Project OmniOsci

[muzik]

Is it at all possible to reduce the bounding boxes for the nrc and nrk examples?

For nrc, the p4, p8 and p12 oscillators we have all neatly fit within a 3x3 bounding box.

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x = 2, y = 3, rule = B2ce/S1c3n
2o$o$bo!

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x = 2, y = 3, rule = B2e3aij/S12k4r
o$bo$bo!

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x = 2, y = 3, rule = B2e3ainr/S014r
2o2$bo!

However, for our current p16, the size jumps up all the way to 7x7:

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x = 3, y = 4, rule = B2cik3a4y/S02ac3j5a
o$2bo2$2bo!

It's known that the maximum period for a 3x3 square is 72 (although the following example has a minimum population of 4, rather than 3), so I'm fairly certain we can find a 3-cell p16 that fits in a bounding box of this size, and if not that then at least a 5x5 bounding box.

Here's a 3x3 p72 found with EPE I would love to call 'Panconfigurational' though it is far from that:

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x = 3, y = 3, rule = B2ei3er4aijny5inqr/S012ak3ajnr4i5ijn6a
3o2$2bo!

I'm pretty sure there's no more higher periods.

As for nrk, we can get to at least p636 while remaining in a 4x4 box. I am quite certain most lower periods could also be filled in with such examples.

p636 in 4x4:

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x = 4, y = 4, rule = B2ek3ajnq4aijkqrw5aeijq6-e/S02eik3ajkqr4-ceknt5-akr6eik
bo2$o$3bo!

Even the 3-cell nrk examples for periods 20 and above exceed this bounding box size, with few exceptions. This 3-cell p120 has a comparatively huge 20x20 bounding box:

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x = 2, y = 3, rule = B2-a3ejq4nrt5aeq6ei7/S01e2ikn3jkn4ijt5ajnr6cek7c8
2o2$bo!

...so if anyone at all can provide improvements for rotating oscillators (or indeed any improvements in general) then input would be appreciated here.

[muzik]

This is old news for those who have been paying attention, but the spreadsheet now has an additional four pages for hexagonal oscillators.

Here's a p/m=6 oscillator with a central gutteroid I just found using apgsearch:

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# p36 - gutteroid p6
x = 4, y = 5, rule = B256/S34H
b2o$3bo$2o$bobo$b3o!

For reference, here are the four other known p/m=6 oscillator periods (in 2-state range-1 isotropic hexagonal rules, that is):

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# p6 - gutteroid p1
x = 3, y = 4, rule = B2o/S2oH
3o$obo2$bo!

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# p18 - gutteroid p3
x = 4, y = 5, rule = B2/S2-mH
2bo$obo$3bo$3bo$2b2o!

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# p30 - central cell not visited
x = 3, y = 2, rule = B2o3o4m5/S2H
bo$3o!

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# p42 - gutteroid p7
x = 4, y = 5, rule = B2o/S2m34H
2b2o$b3o$3bo$4o$2o!

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# p126 - gutteroid p21
x = 2, y = 3, rule = B2/S2-mH
2o$bo$2o!

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# p594 - central cell not visited
x = 5, y = 5, rule = B2-m4-m56/S2m34m56H
3bo$ob3o$3o$bobo$b3o!

There are two time symmetries we currently have no example oscillators for, of any period:
- C3_1 -> D6_1
- C1 -> C3_1

Are there search programs much like EPE that work on hexagonal grids that could be used for filling gaps here? Any and all assistance finding oscillators and reductions appreciated.

[gzaytman]

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#p2 C3_1 -> D6_1
x = 4, y = 4, rule = B4o/S13pH
2obo$2b2o$2bo$3bo!

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#p3 C1 -> C3_1
x = 6, y = 6, rule = B3m4p/S2-o3-p4oH
5o$o2bobo$bob3o$2bob2o$3bobo$4b2o!

[gzaytman]

More hexagonal mod=1 oscillators:

C2_4 -> D4_x4

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x = 4, y = 4, rule = B2m/S1H
2o$2bo$bo$2b2o!

C3_1 -> D6_1o

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x = 5, y = 5, rule = B3m/S12oH
3o$bo$o2b2o$b2obo$2bobo!

C3_1 -> C6

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x = 5, y = 5, rule = B3m5/S12o3pH
obo$2obo$ob3o$2bo$2b3o!

D6_1 -> D12

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x = 5, y = 5, rule = B2p/S1H
3o2$o3bo$bo2bo$2bobo!

We now have a mod=1 object for every hexagonal kinetic symmetry.

[ThePlayzr]

Might be known, but all ones with D2_+1 static and kinetic require 3 cells at most.

