Holstein (B35678/S4678)

[LaundryPizza03]

A self-complementary rule with no still lifes and mostly even-period oscillators. OCA:Holstein at wiki.

Collection of odd-period oscillators I found in February
Holstein odd-period oscillators

p3:

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#C 2023-02-10
x = 10, y = 14, rule = B35678/S4678
4b2o$3b4o$2b6o$2b6o$10o$2b2o2b2o$4b2o$4b2o$2b2o2b2o$10o$2b6o$2b6o$3b4o
$4b2o!

p5:

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#C 2023-02-10
x = 27, y = 9, rule = B35678/S4678
3b3o16b2o$3b5obo10b5o$2b6ob3o2b2ob8o$3b10o2b4ob7o$11obob11obo$3b10o2b
4ob7o$2b6ob3o2b2ob8o$3b5obo10b5o$3b3o16b2o!

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#C 2023-02-15
x = 19, y = 12, rule = B35678/S4678
8b3o$4bob7obo$2bo3b7o3bo$2b15o$19o$ob15obo$ob15obo$19o$2b15o$2bo3b7o3b
o$4bob7obo$8b3o!

p7:

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#C 2023-02-15
x = 18, y = 16, rule = B35678/S4678
8b2o$7b4o$3b12o$4b10o$2b14o$b16o$b16o$18o$18o$b16o$b16o$2b14o$4b10o$3b
12o$7b4o$8b2o!

p9:

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#C 2023-02-15
x = 21, y = 14, rule = B35678/S4678
7b7o$4bob4ob4obo$5b11o$2bob13obo$4b13o$2b17o$21o$21o$2b17o$4b13o$2bob
13obo$5b11o$4bob4ob4obo$7b7o!

p15:

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#C 2023-02-16
x = 41, y = 14, rule = B35678/S4678
10bo19bo$9bobo17bobo$6bob6o13b6obo$4bob10o9b10obo$2bobob29obobo$3b35o$
ob37obo$ob37obo$3b35o$2bobob12o5b12obobo$4bob10o9b10obo$6bob6o13b6obo$
9bobo17bobo$10bo19bo!

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#C 2023-02-17
x = 21, y = 21, rule = B35678/S4678
10b2ob3o$4b3ob6o3bo$2bo3b13o$b17o$2b16obo$ob16obo$ob18o$19o$b19o$20o$
21o$b20o$b19o$2b19o$b18obo$bob16obo$bob16o$3b17o$2b13o3bo$3bo3b6ob3o$
5b3ob2o!

p17:

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#C 2023-02-12
x = 10, y = 10, rule = B35678/S4678
2bo2bo$3b2o$2bob4obo$2b5obo$ob7o$b7obo$bob5o$ob4obo$5b2o$4bo2bo!

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#C 2023-02-16
x = 19, y = 19, rule = B35678/S4678
6bo5bo$5bob5obo$4b5ob5o$4b11o$2b15o$b17o$ob15obo$b17o$b17o$bob13obo$b
17o$b17o$ob15obo$b17o$2b15o$4b11o$4b5ob5o$5bob5obo$6bo5bo!

p21:

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#C 2023-02-15
x = 25, y = 25, rule = B35678/S4678
11b3o$7bob7obo$7bob7obo$6b13o$6b13o$5b15o$3b19o$b23o$3b19o$b23o$b23o$
25o$25o$25o$b23o$b23o$3b19o$b23o$3b19o$5b15o$6b13o$6b13o$7bob7obo$7bob
7obo$11b3o!

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#C 2023-02-16
x = 53, y = 16, rule = B35678/S4678
5b2o39b2o$4b5o35b5o$2bob7o31b7obo$2b11o5b4o9b4o5b11o$2b13o2bo4b2ob3ob
2o4bo2b13o$2b49o$b51o$53o$53o$b51o$2b49o$2b13o2bo4b2ob3ob2o4bo2b13o$2b
11o5b4o9b4o5b11o$2bob7o31b7obo$4b5o35b5o$5b2o39b2o!

p25:

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#C 2023-02-16
x = 51, y = 14, rule = B35678/S4678
10bo13b3o13bo$9bobo2b10o3b10o2bobo$6bob35obo$4bob39obo$2bobob39obobo$
3b45o$ob47obo$ob47obo$3b45o$2bobob39obobo$4bob39obo$6bob35obo$9bobo2b
10o3b10o2bobo$10bo13b3o13bo!

p27:

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#C 2023-02-15
x = 16, y = 9, rule = B35678/S4678
3b10o$4b8o$2b12o$b14o$ob12obo$b14o$2b12o$4b8o$3b10o!

