n-in-m: A Collection of Efficient Synthesis Projects

[bubblegum]

We've sprinkled the Patterns forum with a few threads on synthesising N-bit SLs (in M gliders optionally, where M is usually N or N-1), and since the ones that take the most effort usually run synchronously, I've decided to create a collective thread for them.
this is a terrible idea why did i ever think this was okay
Only strict SLs will be documented here, as not much other data tends to get recorded that well.
Top cost list:

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4:3
5:2
6:4
7:4
8:4
9:5
10:5
11:7
12:7
13:8
14:10
15:12
16:14
17:16
18:30
19:61
20:120
21+:∞

Links to well-defined past/ongoing projects:
12 in 7: https://conwaylife.com/forums/viewtopic ... 35#p125335
15 in 15: https://conwaylife.com/forums/viewtopic.php?f=2&t=2441
16 in 16: https://conwaylife.com/forums/viewtopic.php?f=2&t=2642
17 in 17: https://conwaylife.com/forums/viewtopic.php?f=2&t=3962
18: https://conwaylife.com/forums/viewtopic.php?f=2&t=1467
19: https://conwaylife.com/forums/viewtopic.php?f=2&t=4156
20: https://conwaylife.com/forums/viewtopic.php?f=2&t=4577

iNoMed got a 7G xs12_1784c826, and I got 7G xs12_0j96z346 and 6G xs12_31eg8426, so we discussed a bit in Discord DMs and decided on trying 12-in-7.

Wow, okay, we're already done?

[MathAndCode]

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4:3
5:2
6:4
7:4
8:4
9:5
10:5
11:7
12:8
13:8
14:10
15:12
16:14
17:16
18:30
19:71
20:126
21+:∞

I think that there's a typo. You probably meant 20:136, not 20:126.

[bubblegum]

Yes, I did.

[MathAndCode]

The 33-glider synthesis for xs19_gs2pd2kozx32 looks unnecessarily expensive, the entire right side is only necessary for cleanup and dropping a blinker. I plan to improve it, which will reduce the glider cost of xs19_3p68og4czw1226, which is currently the most expensive nineteen-cell still life.



Edit: It's even worse than that. The region that the modified pi sequence from the right "cleans up" would have died quickly anyway.



Another edit: I have reduced its cost by twelve gliders.

[Entity Valkyrie 2]

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4:3
5:2
6:4
7:4
8:4
9:5
10:5
11:7
12:8
13:8
14:10
15:12
16:14
17:16
18:30
19:71
20:136
21+:∞

I'm seeing those ratios, and I can't stop thinking them in terms of justly intonated musical intervals:

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perfect fourth
octave + major third
perfect fifth
harmonic seventh
octave
just minor seventh
undecimal minor sixth
perfect fifth
tridecimal neutral sixth
narrow tritone
septimal major second
large septemdecimal semitone
major sixth [down]
(octave + 12TET major seventh −18 cents [down])
(two octaves + septemdecimal major sixth [down])

[iNoMed]

xs12_4ai3zx1ac (Remarkably long hook with tail) in 7G (Down from 8G), by figuring out a way to avoid the loaf-cleanup without incurring any additional gliders:

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x = 99, y = 21, rule = B3/S23
95bo$78bobo12b2o$79b2o13b2o$79bo2$98bo$6bo89b2o$5bo91b2o$5b3o$obo$b2o
37b2o39b2o$bo37bobo38bobo$39bo7bo32bo5b2o$38b2o5b2o32b2o4bo2bo$46b2o
37bo2bo$86b2o3$46b3o$46bo$47bo!

9 eight-glider xs12s remain.

[bubblegum]

Update: 7G xs12_ci52zw1246 by Kazyan and Blinkerspawn, and 7G xs12_drz1226 by Kazyan

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x = 68, y = 65, rule = B3/S23
35bo$36bo$34b3o3$38bo$32b3obobo$34bo2b2o$33bo6$26bo$27bo$25b3o4$22b3o
5b3o$24bo7bo$23bo7bo$62b2o$62bobo$62bo25$61bo$61bobo$61b2o3$46bobo$bo
40bo3b2o$2bo5bobo32bo3bo$3o5b2o31b3o$9bo55b2o$65bobo$3bo50b2o9bo$3bobo
48b2o$3b2o$46b3o!