[muzik]

Is it at all possible to reduce the bounding boxes for the nrc and nrk examples?

For reference (and archival purposes?) here are the existing nrk oscillators after period 12, up to period 72:

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# p16, bounding box 7x7
x = 3, y = 4, rule = B2-ae3anq4ery/S01c2ac3j4et5ai
o$2bo2$2bo!

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# p20, bounding box 7x7
x = 2, y = 5, rule = B2-an3i4jy/S12-in3anr4aijr5i6a
o2$bo2$bo!

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# p24, bounding box 7x7
x = 3, y = 4, rule = B2-i3qy4qrt5r6in8/S02-a3cekn4aerw5aejr6-i7
obo3$2bo!

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# p28, bounding box 7x7
x = 2, y = 5, rule = B2ck3-cknr4enrtz5c/S013a4n5ace
o2$bo2$bo!

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# p32, bounding box 7x7
x = 2, y = 3, rule = B2ce3aer4aer6ak/S012a3ci4a5acen6a7c
2o2$bo!

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# p36, bounding box 7x7
x = 2, y = 3, rule = B2ekn3acei4kt5inq6ak/S1e2ae3eijnr4iw5ckn6a
2o$bo$bo!

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# p40, bounding box 7x7
x = 2, y = 4, rule = B2e3aiqr4cwyz5r6k/S012-a3aq5acny
bo$o2$bo!

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# p44, bounding box 9x9
x = 2, y = 5, rule = B2-e3jkqr4ejnq5cinqy6c/S01c2cn3jkq4kwy5ikry6c7c
o2$bo2$bo!

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# p48, bounding box 11x11, empty center
x = 2, y = 3, rule = B2cei3ainr4n5ay6a7c/S1e2-cn3iny4ajn5acn6kn7
2o2$bo!

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# p52, bounding box 9x9, empty center
x = 2, y = 3, rule = B2cek3acjr4krt5cq6ack7/S012ac3nq4ejknqwy5aj6an7e
2o2$bo!

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# p56, bounding box 9x9, empty center
x = 2, y = 4, rule = B2ce3jkq4aijkty5n6ae78/S1e2-ci3eiqr4ir5a6aek
o$bo2$bo!

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# p60, bounding box 9x9, empty center
x = 3, y = 4, rule = B2k3aijn4aqt5n6i/S01e2-an3air4e5ciq6aci
o$2bo2$bo!

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# p64, bounding box 9x9, empty center
x = 2, y = 3, rule = B2ci3ajqr4aeint5jk/S01e2-en3cjknr4jnqry5jqr6kn7c
2o2$bo!

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# p68, bounding box 11x11, empty center
x = 2, y = 3, rule = B2ce4a5-eknq/S012ik3ejnr4ijr5j6k7e
2o2$bo!

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# p72, bounding box 5x5
x = 2, y = 4, rule = B2-ai3ac4anrw5qy6ck7e/S01c2ck3ej4ijnry5cnqy
2o$o$2o$bo!

The fact that a 5x5 bounding box only shows up at period-72, and isn't present at all for any periods below that and after 12, I find very surprising. Is it at all possible to prove that there isn't a configuration with nrc symmetry that fits into a 3x3 box in all phases with a period above 12 and that has at least one phase with exactly three cells? Or is "Panconfigurational" just that much of an outlier? Nonetheless, most of these must be reducible to 5x5, somehow.

For reference, here are all of the 3-cell nrc solutions with periods above 72 so far - these may also be reducible:

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# p76, bounding box 11x11, empty center
x = 2, y = 3, rule = B2-ak3c4acenty5-eknq6i7c/S012ikn3-ckqy4ceijknr5aeijy6-ak
2o2$bo!

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# p96, bounding box 9x9
x = 2, y = 4, rule = B2-ac3aeijn4n5a/S03ijk4qwz
bo$o2$bo!

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# p100, bounding box 5x5, presumably optimal
x = 3, y = 4, rule = B2c3ajq4ejnr5cjr6n/S01c2ci3cijny4aeknrtz5aijr6ace
obo3$2bo!

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# p120, bounding box 13x13
x = 2, y = 5, rule = B2cin3-jknq4ekrtwy5ainr6ik7e8/S02cen3-aknq4acrw5iqry6e
o2$bo2$bo!

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# p128, bounding box 9x9
x = 2, y = 4, rule = B2eik3ijqr4knqrtz5ijkn6-ci7e/S02aei3cijqy4aknq5c6e
bo$o2$bo!

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# p288, bounding box 13x13
x = 4, y = 4, rule = B2-a3ikry4aqwyz5ce6-ae7c/S02aci3ijknq4ar5ciy
o$3bo2$3bo!