Spaceships

A user named Paintless dentisty just reduced the c/4o in Holstein:

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x = 19, y = 34, rule = B35678/S4678
7bo3bo$b6o5b6o$b4ob7ob4o$bo2bobob3obobo2bo$bobo2bo2bo2bo2bobo$bob2o9b2obo$2bo
2b2o2bo2b2o2bo$2bob2o7b2obo$4b11o$b4o2b2ob2o2b4o$bo3b9o3bo$2o3bobo3bobo3b2o$
2o3bobobobobo3b2o$obo4bo3bo4bobo$2bo6bo6bo$5bobobobobo$3b2o2bo3bo2b2o$2b2obo
2b3o2bob2o$4bob3ob3obo$6bo5bo$6bo5bo$6bo5bo$6bo5bo$5b4obo$5b4o$4b4o$8bo$5b3o$
3b2obobo$4b5obo$3b6o$4b5o$2bobob2o$6bobo!

[confocaloid]

Bounded universes, 16x16 torus

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#C p1376
x = 16, y = 16, rule = B35678/S4678:T16,16
bo4bobob4obo$b6obo6bo$b6obo6bo$8o$b6obo6bo$8o$8o$8o$ob4obobo4bo$o6bob
6o$o6bob6o$8b8o$o6bob6o$8b8o$8b8o$8b8o!

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#C p880
x = 16, y = 16, rule = B35678/S4678:T16,16
obo2bobobob2obo$o6bob6o$o6bob6o$8b8o$8b8o$o6bob6o$8b8o$8b8o$bob2obobob
o2bobo$b6obo6bo$b6obo6bo$8o$8o$b6obo6bo$8o$8o!

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#C p800
x = 16, y = 13, rule = B35678/S4678:T16,16
bo$4bo4bo5bo$8o$ob5ob4ob3o$9ob6o$16o$16o$16o$ob14o$4ob4ob5o$8b8o$bo5bo
4bo$9bo!

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#C p752
x = 16, y = 16, rule = B35678/S4678:T16,16
4b4o4b4o$bobobobobobobobo$4b4o4b4o$b3o3bob3o3bo$2obobo2b2obobo$2obobo
2b2obobo$4o4b4o$4o4b4o$4b4o4b4o$obobobobobobobo$4b4o4b4o$3obo3b3obo$ob
2o2bobob2o2bo$ob2o2bobob2o2bo$4o4b4o$4o4b4o!

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#C p488
x = 16, y = 16, rule = B35678/S4678:T16,16
4bo7bo$3ob2ob4ob2obo$3ob2ob4ob2obo$4o2b6o2b2o$b7ob7o$2bo4bo2bo4bo$2bo
4bo2bo4bo$2o6b2o$4bo7bo$3ob2ob4ob2obo$3ob2ob4ob2obo$4o2b6o2b2o$b7ob7o$
2bo4bo2bo4bo$2bo4bo2bo4bo$2o6b2o!

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#C p440
x = 16, y = 16, rule = B35678/S4678:T16,16
7o8bo$5obo6bobo$3ob2o5bo2b2o$obobo4bobob3o$o8b7o$o8b7o$o8b7o$o6bob6o$
7b8o$5bob6obo$3bo2b5ob2o$bobob4obobo$b8o$b8o$b8o$b6obo6bo!

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#C p384
x = 16, y = 16, rule = B35678/S4678:T16,16
o4b3ob4o$3bo4b3ob4o$8b8o$7b8o$8b8o$8b8o$8b8o$6b8o$b4o3bo4b3o$3ob4o3bo$
8o$7o8bo$8o$8o$8o$6o8b2o!

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#C p368
x = 16, y = 16, rule = B35678/S4678:T16,16
b2o6b2o2$3b3o5b3o2$16o$2o3b5o3b3o$16o$5o2b6o2bo$b2o6b2o2$3b3o5b3o2$16o
$2o3b5o3b3o$16o$5o2b6o2bo!

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#C p240
x = 16, y = 16, rule = B35678/S4678:T16,16
obobobobobobobo$obobobobobobobo$obobobobobobobo$3obo3b3obo$3bob3o3bob
3o$bobobobobobobobo$3bob3o3bob3o$obobobobobobobo$obobobobobobobo$obobo
bobobobobo$obobobobobobobo$obobobobobobobo$obobobobobobobo$obobobobobo
bobo$obobobobobobobo$obobobobobobobo!

