Today I asked myself a question: in which rules (1 transition from Life) oscillators with 1-cell rotor are possible (or impossible)?
Since the rotor consists of one cell, state of every cell in neighbourhood of "blinking" cell is constant. Oscillator with 1-cell rotor is impossible in CGoL, since a cell can only be born with three neighbours, but it cannot die due to S3.
Therefore, it is required either to add a birth condition (except for B2x due to S2), or to remove a survival condition with three neighbours (S3-x).
B3/S23-x is easy:
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x = 8, y = 8, rule = B3/S23-c
bo$obob2obo$b2obob2o$3bo$b2o$bo$2bo$b2o!
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x = 7, y = 9, rule = B3/S23-e
2o3b2o$bo3bo$bobobo$2b3o2$7o$o5bo$b3o$3bo!
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x = 6, y = 6, rule = B3/S23-k
bo2b2o$obo2bo$ob3o$bo$2b3o$4bo!
B3/S23-a cannot have such oscillators, because every alive neighbour of the rotor will die due to S3-a or overpopulation (light green=blinking cell, yellow+dark blue=stator, red=any state (showing that no combination will save the stator)):
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x = 4, y = 4, rule = B3/S23-aHistory
4D$B2ED$BCED$3BD!
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x = 7, y = 9, rule = B3/S23-i
2o3b2o$bo3bo$o5bo$2o3b2o$3bo$7o$o5bo$b3o$3bo!
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x = 9, y = 7, rule = B3/S23-n
4bo$3bobo$b3obo$o4b3o$b3o4bo$3bob3o$4b2o!
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x = 9, y = 5, rule = B3/S23-y
2obobob2o$ob2ob2obo$4bo$b2obob2o$b2ob2obo!
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x = 6, y = 8, rule = B3/S23-q
4b2o$3bobo$3bo$2ob2o$obo2bo$2bob2o$2bo$b2o!
B3/S23-j cannot have such oscillators as well as B3/S23-a (W neighbour in this pattern will die regardless of red cells):
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x = 4, y = 3, rule = B3/S23-jHistory
DBEB$DECB$DE2B!
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x = 8, y = 9, rule = B3/S23-r
2b2o$bo2bob2o$b2obob2o$2bobo$obobo$2obob2o$3bo2bo$3bobo$4bo!
I have not yet analyzed all cases with an additional BXx transition. Several examples:
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x = 7, y = 7, rule = B34c/S23
bo3bo$obobobo$b2ob2o2$b2ob2o$obobobo$bo3bo!
According to the classical definition of rotor, in rules with B0 there is no finite oscillator with 1-cell rotor. I didn't come up with an exact redefinition of the rotor in B0 rules so I can only show this:
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x = 13, y = 13, rule = B03/S23
2ob2ob2o2bobo$bob2obo2bob2o$o7b2o$b5o3bob2o$2bo2bo3bob2o$o7b2o$2o4bo4b
2o$3b2o7bo$2obo3bo2bo$2obo3b5o$3b2o7bo$2obo2bob2obo$obo2b2ob2ob2o!
B34e/S23 cannot have such oscillators (if red cell is dead, the stator will break due to B3a; if red cell is alive, the stator will break due to B3k):
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x = 5, y = 5, rule = B34e/S23History
D3BD$2BE2B$BECEB$2BE2B$D3BD!
Oscillators in B1c3/S23 are impossible at all, because every pattern explodes in all directions.
Current results:
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Possible:
B3/S23-(cekinyqr)
B34c/S23
Impossible:
B3/S23Xx
B3/S23-(aj)
B1c3/S23
B2x3/S23
B34e/S23
B3-x/S23
Special status:
B03/S23
Edit: added information from HotdogPi's post:
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Possible:
B3/S23-(cekinyqr)
B34c/S23
B36i/S23
Impossible:
B3/S23Xx
B3/S23-(aj)
B1c3/S23
B2x3/S23
B34(ear)/S23
B35(ckainqj)/S23
B36(cekan)/S23
B37(ce)/S23
B38/S23
B3-x/S23
Special status:
B03/S23
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x = 7, y = 5, rule = B36i/S23
2o3b2o$obobobo$2b3o$obobobo$2o3b2o!