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# p368, bounding box 17x17
x = 2, y = 3, rule = B2ce3iny4aeinqw5acr6-ci7c/S12e3knr4acjrt5ai6e
2o2$bo!

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# p392, bounding box 23x23, empty center
x = 2, y = 3, rule = B2cek3cq4ny5-cnr6c7c/S01e2kn3-aeij4atz5knqr6cin7e
2o2$bo!

All other periods not mentioned either have an example with 4 or more cells during their minimal phase, or no example whatsoever.

Would it be worthwhile to create a "population optimal, but bounding box not (proven) optimal" colour, or would this overcomplicate things?

[gzaytman]

I hooked up a SAT solver onto this problem and it turned out to be rather effective.

[muzik]

Hexagonal oscillator tables (p/m = 1. 2. 3. 6) have been added to the wiki overview page. I've also introduced a new colour for oscillators where the minimum population is as low as it can go, but the bounding box for all phases of the oscillator is still larger than it needs to be (usually when there exists an oscillator of a higher period with that symmetry where the bounding box is smaller).

For history's sake, here are the minimum-population oscillators eligible for replacement:

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# n-c10
x = 3, y = 4, rule = B2-cn5ci/S01e2i3en4n
obo$b2o$b2o$bo!

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# n-c12
x = 3, y = 4, rule = B2cek3aej4ei5n7e/S01c2c3a4jr5ain6a8
obo3$2bo!

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# n-c14
x = 2, y = 5, rule = B2-ai3acjr4eijr5e/S01e2ik3ajnq4w5q6c
o2$bo2$bo!

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# n-c16
x = 4, y = 4, rule = B23acer4ejn5cejr/S03-an4ekrwy5cei
o$3bo2$3bo!

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# n-e12
x = 3, y = 5, rule = B2-ci3a4k5r/S02-ce3n4cer
o2$2bo2$2bo!

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# n-e14
x = 2, y = 5, rule = B2-in3ekr4ejr5cijn/S012ek3aknqr4enqrty5ij6cei
o2$bo2$bo!

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# n-e16
x = 2, y = 4, rule = B2-in3jry4jk5in6aek/S01e2ce3ajkqr4ajy5r6ck
bo$o2$bo!

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# n-e18
x = 3, y = 4, rule = B2aik3ackr4ejy6ce/S01e2ci3ekn4jtwy5ceij6k7e
obo3$2bo!

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# n.e8
x = 3, y = 5, rule = B2-ac3i4aek5q/S012ck3n
o2$2bo2$2bo!

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# n.e10
x = 4, y = 4, rule = B2cn3ainr4er5c/S01c2c3i4enr5i6a
o$3bo2$3bo!

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# n.e12
x = 3, y = 5, rule = B2cik3aen4aer6k/S01e3n4ey
o2$2bo2$2bo!

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# n.e14
x = 3, y = 3, rule = B2ck3ai4kn5cjn7e/S12i3ainqr4eq5ir
obo$2bo$2bo!

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# n.e16
x = 3, y = 5, rule = B2ace3ekr4ijr5cek6acn7e8/S01e2ac3-cijy4ceijk5q6i
o2$2bo2$2bo!

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# n.k6
x = 2, y = 5, rule = B2cik3a4jz5i/S02a4e
o2$bo2$bo!

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# n.k8
x = 3, y = 4, rule = B2ai3ac/S01e2i3e6i
obo3$2bo!

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# n.k10
x = 3, y = 3, rule = B2ack3ar4in7e/S1e2ce3ek4t6e
obo$2bo$2bo!

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# n.k12
x = 4, y = 4, rule = B2-e3ak4a5ry6a8/S01c2e3eijnr4iqy5ciny
o$3bo2$3bo!

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# n.k14
x = 3, y = 5, rule = B2-ck3er4jr5ckn6k7e/S02-ak3ejn4kr5eqr6cek
o2$2bo2$2bo!

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# n.k16
x = 2, y = 4, rule = B2ek3aijk4r/S0
o$bo2$bo!

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# nrk20
x = 3, y = 3, rule = B2-ai3iky4a/S01c2en3iq5n
obo$bo$b2o!

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# nrk24
x = 2, y = 5, rule = B2-a3-aen4jkqryz5ejkqy6-ei7e8/S12ack3jkq4jqwyz5eijkn6ei7c
o2$bo2$bo!

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# nrk32
x = 3, y = 4, rule = B2ik3aijn5c/S01e2ace3j4aky5k
obo3$2bo!