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#C p237
x = 16, y = 16, rule = B35678/S4678:T16,16
7ob7o$2ob4ob2ob4o$7ob7o$b4obo2b4obo$2b2obo4b2obo$3bo7bo$o2bo4bo2bo$3bo
7bo$3ob7ob4o$b2ob4ob2ob4o$3ob7ob4o$2o2bob4o2bob2o$o4bob2o4bobo$7bo7bo$
2bo4bo2bo4bo$7bo7bo!

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#C p236
x = 16, y = 15, rule = B35678/S4678:T16,16
16o$3o2b6o2b3o$4ob7ob3o$o2b6o2b5o$5b2o6b2o$o7bo$o6b2o6bo2$16o$3o2b6o2b
3o$4ob7ob3o$o2b6o2b5o$5b2o6b2o$o7bo$o6b2o6bo!

[DroneBetter]

A user named Paintless dentisty just reduced the c/4o in Holstein:

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x = 19, y = 34, rule = B35678/S4678
7bo3bo$b6o5b6o$b4ob7ob4o$bo2bobob3obobo2bo$bobo2bo2bo2bo2bobo$bob2o9b2obo$2bo
2b2o2bo2b2o2bo$2bob2o7b2obo$4b11o$b4o2b2ob2o2b4o$bo3b9o3bo$2o3bobo3bobo3b2o$
2o3bobobobobo3b2o$obo4bo3bo4bobo$2bo6bo6bo$5bobobobobo$3b2o2bo3bo2b2o$2b2obo
2b3o2bob2o$4bob3ob3obo$6bo5bo$6bo5bo$6bo5bo$6bo5bo$5b4obo$5b4o$4b4o$8bo$5b3o$
3b2obobo$4b5obo$3b6o$4b5o$2bobob2o$6bobo!

This is me (the name is a reference to The Simpsons). I didn't want to reply only to say this while turning up emptyhanded, so I had been searching for a c/6 spaceship in ikpx2, but at present it has adaptively widened from logical width 10 in 58.98 MiB (though this does not provide an exhaustive proof of nonexistence, as far as I know), and then reached 121.05 MiB in width 11 (still fruitlessly).

In other news, I don't think any infinite growth patterns are known (per the wiki page). Overnight ikpx2 searching for c/5 spaceships yielded two distinct wickstretcher mechanisms, the smallest of which has a wick three solid cells wide beneath the 2-cell fuzz on either side, while the larger has only two solid cells.

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x = 29, y = 98, rule = B35678/S4678
21b2o$21b2o$19b6o$18b8o$18b8o$17b2o6b2o$17b10o2$15b14o$16b2o8b2o$18bo6bo2$18b8o$20bo2bo$19b6o$20b4o$19b6o$21b2o$20b3obo$19bob2obo$22b2o$20b5o$21bo$21b5o$22bo$22bobo$21bo2$22b2o2$20bobo$20bobo$20b2obo$21bo$20b2ob2o$23bo$19b6o$21b2obo$20b5o$20b5o$21b3o$21b4o$19b2obo$20bob2o$22bo$21bob2o$21bob3o$22b3o$20b6o$22b3o$21b5o$22b3o$20b6o$23b2o$6b2o12b6o$6b2o15b2o$4b6o11b5o$3b8o11b3o$3b8o10b5o$2b2o6b2o11b2o$2b10o8b6o$23b2o$14o7b5o$b2o8b2o9b3o$3bo6bo9b6o$23b2o$3b8o10b5o$5bo2bo14b2o$4b6o11bob3o$5b4o14bobo$4b6o13bo$6b2o15b4o$5b3obo14b2o$4bob2obo12b4o$7b2o14bobo$5b5o11b3o2bo$6bo18bo$6b5o11b4o$7bo14bobo$7bobo13b3o$6bo16bo$24b5o$7b2o17bo$25b4o$5bobo17b2o$5bobo15b4o$5b2obo$6bo15b3obo$5b2ob2o$8bo14b5o$4b6o14b2o$6b2obo13b4o$5b5o13b3obo$5b4obo11bob3o$4bob4o13b4o$5b6o11b4o$4bob3o15bob2o$6bo2bo13bobo!

