[Game] Most number of minpop 4-cell oscillators

[H. H. P. M. P. Cole]

I have created a game where everyone can participate.

You must create the isotropic non-totalistic rule with the most number of minpop 4-cell oscillators as possible. Still-lifes are counted as oscillators with period 1.

If you find a rule with minpop 4-cell oscillators and (one or more) minpop 4-cell spaceships which satisfy the rules below, let me know!

RULES
- No cheating (no 2-cell B2a replicators or B/Sx rules)
- No infinite series
- No rules where the only 4-cell stable objects are still-lifes
- No objects with minpop lower than 4 in the rule
- No explosive rules
- No 4-cell predecessors of other non-minpop-4 oscillators

I'll start.

6 oscillators (7 configurations) (2 p1s, 4 p2s)

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x = 18, y = 10, rule = B3aeijn4t/S2c3aejky
2o4bo$2o3bobo$6bo4$2o3bo$b2o3b2o2b3o2b3o$6bo4bo$16bo!

[Plasmath]

9 oscillators:

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x = 34, y = 18, rule = B2cik3aejry4aw/S2ci3aeijk4i6i
bo$obo3b2o$bo4b2o3$25bo$12bobo4bo7bo5bo$3o3b2o11b2o3bo6b2o$bo5b2o3bobo
3bo7bo6bo8$obobobo!

2 p1s, 6 p2s, 1 p4

[Entity Valkyrie 2]

Unfortunately, this breaks rules 2 and 4.

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x = 69, y = 43, rule = B2c3ar4in6k/S012ai3jkr4ik
2o26bo2$bo6$2o34bo2$bo6$2o42bo2$bo6$2o50bo2$bo6$2o58bo2$bo6$2o66bo2$bo
!

[toroidalet]

If you find a rule with minpop 4-cell oscillators and (one or more) minpop 4-cell spaceships which satisfy the rules below, let me know!

11, or 12 counting a c/5d spaceship:

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x = 43, y = 48, rule = B2-an3enry4c5ak6a7c/S1c2-en3-cny4ciw5c7c
6bo7b2o3b3o3b2o6bo$14b2o4bo5b2o5b2o$32bo4$5b2o6bobo6bo4bo5bo6b2o$20b2o
3b2o5bobo4bo2bo$13bobo3bo5bo7bo8$14bo$4b3o8bo$12bo$13bo7$12bo$3b4o7bo$
11bo$13bo6$13bo$12bo$2b5o4bobo8$12bo$11bo2bo$6o7bo!

Embarrassingly, I only found the p3 beacon in a soup while trying to confirm that the rule was not explosive and had no 3-cell objects.

By the way, I think it's very unlikely that a rule without any <4-cell objects would have adjustable things (except maybe guns). In order to make an adjustable thing, we would need to have at least two separate components. These must both consist of two cells (if one were 3, the other would be 1, implying the dot was a still life), and the only ways for a two-cell pattern to expand without S0 involve B2a (duoplet, domino, 2k-dots and hollow blinker/2i-dots). All of those except the hollow blinker are out for producing oscillators or spaceships. Since it is a high-period oscillators or counter in many B2a rules, there might be some way to put two hollow blinkers in the plane to make an adjustable oscillator, which would be quite impressive in and of itself.

EDIT: Or it could be in a B0 rule, I guess. Speaking of B0 rules, what happens if a B0 rule has oscillators with a minimum population of 4 cells in odd generations and some in evens? Ugh.

EDIT2:
12 in a rule without the spaceship:

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x = 43, y = 55, rule = B2-an3enry4c5ack/S1c2-en3-cny4ciz5k6e7c
6bo7b2o3b3o3b2o6bo$14b2o4bo5b2o5b2o$32bo4$5b2o6bobo6bo4bo5bo6b2o$20b2o
3b2o5bobo4bo2bo$13bobo3bo5bo7bo8$14bo$4b3o8bo$12bo$13bo7$12bo$3b4o7bo$
11bo$13bo7$12bo$11bo2bo$6o7bo12$15bo$b6o7bo$b6o6bo$b6o5bo$b6o!

Coincidentally, every 4-cell oscillator period in this rule is a factor of the highest such period (24).

[H. H. P. M. P. Cole]

Cool! What did you use to find it?

[toroidalet]

I found it with manual search, I'm afraid. While playing around with different rules to make it the 11th object, I noticed that, in one rule, in generation 5 the three corner cells were forced to be on and it would be easy to make the middle cell turn on as well. But there were some sparks from B4w, so I had to change the leftmost p4 oscillator, which coincidentally made the p3 beacon work as well.

Essentially, the process was to use the draw tool to manually add transitions and hope the result looked like it could be modified to be the original pattern (that's how I found the p6 and p24 as well).

[Entity Valkyrie 2]

EDIT: Or it could be in a B0 rule, I guess. Speaking of B0 rules, what happens if a B0 rule has oscillators with a minimum population of 4 cells in odd generations and some in evens? Ugh.

I have no idea how B0 rules would work with this... maybe in odd generations, that would be ω^2 − 4 (a "surinteger") cells? (though people have criticized my improper use of ω in the past)

ω^2 − 8ω + 12.png (63.44 KiB) Viewed 196 times

[H. H. P. M. P. Cole]

By the way, I think it's very unlikely that a rule without any <4-cell objects would have adjustable things (except maybe guns). In order to make an adjustable thing, we would need to have at least two separate components. These must both consist of two cells (if one were 3, the other would be 1, implying the dot was a still life), and the only ways for a two-cell pattern to expand without S0 involve B2a (duoplet, domino, 2k-dots and hollow blinker/2i-dots). All of those except the hollow blinker are out for producing oscillators or spaceships. Since it is a high-period oscillators or counter in many B2a rules, there might be some way to put two hollow blinkers in the plane to make an adjustable oscillator, which would be quite impressive in and of itself.

I'm not good with B2a rules, so here's my answer in the non-relativistic realm:

If you include my restriction that there should be no other objects with minpop < 4, your 4-cell adjustable osc challenge is impossible.

I have worked out the possible transitions, and concluded that in a rule where no minpop <4 objects exist, B2e and S1c must coexist, B2c implies B2i (but B2i can exist without B2c), and S0 and S1e are impossible as they imply the dot and domino still-lifes, respectively. Taking 3-cell objects into account is far too difficult.

All sequences starting with 2 cells using the constraints above will die or form these objects:

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x = 2, y = 2, rule = B/S3a
2o$2o!

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x = 1, y = 1, rule = B2ci3i/S
obo!

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x = 3, y = 1, rule = B3i/S2i
3o!

If this is what you mean by 4-cell adjustable things, then this is my answer. If you mean >4-cell adjustable things, this is getting less into the realm of rulegolfing and more into the realm of engineering. But I digress.

Here's my result:

13 oscillators (p1, p2, p4, p6, p10)

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x = 22, y = 35, rule = B2-an3ceqry4c5a/S1c2-en3-n5a6a7c
2o4b2o4bo5b3o$2o5b2o4b2o4bo$13bo4$2bo12bo3bobo$2o5b2o4b2o$o5bo2bo2bo6b
obo5$bo$o$3bo$2bo3$bo$3bo2bobo$o6bo$2bo5bo4$obo$bobo5$2bo$o3bo$2bo!

EDIT: Why don't we just use EnumPattEvo, evolution mode, on an RLE of all 4-cell clusters? That will speed the process up!

EDIT2: Why don't we do this with LtL?