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# nrk36
x = 3, y = 4, rule = B2ce3iqy4e5ijn/S01c2ack3a4ry5j6c
o$bo2$2bo!

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# nrk40
x = 2, y = 4, rule = B2cek3aeny4aekrty5q/S01e2aei3iy4jy5e6ae7c
o2$bo$bo!

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# nrk44
x = 3, y = 4, rule = B2ai3ajkqr4-ejntw5eiqry6-ae7/S01e2ikn3eijqr4aeiqrt5-cijq6-ek7c
o$2bo2$2bo!

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# nrk48
x = 3, y = 3, rule = B2ein3ijnq4ai5j6ek/S013ajn4ajnt5aiqy6c
obo2$b2o!

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# nrk52
x = 2, y = 3, rule = B2eik3eijqr4aiy5jkn6k7c/S12ak3-ciky4i5y6i
o$bo$bo!

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# nrk56
x = 5, y = 4, rule = B2ik3ain4akt5acjy6i/S01e2aci3eijqr4aiyz5i7c
o3bo$4bo$4bo$4bo!

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# nrk60
x = 3, y = 5, rule = B2cek3aknqr4iry5cr6c/S01c2in3acejn4w5n6ac
o2$bo$b2o$2bo!

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# nrk68
x = 2, y = 5, rule = B2-a3cenqy4-aertwy5aejkr6ak8/S02en3-acq4aertyz5ejqry6ek7c
o2$bo2$bo!

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# nrk72
x = 3, y = 4, rule = B2-ai3cry4ejqrty5-aeky6ae7e/S012e3-aeiq4acekqw5cjkn6k8
o$2bo2$2bo!

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# nrk76
x = 2, y = 5, rule = B2-ac3-acy4ceinw5-ckqr6ain8/S02eik3-ikry4ceiqz5ejnqr6in7
o2$bo2$bo!

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# nrk80
x = 2, y = 4, rule = B2-an4eit6k/S1e2aek3ry
bo$o2$bo!

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# nrk84
x = 2, y = 3, rule = B2-ae3aejr5jq7c/S01e2-ck3cei4artw5cein6k7c
2o2$bo!

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# nrk108
x = 2, y = 3, rule = B2-an3e4aejntyz5eij6ek/S012ei3-eiqy4krw5jr6ack7c
2o2$bo!

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# nrk120
x = 2, y = 3, rule = B2-a3ejq4nrt5aeq6ei7/S01e2ikn3jkn4ijt5ajnr6cek7c8
2o2$bo!

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# nrk144
x = 2, y = 4, rule = B2-an3ajky4ainqt5-aeny6ck/S02ck3ekry4aijkrz5inr6ak7e
bo$o2$bo!

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# nrk172
x = 2, y = 3, rule = B2ce3en4acntwyz5ei6ei/S12ek3jn4enr5jr6e
2o2$bo!

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# nrk180
x = 2, y = 5, rule = B2cek3ciky4anqr5cknr7c/S01c2ckn3eiry4ajryz5ekry6ae
o2$bo2$bo!

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# nrk224
x = 2, y = 5, rule = B2-ae3aejr4ajkntyz5kq6a7e8/S01c2i3jqr4acijrwz5ceikn6ik7c
o2$bo2$bo!

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# nrk304
x = 2, y = 3, rule = B2cek3ekny4aejnwz5-cjqr6k7c8/S12-ac3iqy4z5ein6ce7e
2o2$bo!

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# nrk412
x = 2, y = 3, rule = B2-ai3jk4eiqt5aciq6aek8/S12cen3-einr4ny5ary6aek8
o$bo$bo!

nrc cases (some of which have already been replaced by minimal bounding box oscillators) are listed earlier at the top of this page. We know that for these nrk cases, we can almost certainly replace them with something that always fits into a 4x4 bounding box, since there exists a p636 oscillator with this exact property and symmetry.

Information that would be useful here is an upper bound (with an example) of the highest known period of an oscillator that fits into a 3x5 bounding box.

[gzaytman]

Is it at all possible to reduce the bounding boxes for the nrc and nrk examples?

Ok I got the nrc down to a 3x3 bounding box for period up to 56 (see the Google doc).
For period 60 the SAT solver assures me there are no min-pop 3 oscillators that fit within a 3x3 box.

Edit: Same for periods 64, 68, 72, 76, 80.

[muzik]

Is it at all possible to reduce the bounding boxes for the nrc and nrk examples?

Ok I got the nrc down to a 3x3 bounding box for period up to 56 (see the Google doc).
For period 60 the SAT solver assures me there are no min-pop 3 oscillators that fit within a 3x3 box.