[DroneBetter]

I created a new section in my userspace page, User:DroneBetter/qfind results#B35678/S4678_(Holstein), in which I found minimal-width odd- and even-symmetric spaceships for c/3, c/5 (for which the width-15 odd is in fact 100 cells, as opposed to the known width-14 even's 110), 2c/6 and c/7 (new speed :-), and that there is no odd- or even-symmetric c/4 thinner than the known width-21 one (though I am away from my most powerful computer so have not found its length to be minimal), here is a collection for it

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x = 173, y = 69, rule = B35678/S4678
3b5o3b5o8b5o3b5o8b5o4b5o8b5o4b5o8b5o5b5o7b5ob5o8b4o10b9o10b4o$2b3ob3ob3ob3o6b3ob3ob3ob3o6b3ob3o2b3ob3o6b3ob3o2b3ob3o9b4obob4o11b2obob2o8b8o7b4o3b4o8b6o$4b3o2bo2b3o10b3o2bo2b3o10b2o3b2o3b2o10b2o3b2o3b2o7b4ob2obobob2ob4o4b13o5b8o7b11o8b6o$2b3o4bo4b3o6b3o4bo4b3o6b2ob3o4b3ob2o6b2ob3o4b3ob2o5bob2obobobobobob2obo6b9o8bo4bo8b4o3b4o6b3o4b3o$b2ob5ob5ob2o9bo5bo9b3o5b2o5b3o4b3o5b2o5b3o4bobo4b2ob2o4bobo3b3o3b3o3b3o3b10o5bob4ob4obo3b4obo2bob4o$3b2o3b3o3b2o8bo2b3ob3o2bo6b4obo6bob4o4b4obo6bob4o6bo3bo5bo3bo9b3ob3o8bo6bo6bobo7bobo5b3o4b3o$4o2b7o2b4o7bob2ob2obo8b2o4b2o2b2o4b2o4b2o4b2o2b2o4b2o5bob3o7b3obo5b3o3bo3b3o4b10o5b2ob2o3b2ob2o4b12o$2o3bobobobobo3b2o4b3ob7ob3o4b4obo8bob4o4b2obo8bob2o9b3o5b3o24b2o8b2o22b4o2b4o$3bo3b5o3bo12bobobo12bo4bo2bo4bo10bobo4bobo9bobo2b7o2bobo11bo8b2obob4obob2o3b3o7b3o3bob2obo2bob2obo$ob2obo2bobo2bob2obo2bobo3b7o3bobo5bo2b2o4b2o2bo8b4o6b4o10b11o10b3o3b3o6bo8bo7b3o3b3o8bobo2bobo$b3o11b3o9bobobobo14bo2b2o2bo11b5ob2ob5o7b3o5bo5b3o9b5o9bob4obo7b3obobob3o4b2ob2o4b2ob2o$3b3o3bo3b3o5b3o2bob5obo2b3o4bob2ob2o2b2ob2obo5b7o4b7o4b3obob3ob3obob3o6bo3bo3bo23b4ob4o7b2o6b2o$3obo2b2ob2o2bob3o4b2obobo3bobob2o7bobo3b2o3bobo7b3o10b3o4b2obo3bo5bo3bob2o5b9o7b8o6b13o3b5o4b5o$b5o3bo3b5o3bo3bobob3obobo3bo3b2o3bo2b2o2bo3b2o4b5o8b5o5bo4bo5bo4bo9bo3bo12b2o11b9o5b3obo4bob3o$3obo2bobobo2bob3o3bo3b2o5b2o3bo3b2obo3b6o3bob2o3b3o12b3o5bo4bo5bo4bo8bob3obo9b6o8b11o4bobo8bobo$2b3o3bobo3b3o4b2ob2ob7ob2ob2o4b2o3bob2obo3b2o5b6ob4ob6o7b2obo5bob2o11bobobo10b6o9b9o7bobo4bobo$6obo3bob6o3b3ob9ob3o11b4o12b5o6b5o7bob2o2bobo2b2obo11b3o10bob4obo8bob5obo5b2obo6bob2o$b5obo3bob5o5bobo2b5o2bobo10bobo2bobo11b14o9b2o2b5o2b2o10b7o8bob4obo10b2ob2o8b3obo2bob3o$ob5o5b5obo4bo2b9o2bo12b4o14bo10bo10bo3bobobo3bo12b3o26bo2bobo2bo4b16o$b4o2b5o2b4o6b2o3b3o3b2o31bobob4obobo14bo3bo16b3o9b2ob4ob2o20b5o6b5o$ob5o5b5obo9b5o37bo6bo14bobo3bobo42bobo5bobo4b3ob6ob3o$b5o2b3o2b5o5bo2bo2b3o2bo2bo35b2o13bo3bobo3bobo3bo24b6o22b2ob2o6b2ob2o$ob3o2bo3bo2b3obo6b11o53b6o3b6o38b4o5b4o8b4o$3ob2o7b2ob3o3b2ob11ob2o34b2o12b3ob2obo3bob2ob3o25b2o26bobo6bobo$2bo13bo7b13o50bo3bo9bo3bo37b4obob4o7b8o$22bo2b11o2bo50b5o5b5o40b3obob3o6bo3b4o3bo$26b3o3b3o56b3o5b3o40b13o4b5o2b5o$25bob3ob3obo55b5ob5o42b9o9bob2obo$26bo7bo56b3o5b3o40b13o5b10o$93b2o3b2o44b9o9bo4bo$90b4ob3ob4o41b9o8b8o$90b3o7b3o43b5o12bo2bo$92b4ob4o46b3o10b10o$94b2ob2o65b2o$94b5o62bobo2bobo$93b7o$95b3o62b2o2b2o2b2o$95b3o61b3obo2bob3o$95b3o62b10o$158bob10obo$96bo61bob10obo$94bobobo60b12o$94b5o61b10o$92bobobobobo60b8o$94bobobo61bobob2obobo$94b5o$94b5o62b8o$93bob3obo$94b5o61b10o$92bob5obo61b2o2b2o$92bob5obo59b2ob4ob2o$90b13o60bo2bo$90b13o$89b15o$89b15o$89b6obob6o$90b5obob5o$90bobo7bobo$90bob3o3b3obo$90bobo7bobo$90bob3o3b3obo2$93b7o$92b2obobob2o$94b2ob2o$92b3obob3o$94bobobo$92b3obob3o$94b2ob2o!

