5bo7bo6bo3b2o5bo$14bo13b
2o7bo21bob2o$36b2o20bobo2bo$16b5o14bo23bo2b2o$14b2ob5o14bobo3b2o$21b2o
14b2o3b2o$14b2o5b2o$15b2obo2b3o8b2o$18bo2b3o7bobo3b2o$18b2o2bo9bo4b2o
$19bo2bo4bo2b2o$20b2o5bo$28bo$24bobo3b2obo$24b2o5bo2bo$25bo6b2o$39bo$
27bobo4b2o2bo$27b2o5b2o2b3o$28bo$31bobo$31b2o$32bo!

Also, I know that the Vacuum p46 double-barred glider gun can perform a block pulling reaction similar to sawmills. Can this be useful?

Code: Select all

x = 49, y = 43, rule = B3/S23
b2o23b2o$b2o23bo$24bobo$15b2o7b2o$2o13bobo$2o13bob2o$16b2o$16bo$44b2o
$16bo27b2o$16b2o$2o13bob2o13bo3bo$2o13bobo13bo5bo7b2o$15b2o14bo13b2o$
31b2o3bo$b2o30b3o$b2o$33b3o$31b2o3bo$31bo13b2o$31bo5bo7b2o$32bo3bo2$44b
2o$44b2o5$37b2o$37bobo7b2o$39bo7b2o$37b3o$22bobo$21b3o$21b3o$21bo15b3o
$25bobo11bo$21b2o4bo9bobo$16b2o4bo3b2o9b2o$15bobo6bo$15bo$14b2o!

[Disaster16439]

Right now, the NW section of the 10k cell diehard is wasted mostly on glider OTTs to delay a tiny bit of the time. Could we put a crabstrecher, add some delay mechanism, then have it burn a fuse, then make a glider which comes back and triggers the start. Or, maybe we can add a mechanism for the fuse-stopping before crabstretcher and reactivating?

An unrelated sidenote, has someone made an exact recursive recursive recursive formula for the current diehard length?

[Pavgran]

Right now, the NW section of the 10k cell diehard is wasted mostly on glider OTTs to delay a tiny bit of the time. Could we put a crabstrecher, add some delay mechanism, then have it burn a fuse, then make a glider which comes back and triggers the start. Or, maybe we can add a mechanism for the fuse-stopping before crabstretcher and reactivating?

Potentially, we can do that. There are some drafts early in the thread.

An unrelated sidenote, has someone made an exact recursive recursive recursive formula for the current diehard length?

So, reusing some parameters and formulas from my post:

Pattern with the help of which I measured all the parameters:

Code: Select all

x = 1565, y = 510, rule = LifeSuper
168.S$166.S.S$167.2S91.2M158.2M158.2M158.2M158.2M158.2M158.2M158.2M
158.2M$180.2S78.2M158.2M158.2M158.2M158.2M158.2M158.2M158.2M158.2M$
163.2S11.2S.S2.S33.2M20.2M27.O108.2M20.2M27.O108.2M20.2M27.O108.2M20.
2M27.O108.2M20.2M27.O108.2M20.2M27.O108.2M20.2M27.O108.2M20.2M27.O
108.2M20.2M27.O$.S.2S2.3S6.S.5S3.4S6.2O7.2O6.M.4M7.M9.M.2MQ2.Q4.Q7.M.
6M2.2O4.2O48.S2.S3.3S4.2S.2S.S13.2O7.2O10.M20.M27.O.O3.2O30.V.3V.3V
43.2O7.2O10.M20.M27.O.O3.2O82.2O7.2O10.M20.M27.O.O3.2O30.V.3V.3V43.2O
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84.2O2.2O.2O2.2O8.2M4.M3.M7.2M3.2Q2.2Q10.2M8.2O3.O$.S5.2S5.S2.2S.S3.S
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2M52.2W33.2M5.M10.2H42.2M7.2M52.2W12.2H3.H77.2M7.2M52.2W33.2M5.M22.2U
30.2M7.2M$4.W36.2M3.M.2M9.2U2.3U2.U4.U3.UM.2M2.M5.M2.M9.2M4.M2.M.2M
85.2M3.M2.M15.U6.2U2.U.U24.2M7.2M87.2M3.M2.M15.U6.2U2.U.U24.2M7.2M87.
2M3.M2.M15.U6.2U2.U.U24.2M7.2M87.2M3.M2.M8.H2.H3.U6.2U2.U.U24.2M7.2M
87.2M3.M2.M53.2M7.2M149.2M7.2M87.2M3.M2.M8.H2.H41.2M7.2M49.H99.2M7.2M
87.2M3.M2.M15.U6.2U2.U.U24.2M7.2M$40.M3.M4.M.M8.3U.U4.3U2.2U5.2M3.5M.
M14.M2.M98.M15.U.U4.2U2.2U131.M15.U.U4.2U2.2U131.M15.U.U4.2U2.2U4.2M
125.M8.H.H4.U.U4.2U2.2U131.M192.2M125.M8.H.H101.H.H77.2M125.M15.U.U4.
2U2.2U$42.M2.M13.U3.U2.2U4.U.2U7.2M2.M5.M6.2M5.M.3M3.M87.2M.M2.M15.2U
5.U4.U22.2M5.2M94.2M.M2.M15.2U5.U4.U22.2M5.2M94.2M.M2.M15.2U5.U4.U3.M
.M16.2M5.2M94.2M.M2.M9.H5.2U5.U4.U22.2M5.2M94.2M.M2.M50.2M5.2M132.M.M
16.2M5.2M94.2M.M2.M9.H40.2M5.2M52.H2.H76.M.M16.2M5.2M94.2M.M2.M15.2U
5.U4.U22.2M5.2M$46.M.2M14.U.2U.U.3U2.U3.M5.M2.2M9.2M6.M.3M.2M90.M2.M
19.U30.2M5.2M6.2M89.M2.M19.U30.2M5.2M6.2M89.M2.M19.U13.M16.2M5.2M6.2M
84.2H3.M2.M19.U30.2M5.2M6.2M89.M2.M50.2M5.2M6.2M58.2pA66.M16.2M5.2M6.
2M84.2H3.M2.M50.2M5.2M6.2M45.2H79.M16.2M5.2M6.2M89.M2.M19.U30.2M5.2M
6.2M$46.4M10.3U3.U.4U3.U3.M.M5.3M.M2.M13.M.2M.M.M91.2M20.U.U43.2M90.
2M20.U.U43.2M90.2M20.U.U43.2M84.2H4.2M20.U.U43.2M90.2M66.2M58.2pA98.
2M84.2H4.2M66.2M158.2M90.2M20.U.U43.2M$207.2M20.2U136.2M20.2U136.2M
20.2U136.2M20.2U136.2M318.2M318.2M20.2U2$691.2H278.2pA198.2H$257.2M
158.2M158.2M111.H2.H43.2M158.2M72.2pA84.2M111.H2.H43.2M56.H101.2M158.
2M$257.M.M157.M.M157.M.M110.H.H44.M.M157.M.M157.M.M110.H.H44.M.M54.H.
H100.M.M157.M.M$259.M159.M159.M111.H47.M159.M159.M111.H47.M53.H2.H
102.M159.M$259.2M158.2M158.2M92.2H64.2M158.2M66.2pA90.2M92.2H64.2M53.
2H11.2H90.2M158.2M$673.2H292.2pA184.2H132.2H3$683.2H278.2pA198.2H$
682.H2.H277.2pA197.H2.H101.H$682.H.H477.H.H101.H.H$683.H479.H101.H2.H
$665.2H292.2pA184.2H119.2H11.2H$665.2H292.2pA184.2H132.2H3$675.2H278.
2pA198.2H$674.H2.H277.2pA197.H2.H101.H$674.H.H477.H.H101.H.H$675.H
479.H101.H2.H$657.2H292.2pA184.2H119.2H11.2H$657.2H292.2pA184.2H132.
2H3$532.2M133.2H278.2pA198.2H$531.M.M132.H2.H277.2pA197.H2.H101.H$
533.M132.H.H477.H.H101.H.H$667.H479.H101.H2.H$943.2pA305.2H11.2H$943.
2pA318.2H3$939.2pA$939.2pA302.H$1242.H.H$1241.H2.H$935.2pA305.2H11.2H
$935.2pA318.2H3$931.2pA$931.2pA302.H$1234.H.H$1233.H2.H$927.2pA305.2H
11.2H$927.2pA318.2H3$923.2pA$923.2pA3$919.2pA318.2H$919.2pA318.2H$
502.2M$501.M.M$503.M28$472.2M$471.M.M$473.M28$442.2M$441.M.M$443.M28$
412.2M$411.M.M$413.M28$382.2M$381.M.M$383.M28$352.2M$351.M.M$353.M28$
322.2M$321.M.M$323.M28$292.2M$291.M.M$293.M28$262.2M$261.M.M$263.M28$
232.2M$231.M.M$233.M28$202.2M$201.M.M$203.M28$172.2M$171.M.M$173.M28$
142.2M$141.M.M$143.M!

