I may have found a finite stable constellation that requires exactly three gliders for clean destruction. I haven't been able to find a previous construction of this kind, but I may have missed one.
The target consists of three widely separated blocks, with origins at
Code: Select all
(0, 0)
(100000, 300000)
(300000, 800000)
so the target has population 12.
Code:
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x = 300002, y = 800002, rule = B3/S23
2o$2o299999$100000b2o$100000b2o499999$
300000b2o$300000b2o!
I have a computer-assisted proof that no pair of arbitrarily rewindable input gliders can cleanly destroy this constellation.
The proof is not based on searching a finite range of launch positions or time offsets. Instead, it reduces all possible interactions to finitely many local cases and then uses geometric separation to lift the result to unrestricted positions and time differences.
A few details:
- All one-glider interactions with a block are exhaustively classified.
- Cases where the two input gliders collide with each other before reaching a block are also included, including collisions that produce additional escaping gliders.
- Cases where both gliders interact with the same block reaction, including arbitrarily late interactions with periodic ash, are included.
- 103,661 local cases were checked in total.
- 4,384 terminate in the empty pattern.
- 99,277 are certified as unions of mutually noninteracting periodic/translating components.
- 0 cases remain unresolved.
- The maximum verification time was 65,536 generations and the maximum certified period was 30.
The resulting certified locality bound is R = 65,778. The three block positions were chosen so that their separations in x, y, x-y, and x+y are all large enough relative to this bound. This guarantees that any reaction involving both input gliders can affect at most two of the original blocks. Therefore at least one original block survives unchanged forever.
There is also an explicit three-glider destruction: one glider is sent to each block, and the entire pattern is empty after 32 generations. Thus the glider destruction cost of this constellation is exactly 3.
I am deliberately calling this a stable constellation rather than claiming that it resolves the LifeWiki question about the smallest still life without a two-glider destruction, since that question may be intended to mean a single still-life object (or a strict still life).
Does anyone know whether an equivalent construction, or a more general argument of this kind, has been posted before?
I have the verification code, generated certificates, and reproduction procedure available, and I would be happy to share them if there is interest in checking the result.