despite my c/7 I have not found a c/6, and suspect that in general, of speeds with odd displacements, those with odd period will be easier to find in this rule, since spaceships seem often to contain internal lateral zebra stripes (which strobe unlike Life, due to B6 and no S2), so even period would require working without them or phase-inverting them each period

[DroneBetter]

some more results from another year of occasional sporadic searching:
the thinnest possible c/3, which seems to be the smallest of any speed by population

Code: Select all

x = 22, y = 7, rule = B35678/S4678
3b3o2b6o2b3o$b5ob3o2b3ob5o$b7o6b7o$ob2obo3bo2bo3bob2obo$bob5o6b5obo$bobo14bobo$4b2o10b2o!
[[ TRACK 0 -1/3 ]]

also, a c/5 diagonal; the first diagonal spaceship and also the fastest known in the taxicab metric (though a 2c/5 orthogonal seems most likely impossible (at least at true period); LLSSS has disproven up to width 128, though attempts at outright proofs seem to just grow indefinitely until they're killed for excessive RAM use)

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x = 27, y = 27, rule = B35678/S4678
7b5o$6b2ob4o$8b6o$10b3o$7b7o2bo$6b2o4bobobo$bo3bob6ob3o$2o2b6o$obobob12o$3obob5obo4bobo$5obob7o2b3obo$5obobobo3bob4obo$b6ob3obobob6obo$2bobo3bobo6b3o3bo$5b2obob3o5b2ob6o$6bobo13b2obo$4b3obo2b2o7bob3obo$8b6o8b4o$10b5o5b4o$9b6o5b4o$12bo3bob2obo$10b3obo3b4o$14b6o3bo$12b8o2bo$14bob2o$14b2obo$14bobo!
[[ TRACK -1/5 -1/5 ]]

a c/7d wickstretcher; my week-long 128×128 rlifesrc search was getting congested by trying to find backends for it, but thankfully in an inert reference frame the wick is p2 and easily completable