Edit: Same for periods 64, 68, 72, 76, 80.

Very good! I assume this means that nrc periods 60 up to some higher undiscovered limit can be attained in a minimum 5x5 bounding box (which the period 60, 72 and 100 examples already do).

Here's an interaction between two nrc56s that creates a n.c16 oscillator:

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x = 7, y = 5, rule = B2ei3ajq4in5r/S01e2aik3-ckny4in5ij6a
bo$o$b6o$4bo$5bo!

How easy would it be to prove an upper period limit for nrk in a 4x4 box? As mentioned earlier, 636 is our current record. I feel this may be beatable, though I don't know if the maximum period (at least for isotropic rules) is less than 1536. Even then, we could definitely fill a lot of gaps in with optimal examples below 636 in the process.

[muzik]

What we have (or had) for nrk, population > 3:

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# 8nrk88
x = 7, y = 9, rule = B2-a3c/S12-i
obo$2bo$o3$6bo$3bo2bo2$4bo!

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# 27nrk92
x = 29, y = 7, rule = B3-ckr4z5ky7c/S2-i3-aek4i6a
3bo21bo$2b3o19b4o$bo2bo19bo3bo$obob2o19bo2bo$bo3bo20b3o$2bobo21bo$3bo!

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# 9nrk104
x = 4, y = 5, rule = B3/S23-c4w
b2o$bobo$o2bo$b2o$bo!

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# 57nrk116
x = 28, y = 21, rule = B34c6i/S2-in3-a4iqy5ai6n
7b3o$7bo2bo$7bob2o$9bo4$4b2o$3bo3bo$2bobo2bo$b2o4bo$bobo$2bo3bo$5bob2o$2o5b2o
$2obob2o18b2o$o3b2o16b2o2b2o$2o3bo15b2o2bo$bo2bo17b2obo$27bo$27bo!

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# 8nrk124
x = 4, y = 5, rule = B2in3aceir4ey5ik6i7e/S2-i3-ak4ity
b3o$o2bo$3bo$2bo$2bo!

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# 8nrk132
x = 3, y = 5, rule = B34i/S2-c34-airy
bo$2bo$3o$b2o$2bo!

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# 6nrk136
x = 3, y = 4, rule = B3aijry4z5kry6ci7/S2-c3-aky4-ajkqw5cer6cn78
3o$2bo$2bo$bo!

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# 5nrk140
x = 3, y = 5, rule = B2e3cin4jkwz5kqy6k/S12acn3aejqr4ajkqz5cjry
3o$b2o$bo2$bo!

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# 95nrk148
x = 36, y = 36, rule = B3-ckq4z/S2-c3-a4iq5k6k
15bobo$15b3o2b2o$21bo$15b5o6b2o$15bobobo6bobo$27b2o2$4b2o$3bobo$3b2o5$b2o$bo
6b3o20b2ob2o$3b2o2bo2bo21bobo$3bo3bo2bo20b2ob2o$2ob2o2b2o23bo$bobo4bo22b2o$2o
b2o29bo$33b2o5$31b2o$30bobo$30b2o2$7b2o$7bobo6bobobo$8b2o6b5o$14bo$14b2o2b3o$
18bobo!

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# 7nrk152
x = 3, y = 4, rule = B34y/S2-i3
3o$2bo$obo$bo!

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# 6nrk160
x = 2, y = 5, rule = B3/S2-i34q
2o$2o$bo$bo$bo!

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# 7nrk176
x = 3, y = 5, rule = B2n3aijr4jk5ek6cin7e/S23-ay4einrtyz5kn6cen7
2bo$b2o$2o$bo$2bo!

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# 19nrk184
x = 7, y = 9, rule = B35n78/S2-c3-ky4it6a
3obo$3ob2o$2b2ob2o$2b2obo2$3bo$o$o2bo$3o!

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# 8nrk196
x = 3, y = 5, rule = B367/S2-k34-akr78
3o$3o$obo$b2o$b2o!

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# 12nrk200
x = 4, y = 9, rule = B3-j6i/S23-c4i
b3o$bobo$3bo3$bo$obo2$3o!

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# 16nrk204
x = 6, y = 11, rule = B3-cnqy4cijny5r6i/S2-i3-a4a
b3o$5o$obo2bo$3bobo$4b2o4$o$2o$o!

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# 9nrk208
x = 8, y = 3, rule = B2ce3anry4-ekrw5ajry6-ik/S2-e3-ijr4ackt5eikny6ak
3o3bo$bo5bo$b2o3bo!

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# 7nrk212
x = 3, y = 4, rule = B3-ckr4tz5ky/S2-i3-aek4cei5c6ck7e
2o$obo$2bo$b2o!

