Is it possible to live with false eyes, only?

Question

Sometimes, in a game of Go, we get a life where there is a false eye involved: There are two groups with a single real eye, each, but they are connected to one another via a false eye, allowing their individual eyes to join forces to ensure life. Something like this:

$$Bcm0
$$ .........
$$ .OOOOO...
$$ .OXXXO...
$$ .OX.XOOO.
$$ .OXXaXXO.
$$ .OOOX.XO.
$$ ...OXXXO.
$$ ...OOOOO.
$$ .........

While the false eye at a is not necessary (black could fill it and still live), it does not need to be filled either, directly providing a liberty to each of the two black chains. I wonder whether it's possible to construct a live that entirely relies on such false eyes?

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In this question, I use a purely local definition of a false eye: A false eye is a free spot that is a freedom of two distinct chains. I am aware that there are other definitions of false eyes that would call the example above a living group with three eyes.

Answer 1

Yes, this is possible. It requires the living group to circle back on itself like this:

$$Bcm0
$$ ............
$$ ...OOOOOO...
$$ ..OOXXXXOO..
$$ .OOX.XX.XOO.
$$ .OX.XOOX.XO.
$$ .OXXO.OOXXO.
$$ .OXXOO.OXXO.
$$ .OX.XOOX.XO.
$$ .OOX.XX.XOO.
$$ ..OOXXXXOO..
$$ ...OOOOOO...
$$ ............

Here, each of the eight black eyes is locally a false eye, but because the chains that they connect are also connected via the other false eyes, none of the false eyes can ever be attacked. The term for such a construction is "two headed dragon", and they are described in Sensei's Library at https://senseis.xmp.net/?twoheadeddragon (thanks to Karl Knechtel for the link).

The above example makes the circle explicit. However, much smaller forms exist which do not require encircling an opponents living group, where the two eyes are also false by the definition you gave:

$$Bcm0
$$ ........
$$ .OOOO...
$$ .OXXOOO.
$$ .OX.XXO.
$$ .OXX.XO.
$$ .OOOXXO.
$$ ...OOOO.
$$ ........

Here too, we have two eyes which are shared liberties of two separate chains, and again, the whole thing is only alive because those two chains are connected via the other eye. Most Go players would call these two real eyes, though. The two eyes are just so close together, and both groups rely on the exact same two liberties, so it's hard to call these eyes false, even though they fit the definition you gave.