Code: Select all

x = 101, y = 101, rule = B35678/S4678
8b2o$8b3o$7b4o$6b3obo$4bob3o$6b3o$3b3ob2obo$2b7obo$8obobo$3o5b2obobo$b3o2b2o3bobobo$8b6o$11b4obo$9b4o$12bo2b2o$10bo3bo3b3o$12bobo2bob2o3bo$16b5obobo$15bob2obobobobo$15b3ob2obobo2bo$15b5o2b6o$21bob3o2bo$17b4obobobobo$20b2o2b2o2b2o$16b8o3bobo$20b2obobo3b3o$18bobobo3b2o4b2obo$19b2o3bob6obo$21b3o3bob7o$23b3ob9obo$25bob9obobo$25bob9obobo$26bob8obobo$26b12obo$28b10obo$26bob12o$33b9o2bo$29b12obo$35b6o2b2obo$30b11o$36b11o$36bo3b5o2bo$37bo2b11o$38bob9ob3o$36bobob11o$40bob14o$38bobob14o$41b16o$42b15obo$42bob13o$42b15o$43bob14o$43bob14o3bo$45b17o$45b18o$45b17ob2o$47b15o3bo$51b16o$48bo2b12obo$53b12obo$53b8obo2b4o$53b7obob6o$52bobo2b4o5bo$55bobobobobob5o$55bob3obo3bo2bo$56b2o2b2ob3obo$57bob5o3bo3bobo$60b2obob2o4bobobo$60b2ob2o3b2obo4bo$63bo4b4ob3o$69bobo2bob2o$66b7o2b4o$71bob2ob2obo$66b2obo2bo3b2obo$69b2obo5bob2o$67bobobo3bob4ob3o$68bob4o3b5o$70b4ob9o$71bo2b7o$72b2ob6obo$74b7obobo$74bob2o3b3o$75bobob3obo$75bobo3b2o2b2o$75bo4bo3b2o$83b2obobo$83bob7o$86b5o$85b8o$86b3obo$86b10o$86bobobo2bo$88bobob5o$90b4obobo$90bobob2obo$90bob4o3bo$92bo6bo$93b2o5bo$99bo$95b2ob2o$97bo!
[[ TRACK -1/7 -1/7 ]]

[LaundryPizza03]

Some spaceships and oscillators in the non-strobing vertical-stripe dual of this rule.

Code: Select all

x = 99, y = 42, rule = MAPBElJlkmWlm1JlpZtlm1t3wAEBEkESUmWBElJlkmWlm1JlpZtlm1t35Ztbd9t39//BElJlkmWlm1JlpZtlm1t3w
41bo9bo6bo17bo19bo$41bo11bo2bo17bobo18b2o$41bo9bobo2bobo14bo18bobobobo
$41bo9bo2b2o2bo17bo14b3ob3o$49bobob4obobo15bo15bo2b2o$37bobo7bo2b2o2b
2o2b2o2bo11bo3bo10bobo2bo$48bobob6obobo9b2o3b2o15bo$37bo12b2ob4ob2o10b
2obobo12bo2b2obo$50bob2o2b2obo14bo12bob3obobo$48bobo3b2o3bobo6bo4bobo
12bob3o$67bo17bobobobobobo$48bo4b4o4bo9bobo10bobobobobobo$27b2o24bo2bo
15bo10bobobobobo$52b2o2b2o13bo12bobobobobo$53bo2bo24b3obo3bo$27bo25bo
2bo27bo3bobo$28bo21bobob2obobo19bo3bo$49b2ob6ob2o23bo$54b2o26bo$52b2o
2b2o23b3o$27bobo21bobo2bobo21bobo$28bo54bo$51bo6bo23bo$81bobo$27bo$29b
o$29bo$27bo4$9b2o17bo$10bo10b2o2bo$7bobo2bobo5bo2bobo4bo$5bo3bobo3bo2b
o2bob3ob2o$2bobo9bobobo5b5o$bo9bobo4bobobob3o$o14b3obo4bobo$9bo6bob2ob
o$9bobo3b2ob2obo$9bobob2o$13bo!

Equivalently:

Code: Select all

x = 128, y = 128, rule = B35678/S4678:T128,128
$128o2$128o2$128o2$128o$46bo$47ob80o$48bo$128o$48bo$49ob78o2$50ob77o2$
43o3b3o2bob75o$51b2o$43o2b2obob78o$50bo$42o2b3ob80o$43bo4bobo$44obob2o
b78o$44b3obo$46ob81o$44b2ob3o$44o3b81o$47bo2bo$44o2b3obob76o$47bo3bo$
49o2b77o$46b4o$47o5b76o$46b3o$48o2b6ob2ob3ob4ob2ob56o$48b2o2bo9bo4bo3b
o$57o2b4ob64o$50bo$128o2$128o2$128o2$76obob49o2$78ob49o2$80o4b44o2$
128o2$128o2$78ob49o$72bobo2bo$66ob12ob48o$66b2o6b5o$61obob12obo4bob44o
$64bob2o2bo4bobo$63o2bobo5b2o3bobo2b45o$65b3o4bobob5o$65o3b4obobo5b47o
$63b2obob5o2b3obob2o$64obo2b2ob4obob50o$61bobo12bob4obo$66o2b6o5b49o$
66bo12bo$72obob2ob50o$78bo$128o2$128o2$73ob54o$74bo$128o$76bo$70obob3o
2b50o$71bo5bo$72obobob4ob46o$75bo2bo3bo$74obob51o$77bob2ob2o$77ob50o$
78bo$67ob60o$63bo$60ob3ob4ob58o$61bob3o3bo$60obobob2obobob56o$66bobobo
bo$69obobob54o$70bobo$71obobob52o$68bobobobobo$69obobobob2ob49o$68bobo
bob2o$71obo4bobob47o$70bobobobo2b3o$73obobob2ob47o$72bo3bobo2bo$73obob
3o2bob45o$79b5o$80ob47o$81bo$128o2$128o2$128o2$128o2$128o2$128o2$128o
2$128o2$128o2$128o2$128o!