[Kazyan]

Integral with cis-hook in 7G:

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x = 27, y = 26, rule = B3/S23
15bo$14bo$obo11b3o$b2o$bo16bobo$18b2o$19bo2$15bo$16b2o$15b2o6$9b2o$8bo
bo$10bo3$6b3o$8bo$7bo17bo$24b2o$24bobo!

[AGreason]

It may also be useful to list how many still lives of a given bitcount (20 and below) saturate that bound - i.e. how many synths would need to be improved to lower the cost limit for that bitcount.

EDIT: Also, reduced xs20_0bqi4oz320254c to 124 (synth boxed) so that's the new 20-bit limit.

[bubblegum]

It may also be useful to list how many still lives of a given bitcount (20 and below) saturate that bound - i.e. how many synths would need to be improved to lower the cost limit for that bitcount

That may not be feasible to maintain manually, given it'd probably need to be updated a fair percentage of the times a synthesis is bøxed.

[Kazyan]

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x = 100, y = 41, rule = B3/S23
10bo$8bobo$9b2o12$o$b2o$2o12$96b2o$97bo$95bo$20b2o15bo57b2o$19bobo6bo
7b2o58bo$21bo7bo6bobo57bob2o$27b3o67bobo$92b2o$92b2o$27b2o$28b2o60b2o$
27bo61bobo$91bo!

[Goldtiger997]

xs12_3123c4go and xs12_8o6413z32 in 6G and 5G respectively:

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x = 69, y = 20, rule = B3/S23
8bo57bobo$6bobo57b2o$bo5b2o58bo$2bo$3o62b3o$65bo$66bo$56b3o$58bo$21bo
35bo$11bobo6bo$12b2o6b3o$12bo4$4bo42b3o$4b2o21b2o20bo13bo$3bobo21bobo
18bo13b2o$27bo34bobo!

I believe that means 12 in 7 is done!

[bubblegum]

14-in-9 seems possible. Here are the six remaining 10G 14-bitters:

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x = 98, y = 10, rule = B3/S23
b2o18b2o5bobo28b2o3bo7b2o9b2o8b2o$2bo18bobo3bob2o28bo3bobobo4bo4b2o4b
o10bo$bo22bo2bo32bo2bo2b2o6bo2bo6bo8bo$2b3o20bobo33bobo9b2o3bo6bo7b2o
$5bo20bo35bo12b3o6b2o9bo$4bo70bo10b2o7bo$3bo82bo9b2o$obo84bo9bo$2o86b
o7bo$87b2o7b2o!

Fun.

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x = 32, y = 32, rule = B3/S23
17bo3bo3bobob2o$18b4o3b4o$18b3o3b2obo3bo$17b4o2b2o3b2o$17b2obobobo3bob
2o$17b2o3bob5ob2o$16bo2bobob2ob5o$16b8obo4b2o$21bo5bo2bo$16bob3ob2o2bo
4bo$16bob4ob3o2bobo$16b4ob2ob2o4b2o$21b6ob3o$18b3o2b2o3b4o$16bo3bo3bo
2b3obo$16b2o6b2o$6b2o6b2o$ob3o2bo3bo3bo$4o3b2o2b3o$b3ob6o$2o4b2ob2ob4o
$bobo2b3ob4obo$o4bo2b2ob3obo$bo2bo5bo$2o4bob8o$b5ob2obobo2bo$2ob5obo3b
2o$2obo3bobobob2o$2b2o3b2o2b4o$o3bob2o3b3o$3b4o3b4o$b2obobo3bo3bo!

Here's an 8G xs14_wg853zgb6z11, synthesis by Kazyan with the first 1G cleanup that turned up (which happened to be mine):

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x = 39, y = 46, rule = B3/S23
bo$2bo$3o14$30bo$29bo$29b3o12$22bo$20bobo13b3o$21b2o9b2o2bo$31b2o4bo$
33bo$22bo$20bobo$21b2o2$23b3o$25bo$24bo2$9b2o$10b2o$9bo!