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# 10nrk216
x = 4, y = 5, rule = B2n3aeijr4jk5cey6cn7e/S23-ak4einrtyz5n6cn7
2bo$b2o$2obo$bobo$2b2o!

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# 18nrk232
x = 12, y = 5, rule = B3-ckr5ky7c/S2-i3-aek4ceik5c6cek7
7b3o$3o4b2ob2o$o3bo3bo2bo$bobo5bobo$2bo6b2o!

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# 7nrk236
x = 7, y = 3, rule = B2cek3acnry4aiqrtz5-anry6ace7c/S1e2ae3ejnqy4cirtwz5cenr6ac
2o4bo$bo3bo$bo4bo!

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# 12nrk240
x = 6, y = 5, rule = B3-jknr4city5ijk8/S23-a4city
bob2o$6o$b2o$2b3o$3bo!

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# 6nrk260
x = 3, y = 3, rule = B2n3aeijr4ejk5ek6cin7e/S2-c3-a4einrtyz5ekn6cn7c8
3o$2bo$b2o!

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# 6nrk292
x = 3, y = 3, rule = B2n3aijr4jkz5c6cn7e/S23-ac4einrtyz5n6cn7
3o$2bo$b2o!

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# 4nrk308
x = 2, y = 3, rule = B2cei3ai4ekz5ckqr6cik/S2-n3-ce4cikny5cinq6ik
2o$o$bo!

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# 10nrk320
x = 6, y = 5, rule = B2kn3-enq4eiwz/S2-n3-ae4-aejrw
2o2bo$bo2bo$5bo$2bobo$3b2o!

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# 6nrk324
x = 2, y = 5, rule = B3-cnqy4jz5ckr6cn7c8/S2-i3-ak4einrtyz5-inqy6-kn7c
2o$o$2o2$bo!

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# 7nrk340
x = 3, y = 4, rule = B3-cknq4ez5cer6c7/S2-ci3-ak4einrtyz5cnr6-ak7
b2o$2o$b2o$2bo!

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# 6nrk352
x = 3, y = 3, rule = B3-cknq4z5ky6c7/S2-c3-aky4-ajkqw5cjkr6cin7
3o$2bo$b2o!

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# 7nrk368
x = 3, y = 4, rule = B3-cnqy4jqyz5kr/S2-i3-ak4eiqrtyz5cejkr6i7c8
b2o$2o$b2o$2bo!

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# 9nrk468
x = 3, y = 5, rule = B2n3aijr4ejk5ceky6cin7e/S23-aky4einrtyz5ekn6cn7c
b2o$obo$o$3o$2bo!

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# 8nrk488
x = 4, y = 4, rule = B2in34ce5q/S2-n34cejnyz
b2o$o2bo$3bo$b3o!

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# 6nrk656
x = 2, y = 5, rule = B34eqz6ei7c8/S234e5e
2o$o$bo$bo$bo!

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# 16nrk860
x = 8, y = 4, rule = B2n3aijr4ejkz5ck6cn7e/S23-acky4einrtyz5n6-ak78
b2o2b3o$3o4bo$2o3bobo$4b3o!

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# 19nrk876
x = 9, y = 9, rule = B2n3-j4i6n7c/S2-i3-ay4ciqr5i6c7
b3o$o2bo$o2bo$b2o$3o$7b2o$8bo$5bo2bo$5b3o!

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# 8nrk1372
x = 4, y = 5, rule = B2e3-cjqr4eiknqwz5cijkn6ak7e/S01c2-ck3cjkny4acijqwy5iknr6ckn7c8
ob2o$b3o$3bo$bobo$2bo!

[muzik]

These oscillators up to period 6 have been marked as green for a long time; is it possible for us to prove that they are, in fact, the smallest possible for these specific symmetries and periods, or are any reducible?

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# 4n.c2
x = 3, y = 3, rule = B2e/S1c2a3q
o$b2o$2bo!

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# 5n.k2
x = 2, y = 4, rule = B2e/S1c2ac3n
2o$bo$o$bo!

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# 6-e+e2
x = 2, y = 4, rule = B3i/S2a4r
bo$2o$2o$bo!

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# 6.c+c2
x = 3, y = 5, rule = B2k/S1
bo$o$obo$2bo$bo!

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# 7.crc2
x = 5, y = 5, rule = B4n/S01c2i
3bo$obo$2bo$2bobo$bo!

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# 6.krk2
x = 4, y = 4, rule = B2k/S1
b2o$o$3bo$b2o!

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# 6xc*c2
x = 3, y = 3, rule = B2e/S3n
2o$obo$b2o!