[LaundryPizza03]

(2,1)c/4 found by sylvani on april 12 (original post):

Code: Select all

x = 16, y = 16, rule = MAPBElJlkmWlm1JlpZtlm1t3wAEBEkESUmWBElJlkmWlm1JlpZtlm1t35Ztbd9t39//BElJlkmWlm1JlpZtlm1t3w
2bo$o$o$o$bob4o$2b5obo$3b6obobo$bob4o3bobobo$3bo2bo$14bo$8bo$9bo$9bobo
$7bobobo$9bobo3bo$11bo!

[Disaster16439]

(2,1)c/4 found by sylvani on april 12 (original post):

Code: Select all

x = 16, y = 16, rule = MAPBElJlkmWlm1JlpZtlm1t3wAEBEkESUmWBElJlkmWlm1JlpZtlm1t35Ztbd9t39//BElJlkmWlm1JlpZtlm1t3w
2bo$o$o$o$bob4o$2b5obo$3b6obobo$bob4o3bobobo$3bo2bo$14bo$8bo$9bo$9bobo
$7bobobo$9bobo3bo$11bo!

I'm confused. That map rule doesn't seem to correspond to anything, and indeed, switching the rule to B35678/S4678 makes it fail?

[WhiteHawk]

(2,1)c/4 found by sylvani on april 12 (original post):

Code: Select all

x = 16, y = 16, rule = MAPBElJlkmWlm1JlpZtlm1t3wAEBEkESUmWBElJlkmWlm1JlpZtlm1t35Ztbd9t39//BElJlkmWlm1JlpZtlm1t3w
2bo$o$o$o$bob4o$2b5obo$3b6obobo$bob4o3bobobo$3bo2bo$14bo$8bo$9bo$9bobo
$7bobobo$9bobo3bo$11bo!

I'm confused. That map rule doesn't seem to correspond to anything, and indeed, switching the rule to B35678/S4678 makes it fail?

Read the preceding posts. This is a simulation of the rule if the plane was covered in a p2 (EDIT:stripes) agar.

[LaundryPizza03]

(2,1)c/4 found by sylvani on april 12 (original post):

Code: Select all

x = 16, y = 16, rule = MAPBElJlkmWlm1JlpZtlm1t3wAEBEkESUmWBElJlkmWlm1JlpZtlm1t35Ztbd9t39//BElJlkmWlm1JlpZtlm1t3w
2bo$o$o$o$bob4o$2b5obo$3b6obobo$bob4o3bobobo$3bo2bo$14bo$8bo$9bo$9bobo
$7bobobo$9bobo3bo$11bo!

I'm confused. That map rule doesn't seem to correspond to anything, and indeed, switching the rule to B35678/S4678 makes it fail?

Read the preceding posts. This is a simulation of the rule if the plane was covered in a p2 checkerboard agar.

No, it's for a stripe agar.

[MDA]

I assume this is the correct place to post a proof against the existence of c/2o spaceships in Holstein for future reference:

Suppose a c/2o spaceship exists in Holstein.

1. We will, for the moment, assume the spaceship has only one cell in its inactive phase (immediately after advancing one cell forward).

We can represent such a frontend by the shape

Code: Select all

x = 3, y = 2, rule = B35678/S4678Super
.A$3B!

where the blue cells correspond to the second row.

For this spaceship to reappear one cell forward, there must be a suitable B3i arrangement of cells to cause a birth in the proper cell.