Exact formulas with parameters:

Code: Select all

a=152 (first sawtooth block offset)
b=169 (initial delay in multiples of 120)

Formula for first sawtooth timings before wick triggering:
y_n = (a+120b+5)*(121^(n+1)-1)+120(b-4)

c=21 (first sawtooth block-eating offset)
d = y_15/24+5 (number of first sawtooth loaves)

Formula for first sawtooth timings during loaf absorbtion:
t_n = (a-c+16d+3*120+9+8/120)*(121^n-1)-8n+24d+3*120

Formula for first sawtooth timings after loaf absorbtion:
z_n = (a+5+t_d)*(121^(n+1)-1)+t_d-120*4

e = 152 (second sawtooth block offset)

s = [32, 28, 30, 23, 24, 19, 20, 15, 16, 11, 12, 7, 8, 3] (block-eating offsets for second sawtooth for sacrifitial still lives)
Formulas for second sawtooth timings before wick triggering:
u_0 = 2
u_(i+1) = z_(u_i)+u_i+1+e-s[i]

h = 45 (bloack-eating offset for second sawtooth)
j = z_(u_14)/24+7 (number of second sawtooth loaves)

Formulas for second sawtooth timings during loaf absorbtion:
v_0 = u_14
v_(i+1) = z_(v_i)+v_i+1+e-h-8(j-i-1)

f = 7692 (death timing offset)

Final formula:
v = z_(v_j)+v_j+1+e
Lifespan = 4*z_v+f

Exact formulas with (most of) substitutions done:

Code: Select all

d=1797926063936542177945655789595339790
t_d = (16*d+500)*121^d-140+8/120*(121^d-1)
z_n = (t_d+157)*121^(n+1)-637
u_0 = 2
u_1 = z_(u_0)+u_0+121
u_2 = z_(u_1)+u_1+125
u_3 = z_(u_2)+u_2+123
u_4 = z_(u_3)+u_3+130
u_5 = z_(u_4)+u_4+129
u_6 = z_(u_5)+u_5+134
u_7 = z_(u_6)+u_6+133
u_8 = z_(u_7)+u_7+138
u_9 = z_(u_8)+u_8+137
u_10 = z_(u_9)+u_9+142
u_11 = z_(u_10)+u_10+141
u_12 = z_(u_11)+u_11+146
u_13 = z_(u_12)+u_12+145
u_14 = z_(u_13)+u_13+150

j = z_(u_14)/24+7
v_0 = u_14
v_(i+1) = z_(v_i)+v_i+116-8j+8i
v = v_(j+1)+37

Lifespan = 4*z_v+7692

Some approximations:

Code: Select all

d > 121^17.4 (that's the number of first sawtooth loaves)
t_d > 121^121^17.4
z_n > 121^(n+121^17.4)
u_1 > 121^121^17.4
u_2 > 121^121^121^17.4
...
v_0 = u_14 > 121^121^121^121^121^121^121^121^121^121^121^121^121^121^121^17.4 > 121^^15
j > 121^121^121^121^121^121^121^121^121^121^121^121^121^121^121^121^17.4 > 17^^17 > 121^^16 (that's the number of second sawtooth loaves)
v_1 > 121^^16
v_2 > 121^^17
...
v > v_(j+1) > 121^^(17^^17) > 17^^^3
Lifespan > 17^^^3

[Pavgran]

Also, adopting linear approximation for real heights, that is:
a^^x = a^(a^^(x-1)) for x >= 1,
a^^x = a^x for 0 <= x < 1,
we can make some more precise bounds:

Code: Select all

d > 121^17.406868871 (first sawtooth loaves)
v_0 = u_14 > 121^^15.595702598
j > 121^^16.595702598 (second sawtooth loaves)
Lifespan > 121^^121^^16.595702598

Using similar approximation for tetration itself (I doubt it behaves nicely from mathematical point of view), that is:
a^^^x = a^^(a^^^(x-1)) for x >= 1
a^^^x = a^^x = a^x for 0 <= x < 1,
we can calculate the following:

Code: Select all

Lifespan > 121^^121^^121^^0.585751974 = 121^^^2.585751974

Varying the base instead, we can say:

Code: Select all

Lifespan > 121^^121^^16.595702598 > 121^^17.062596946^^17.062596946 > 17.062596946^^^3

[Pavgran]

So, about that upper-left corner.

If we modify self-destruction procedure, we can eat more blocks from the second sawtooth with this contraption:

Code: Select all

x = 207, y = 170, rule = LifeSuper
182.2M$182.2M$138.2M20.2M27.O$118.2O7.2O10.M20.M27.O.O3.2O$118.O2.2O.
O.O2.O.O7.M.M6.M.M7.M.M18.2M7.O2.O.O.O$119.2O2.2O.2O2.2O8.2M4.M3.M7.
2M3.2Q2.2Q10.2M8.2O3.O$121.O24.M16.Q2.Q2.Q$118.O.O13.3O8.M4.M14.2Q.Q
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22.O7.2M$124.2Q22.M.M38.MO$157.2O3.5Q$119.2M25.M7.2O2.O3.Q3.Q27.2M$
119.2M5.O20.2M4.O.O.O7.Q28.2M$126.2O18.2M5.O3.2O6.2Q$126.2O26.3O2.O
37.2M$119.M7.O28.O.O21.2M15.2M7.M$118.3M36.O.3O21.M20.3M$117.M3.M37.O
2.O17.M2.M19.M$119.M39.O.O19.2M20.2M$116.M5.M37.O20.2M$116.M5.M57.M2.
M$85.2S30.M3.M58.M$85.2S31.3M41.M18.3M$161.M9.2M$137.2Q22.3M7.2M23.2M
$88.2S47.2Q58.2M$88.2S60.2M22.2M20.M$150.M23.2M$85.2S53.2M6.M.M40.3M$
77.S7.2S51.M2.M6.2M41.4M$77.3S36.2M11.2M6.M33.2M17.M2.M.2M$80.S36.M
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M47.M5.M$140.2M51.2M$173.M15.M.M$84.3S22.S9.S51.2M8.M$85.S.3S18.S.S6.
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3.2Q$111.2S5.2W64.Q.Q3.Q.Q$96.2S7.2W4.S6.2W12.M43.M8.Q5.Q.Q$96.2S7.2W
5.3S4.W11.M43.M6.3Q8.Q2.2Q$114.S9.M.M4.3M41.M6.Q10.Q.Q.Q$108.2W14.2M
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10.2W24.S32.Q$90.W.W.W3.W30.3S$93.2W3.2W31.S53.3M$98.2W41.M43.M$99.W
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84.W.W13.2M62.M20.2M$83.W.W14.2M$82.W.W88.M8.2M5.2M$81.W.W90.2M6.2M5.
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2M$75.W.W14.2M85.M.M$74.W.W104.M$73.W.W105.2M$72.W.W13.2M49.2pA$71.W.
W14.2M21.pAM27.pA$70.W.W37.pA.pA25.pA19.2pA$69.W.W38.pA27.2pA18.pA5.
2pA$68.W.W13.2M23.2pA4.pAM19.2pA3.pA14.pA.pA5.2pA$67.W.W14.2M30.pA18.
pA2.4pA14.2pA$66.W.W44.3pA19.2pA.pA6.2pA$65.W.W45.pA22.pA.pA.2pA3.2pA
20.2pA$64.W.W13.2M54.pA.pA.2pA25.2pA$63.W.W14.2M53.2pA.pA$62.W.W73.pA
25.2pA$61.W.W74.pA.pA23.2pA$60.W.W13.2M61.2pA$59.W.W14.2M68.2pA$58.W.
W85.pA$57.W.W83.2pA.pA3.2pA$56.W.W13.2M67.pA2.pA.2pA2.2pA$55.W.W14.2M
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3pA$51.W.W14.2M79.2pA12.2pA$50.W.W110.pA2.3pA$49.W.W$48.W.W13.2M97.pA
$47.W.W14.2M97.pA3.2pA13.2pA$46.W.W115.pA2.pA14.pA$45.W.W116.pA.pA8.
2pA6.3pA$44.W.W13.2M113.2pA8.pA$43.W.W14.2M$42.W.W127.2pA$41.W.W128.
2pA$40.W.W13.2M$39.W.W14.2M$38.W.W134.2pA$37.W.W135.2pA$36.W.W13.2M$
35.W.W14.2M$34.W.W$33.W.W$32.W.W13.2M$31.W.W14.2M$30.W.W$29.W.W$28.W.
W13.2M$27.W.W14.2M$26.W.W$25.W.W$24.W.W13.2M$23.W.W14.2M$22.W.W$21.W.
W$20.W.W13.2M$19.W.W14.2M$18.W.W$17.W.W$16.W.W13.2M$15.W.W14.2M$14.W.
W$13.W.W$12.W.W$11.W.W$10.W.W$9.W.W$8.W.W$7.W.W$6.W.W$5.W.W$4.W.W$3.W
.W$2.W.W$.W.W$W.W$2W!

Here, red cells emulate the first sawtooth, so that the second sawtooth fires every 4 cycles.
After eating a few blocks with previous mechanisms, the second sawtooth eventually opens up a new glider gun, which helps eat even more blocks.
I found a reaction that cleanly eats a block and fires backwards glider. It can be eaten a few times, and then it cleanly destroys the new glider gun, clearing the way for the last block-eating reaction that starts the second fuse.

The space for potential reactions is quite big, and it's possible that we can find something even better than this (I've seen a few crystal-growing reactions, some herschel-producing reactions, some reactions firing gliders in different directions).

But I had to get rid of the SL that is involved in self-destruction, placing an essential eater in its place. So, if we go that road, another cleanup problem would have to be solved.