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# 5.crc4
x = 3, y = 5, rule = B2ac4a/S1e2i4z
o2$3o2$2bo!

The hexagonal cases haven't been up for as long, so there's more of these:

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# 5.c2
x = 3, y = 5, rule = B2o3-o4-m56/S2m3o4-o56H
o$bo$bo$bo$2bo!

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# 6.e4
x = 2, y = 5, rule = B23p/S34H
bo$o$2o$bo$o!

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# 4n-2
x = 3, y = 5, rule = B2p/S01H
o2$2bo$2bo$2bo!

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# 4n/2
x = 2, y = 4, rule = B3m/S01H
bo$2o2$bo!

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# 6n.c2
x = 3, y = 5, rule = B3m/S12oH
2o$bo$bo2$b2o!

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# 5n.e2
x = 2, y = 5, rule = B3m/S012oH
o2$2o$bo$bo!

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# 5/+2
x = 3, y = 5, rule = B2m/S02oH
o2$bo$b2o$2bo!

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# 6.ex2
x = 3, y = 4, rule = B2m/S1H
2o$o$2bo$b2o!

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# 7rc*/2
x = 4, y = 4, rule = B4o/S13pH
2obo$2b2o$2bo$3bo!

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# 6rk*k2
x = 4, y = 4, rule = B2o/S3mH
o$b3o$2bo$2bo!

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# 9*/*2
x = 5, y = 5, rule = B2p/S1H
3o2$o3bo$bo2bo$2bobo!

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# 18r*
x = 7, y = 7, rule = B2m/S2m3pH
2bo$2obo$2b2obo$b2ob2o$bob2o$3bob2o$4bo!

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# 6/+4
x = 6, y = 4, rule = B25/S34mH
o4bo$b5o2$4bo!

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# 9*/*4
x = 7, y = 7, rule = B2-o/S02oH
o2bo$2bo2$o5bo$2bo2bo2$3bo2bo!

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# 18r*4
x = 9, y = 9, rule = B256/S34H
bo$3bobo$bo$o2b2obo$3bobo$2bob2o2bo$7bo$3bobo$7bo!

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# 6nr6
x = 3, y = 4, rule = B2o/S2oH
3o$obo2$bo!

There are also several hexagonal symmetries marked in yellow, indicating that optimization is almost certainly possible: nrc3, nrk3, .cr3, -4, /4, rk4, *4, n-4, n/4, n.e4, -x4, .ex4, rk*k4, /5, /6, +6, *k6, *6, -+6, -x6, /x6, *-*6, r*6, .cr6

[gzaytman]

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#C 4n.k2
x = 3, y = 3, rule = B2c3ir4n/S
obo$2bo$2bo!

[muzik]

Following up on previously, here are oscillators marked as "very likely to be optimal"; i.e. we don't have explicit proofs that these are optimal, but extensive searching hasn't brought up anything smaller yet:

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# rk3
x = 6, y = 6, rule = B2k3r/S01e5n
3bo2$ob2o$2b2obo2$2bo!

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# rk4
x = 6, y = 6, rule = B2k3r5a/S015n
3bo2$ob2o$2b2obo2$2bo!

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# rk5
x = 6, y = 6, rule = B3jr5a/S012a3-ceny4q
3bo2$ob2o$2b2obo2$2bo!

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# n-c2
x = 2, y = 3, rule = B2e/S1c2a3n
2o$o$bo!

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# n-e2
x = 2, y = 4, rule = B3ej/S1
bo$o$bo$bo!

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# n/2
x = 2, y = 3, rule = B2c/S2ae3n
2o$o$bo!

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# .c+c4
x = 3, y = 5, rule = B2cn/S1c
bo$o2$2bo$bo!

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# .c+c6
x = 3, y = 5, rule = B2n3ei/S1c2acn
bo$o2$2bo$bo!

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# .cxc2
x = 3, y = 5, rule = B2c3c/S
o2$obo2$2bo!

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# .cxc4
x = 3, y = 3, rule = B4az/S1e2a
2o2$b2o!

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# .crc6
x = 3, y = 5, rule = B2aci4k/S03ij
o2$obo2$2bo!

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# .crc8
x = 3, y = 5, rule = B2e3acin/S02ae3c5j
bo$2o2$b2o$bo!

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# +c*c6
x = 1, y = 5, rule = B2i3i6i/S01e
o2$o2$o!

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# xc*c4
x = 3, y = 3, rule = B2e6n/S1c
o$bo$2bo!

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# xc*c10
x = 5, y = 5, rule = B2en3c/S06n
o2$2bo2$4bo!