Thus, the next generation of the pattern must have a B3i configuration, which means the front cell of the spaceship must survive from one cell to the next. However, it can only have 3 neighbors for the lateral cells to be born, resulting in a contradiction, since there is no S3 condition in Holstein:

Code: Select all

x = 31, y = 12, rule = B35678/S4678Super
15.D$15.D$15.D$13.D.D.D$14.3D$15.D2$9.D13.D5.A$.A8.D3.ADA7.D3.3B$3B2.
7D2.3B2.7D$10.D13.D$9.D13.D!

This necessaily implies that if any c/2o spaceships exist in Holstein, they must have two front cells in their inactive phase.

2. We will now disprove the existence of c/2o spaceships in Holstein which have two front cells in their inactive phase.

We can reapply our initial reasoning for section 1 to represent this frontend by the pattern

Code: Select all

x = 34, y = 5, rule = B35678/S4678Super
10.D14.D5.2A$.2A8.D3.4A7.D3.4B$4B2.7D2.4B2.7D$11.D14.D$10.D14.D!

We can then conclude that it is impossible for the two front cells to survive with 6 or more neighbors, leaving 4 as the only possibility. Thus, the frontend of the first generation of this c/2o spaceship must be:

Code: Select all

x = 4, y = 2, rule = B35678/S4678Super
.2A$4I!

Updating our frontend map:

Code: Select all

x = 36, y = 5, rule = B35678/S4678Super
10.D14.D6.2A$.2A8.D3.4A7.D4.4I$4I2.7D2.4B2.7D2.6B$11.D14.D$10.D14.D!

From this diagram, we can conclude that every front cell in generation 2’s frontend must survive to the next generation. Again, we can prove that it is impossible for such cells to survive on 6 or more neighbors, leaving 4 as the only possible choice. Specifically, for the leftmost and rightmost front cells, S4a is the only way to support them. However, this is a contradiction, because, as the pattern below demonstrates,

Code: Select all

x = 22, y = 5, rule = B35678/S4678Super
12.D$13.D5.2A$.A2UA3.7D3.I2BI$2I2K2I7.D$12.D!

the middle green cells have the S5i configuration, which is not a valid survival condition in Holstein. Therefore, no c/2o spaceships can exist with 1 or 2 cells in their inactive phase’s frontend.

3. We can prove that it is non-viable for any inactive frontend to have more than 3 front cells in a line, due to B3i, which advances the frontend, disqualifying it from being inactive. We can also prove the nonexistence of c/2os with more than 2 frontends separated by 1 or 2 blank cells (for 3 or more blank cells, the reasoning is identical to cases 1 and 2) by observing that:

- With 2 frontends one cell apart, the only way to trigger B3a on each frontend triggers B5i in the middle, making such a frontend pair advance more than one cell in its period, thus making it impossible for such a frontend pair to originate a c/2o spaceship.

- With 2 frontends 2 cells apart, the only way to trigger B3a on each frontend triggers B4a on both cells in the middle, making such a frontend pair advance more than one cell in its period, thus making it impossible for such a frontend pair to originate a c/2o spaceship.

4. The reasoning for trios and further combinations of frontends (including multiple offset frontends) is identical to section 3 and none originate valid c/2o spaceships.

Conclusion: No c/2o spaceships exist in Holstein.

[NNlk05]

*snip*
Conclusion: No c/2o spaceships exist in Holstein.

If I am not mistaken (Which is likely, either by my horrible memory or mixing up the countless search programs with the format L*nS*m), we can prove the same fact using LLS.

[MDA]

[…]

If I am not mistaken […], we can prove the same fact using LLS.

This is probably true, but, in my opinion, the kind of proof I posted is more explicit and is more intuitive to others, especially newcomers, than “There are no c/2o spaceships because LLS(SS) says so”.

Edit (June 20th, 2026, 9:57 PM):

[…]

If I am not mistaken (Which is likely, either by my horrible memory or mixing up the countless search programs with the format L*nS*m), […]

I think you meant LLSSS. I don’t think you could exhaustively disprove c/2o spaceships in Holstein using LLS other than trying out different bounding boxes and checking whether they return UNSAT or not. As for LSSS, it has been mentioned multiple times that it is buggy and unreliable at times, so it wouldn’t be the ideal tool for disproving c/2os in Holstein, thus leaving LLSSS as the only valid option, especially since it is frequently used for spaceship proofs and disproofs.

[I6_I6]

c/4 diagonal and c/2 orthogonal spaceships are impossible, c/3 is the orthogonal speed limit and c/5 is the diagonal speed limit. [3]

, citing MDA's proof against c/2os, but where's the evidence behind the other statements? Couldn't there be speeds between c/2 and c/3o or between c/4 and c/5d?