I am working on extending the proof to Gardens of Eden, i.e. cells outside the four rows in the target are specified as dead.

If I take a typical parent pattern that produces a given four row high sub-pattern it will often have a few stray live cells produced outside the four target rows. But if I then take that parent pattern and run it between "rails" the stray cells are mostly suppressed:

Code: Select all

```
x = 124, y = 53, rule = B3/S23
2$32b2o30bo2$30bo4bo30bo$34bo39bo3b2ob2o2b3o2bo3bo3bo$59bo2bo2bo2bo5b
2ob2obobobobob3ob4ob2o$33bo40b2ob2o3bo2b2obobo2b3ob2o$33bo32bo7b2ob2ob
obobob2o2bo3b2ob2o2$64bo$33bo11$64bo$95bo4b3o$5b2obo3b2o7b2ob3o2b5ob2o
bo27bo8b3obo11b5o2bo2b3o2b2o$5b2o2b2o5b2o3bo2bo3b3o3b3o3bo3bo3bo3bo21b
o3b2ob2o2b3o2bo3bo3bo$5bo2bo3bobobobobobo10bo3bo3bo3bo3bo3bo5bo2bo2bo
2bo5b2ob2obobobobob3ob4ob2o24bo$5bobo2bob2obo2bo2bobob2obob24o20b2ob2o
3bo2b2obobo2b3ob2o24bo$5bo2bobo3bo2bo2bobobo3b4o2b2o2b2o2b2o2b2o2b2o
14bo7b2ob2obobobob2o2bo3b2ob2o$5bo2bo2bo2bobo4b2o56bo5bo5b3o5b2o$64bo
11$71b52o$54o10bo5bo52bo$54o16bo52bo$5b2obo3b2o7b2ob3o2b5ob2obo27bo4b
3o15bo18bo3bo3bo3bo2bo$5b2o2b2o5b2o3bo2bo3b3o3b3o3bo3bo3bo3bo21bo3b2ob
2o2b3o2bo3bo3bo$5bo2bo3bobobobobobo10bo3bo3bo3bo3bo3bo5bo2bo2bo2bo5b2o
b2obobobobob3ob4ob2o24bo$5bobo2bob2obo2bo2bobob2obob24o20b2ob2o3bo2b2o
bobo2b3ob2o24bo$5bo2bobo3bo2bo2bobobo3b4o2b2o2b2o2b2o2b2o2b2o14bo7b2ob
2obobobob2o2bo3b2ob2o$5bo2bo2bo2bobo4b2o48b3o22bo$54o10bo5bo52bo$54o
16bo52bo$71b52o!
```

By itself this isn't enough, but what I can do is modify the definition for the proof set so that the stray cells in the target generation are always suppressed when run between rails. The cells produced where the rails stop can be dealt with by just running the rails out to infinity:

Code: Select all

```
x = 49, y = 43, rule = B3/S23
24bo2$24bo2$24bo3$8bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2$7b35o$7b35o$8b
2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o4$8bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo
2bo2$7b35o$7b35o$8b2o2bob2ob2o10b2obobobo3b2o$13b2obo2bobob2obo2b2o2b
2o2bo$obobo8bo5bo4bo2bo2bo2bobo8bobobo$13bobobo4bobobobobo2bo2bo$8bo2b
o2b2o2b3o4bo2bo3bo2bo3bo$13bo5b2ob2o4bo2b2o3bo$7b35o$7b35o$8b2o2b2o2b
2o2b2o2b2o2b2o2b2o2b2o2b2o4$8bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2$7b
35o$7b35o$8b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o3$24bo2$24bo2$24bo!
```

A schematic of what's going on:

Code: Select all

```
.
.
.
**********************************************
**********************************************
(solution producing empty target here)
**********************************************
**********************************************
... (solution producing desired target here) ...
**********************************************
**********************************************
(solution producing empty target here)
**********************************************
**********************************************
.
.
.
```

For B3/S23 it's not too hard to find a finite stabilization for the rails for a finite target population, and I expect this will be possible to prove. But this extra step isn't necessary to disprove Gardens of Eden:

Code: Select all

```
x = 46, y = 14, rule = B3/S23
o44bo$bo2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2bo$46o$46o$bo2b2o2bob
2ob2o10b2obobobo3b2o2b2o2bo$o8b2obo2bobob2obo2b2o2b2o2bo11bo$9bo5bo4bo
2bo2bo2bobo$9bobobo4bobobobobo2bo2bo$o3bo2bo2b2o2b3o4bo2bo3bo2bo3bo2bo
2bo3bo$bo7bo5b2ob2o4bo2b2o3bo11bo$46o$46o$bo2b2o2b2o2b2o2b2o2b2o2b2o2b
2o2b2o2b2o2b2o2bo$o44bo!
```

My hope is that the extra requirement on the proof set is mild enough that a proof set can be found without an excessive amount of computation. It's not looking good at the moment, but I'm not throwing in the towel yet.