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# xc*c12
x = 5, y = 5, rule = B2-ak/S02cek
o2$2bo2$4bo!

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# xc*c14
x = 5, y = 5, rule = B2en3a4e/S03c4c6e
o2$2bo2$4bo!

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# xc*c16
x = 5, y = 5, rule = B2en8/S01c5a
o2$2bo2$4bo!

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# rk*k6
x = 6, y = 6, rule = B2k3a5a/S1e3qr
3bo$3bo$2o$4b2o$2bo$2bo!

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# rk*k8
x = 6, y = 6, rule = B2ek3r5a/S01e5n
3bo2$ob2o$2b2obo2$2bo!

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# rk*k10
x = 6, y = 6, rule = B2k3jr5a/S01e2a3-ceny4q
3bo2$ob2o$2b2obo2$2bo!

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# +4
x = 3, y = 5, rule = B25/S34-oH
o2$bo2$2bo!

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# r*4
x = 7, y = 7, rule = B2o3m/S02m4pH
2bo$o$2b2obo$2bobo$bob2o$6bo$4bo!

Most of these share columns with others.

[gzaytman]

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#C 5.c+c2
x = 3, y = 5, rule = B2c/S1c2n
bo$o$bo$2bo$bo!

[R2INT]

+c*c6 with slightly reduced maxpop:

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x = 5, y = 1, rule = B2i3i6i/S0
obobo!

EDIT:

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#+c*c8
x = 3, y = 1, rule = B2ai5i/S01e2ci3e
obo!

[muzik]

Following up on previously, here are oscillators marked as "very likely to be optimal"; i.e. we don't have explicit proofs that these are optimal, but extensive searching hasn't brought up anything smaller yet:

We're now down to just eight:

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# rk3
x = 6, y = 6, rule = B2k3r/S01e5n
3bo2$ob2o$2b2obo2$2bo!

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# rk4
x = 6, y = 6, rule = B2k3r5a/S015n
3bo2$ob2o$2b2obo2$2bo!

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# rk5
x = 6, y = 6, rule = B3jr5a/S012a3-ceny4q
3bo2$ob2o$2b2obo2$2bo!

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# n-c2
x = 2, y = 3, rule = B2e/S1c2a3n
2o$o$bo!

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# n-e2
x = 2, y = 4, rule = B3ej/S1
bo$o$bo$bo!

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# n/2
x = 2, y = 3, rule = B2c/S2ae3n
2o$o$bo!

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# +4
x = 3, y = 5, rule = B25/S34-oH
o2$bo2$2bo!

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# r*4
x = 7, y = 7, rule = B2o3m/S02m4pH
2bo$o$2b2obo$2bobo$bob2o$6bo$4bo!

[muzik]

I imagine r*4 is already optimal? I can't seem to think of a chiral hexagonal arrangement of cells that can interact with each other such that they appear flipped in two generations.

[muzik]

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# rk3
x = 6, y = 6, rule = B2k3r/S01e5n
3bo2$ob2o$2b2obo2$2bo!

Here's a version with a smaller minimal phase bounding box, however the minimum population is still 8, rather than 4:

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x = 4, y = 4, rule = B2c3jn4a6c/S1e4k5n6k
2obo$3bo$o$ob2o!

Is it at all possible for this to be a p3, p5 or non-flipping p4?:

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x = 4, y = 4, rule = B/S0
bo$3bo$o$2bo!

[yyh_baboon]

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# rk3
x = 6, y = 6, rule = B2k3r/S01e5n
3bo2$ob2o$2b2obo2$2bo!

Here's a version with a smaller minimal phase bounding box, however the minimum population is still 8, rather than 4:

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x = 4, y = 4, rule = B2c3jn4a6c/S1e4k5n6k
2obo$3bo$o$ob2o!

Is it at all possible for this to be a p3, p5 or non-flipping p4?:

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x = 4, y = 4, rule = B/S0
bo$3bo$o$2bo!

I think no. The generation 1 is determined(If no B2k it will be stable, if S0 the symmetry changes)and in gen 2, a little discussion will reveal that it must be P2 or P4 flipping.
Edit: maybe I’m wrong:(strange stuff)

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 x = 4, y = 4, rule = B2ack3n/S2a3q4ar
bo$3bo$o$2bo!

[1and0nlycordershippr]

I bet the kinetic symmetry D4_1->D8_4 has an optimal osc already:
As P2 (+c*c2, 3x3 bounding box):

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x = 3, y = 1, rule = B2c/S
obo!

As P4 (+c*c4, 3x3 bounding box):

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x = 3, y = 1, rule = B2-a/S1
obo!