Catalysts as "natural" motifs

[MathAndCode]

The recent effort to synthesize the drifty eater 3 from a relative of the catalyst at the heart of the Snark made me realize how similar the two catalysts are. For one thing, they share about two-thirds of their cells (twenty-one to be exact (although one can be considered coincidental)) (which was part of the reason why I had the idea of synthesizing one from the other).

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x = 13, y = 8, rule = DoubleB3S23
2.2C$2.C.C2.2A$4.C3.A2B$4C.2CA.B2.B$C2.C.C.B.B.2B$3.C.C.CAC$4.2C.C.BA$8.B2A!

The similarities go deeper than that, though. While the drifter catalyst has a variety of recovery mechanisms, one of them looks similar to that of the Snark catalyst. Here's an example:

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x = 13, y = 24, rule = B3/S23
2b2o$2bobo2b2o$4bo3bo$4ob3o$o2bobo$3bobob3o$4b2obo2bo$9b2o7$5b4obo$4b
obob5o$4b2ob4obo$7bobo2bo$3bo2bo2b2o$4bobo2b2o$3b6ob2o$5b8o$4b9o$3b2o
5bobo!

For reference, here is the relevant Snark catalysis:

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x = 16, y = 18, rule = B3/S23
5b2o$5bobo$7bo4b2o$3b4ob2o2bo2bo$3bo2bobobobob2o$6bobobobo$7b2obobo$11b
o$7bo$6b3o$5bobobo$5bo3bo$2ob2o5b2o$3o4bo4bo$b3o2bob2o2b2o$2bob6obo$3b
2o$4bo!

This catalysis involves the birth of a cell that does not change the general recovery mechanism but speeds the recovery up by one generation. Here is a version without that:

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x = 14, y = 16, rule = B3/S23
3b2o$3bobo$5bo4b2o$b4ob2o2bo2bo$bo2bobobobob2o$4bobobobo$5b2obobo$9bo
$5bo$4b3o$3bobobo$3bo3bo$3o5b2o$10bo$6b2o2b2o$6b2obo!

The speed-up mechanism also works with the drifter catalyst.

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x = 11, y = 13, rule = B3/S23
2b2o$2bobo2b2o$4bo3bo$4ob3o$o2bobo$3bobob3o$4b2obo2bo$9b2o$6bo$5bobo$
3b2o2bo$3bo$3b2o!

The fact that the speed-up mechanism works with both catalysts hints at the fact that their recoveries are similar.

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x = 26, y = 8, rule = B3/S23
2b2o11b2o$2bobo2b2o6bobo$4bo3bo8bo4b2o$4ob3o5b4ob2o2bo2bo$o2bobo7bo2b
obobobob2o$4bob4o7bobo2bo$10bo9bobo$9b2o10bo!

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x = 13, y = 8, rule = DoubleB3S23
2.2C$2.C.C2.2A$4.C3.A2B$4C.2CA.B2.B$C2.C.C.B.B.2B$4.C.C2AC$7.B.BA$8.B2A!

Knowing that the Snark catalyst is a cousin of the drifty eater 3 raises the question of why. The answer is that that motif tends to form, which makes it more likely to form from state that it may be perturbed into. As an example, xs28_oggs2uge1egozw643, which the Snark catalyst synthesis is based on (due to the fact that xs28_oggs2uge1egozw643 is almost the same as the Snark catalyst, including any cells within a von Neumann distance of three of any cells that change in the Snark catalyst) has a C1 soup (which is only true for 0.0058% of all 28-cell still lives) that yields a nine-glider synthesis. The formation xs28_oggs2uge1egozw643 in the soup and corresponding synthesis bears a resemblance to the recovery of the Snark catalyst.

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x = 37, y = 13, rule = B3/S23
3b2o$2b2o2bobo$3bo2bo2bo$3b6o4bo13b2o$8bo2b2obo12bobo$4b2o4b2o2bo14bo
$2b3o6bobob2o3b2o3b4ob2o3b2o$bo3bo2b3o3bo2bobobo3bo2bo3bobobo$2ob2o2b
3o4b2obobo9b2obobo$2bobob2o7bob3o10bob3o$7bo$b2o2b2o$b2o!

These types of motifs are rare because ConwayLife is not a chemistry rule, and they are valuable for catalysts. Therefore, one such motif can be used for multiple catalysts because it's useful for catalysts, and there aren't many options for that type of catalyst.
Here is a catalyst that builds back that same motif in the same way (although in this case, it isn't the end of the recovery):

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x = 15, y = 13, rule = B3/S23
9b2o2b2o$5b2obo2bo2bo$2o2bobobob2obo$o3bobobo2bob2o$b3obobobo2bo$3bobo
bob2obo$3bobobo4b2o$4b2ob4o2bo$11b2o$2b2o5b2o$2b2o6bo$2b2o5bo$9b2o!

There are two more catalysts with that motif, and while they don't share the recovery just discussed, they do share the recovery characteristic of the drifty eater 3.

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x = 30, y = 13, rule = B3/S23
3bo$3b3o15b2o$6bo2b2o10bobo2b2o$b4obo3bo12bo3bo$o2bobob3o9b4ob3o$2o3b
obo11bo2bobo$5bobob3o10bobob3o$6b2obo2bo10b2obo2bo$11b2o7b2o6b2o$3b3o
15bo$6bo2$3b3o!

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x = 33, y = 15, rule = B3/S23
2b2o7bo$2bo7bobo$4bo5bobo11b2o$3b4o2b2ob2o10bobo2b2o$7bo5bo12bo3bo$3b
4ob5o9b4ob3o$o2bo2bobo13bo2bobo$2o4bobob3o12bobob3o$bo5b2obo2bo12b2obo
2bo$bo10b2o17b2o2$5b2o14b2o$5bo16b2o$6bo14bo$5b2o!

The only other good example that I can think of (although there is probably at least one that I don't know about or am forgetting) is the eater 4, which typically recovers from this quadruply great grandparent, which is four cells smaller:

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x = 14, y = 14, rule = B3/S23
3b2o$3bo$2obo$o2b2o$b2o$3b3o$3bo2bo$4b2obo$6bobo$6bobobo2bo$7b2ob4o$9b
o$9bobo$10b2o!

The eater 4 has a (clearly soup-based) twelve-glider synthesis (which is rather low for a 42-cell still life, even for a symmetric 42-cell still life) that goes through that quadruply great grandparent.
Also, while this isn't as good of an example, the eater 2 is more common than over 99% of the other nineteen-cell still lives and is cheaper than over 99.5% of the other nineteen-cell still lives, and in both of its six-glider syntheses, the grandparent is the eater 2's main recovery grandparent.
Of course, because build-back reactions are rare, not all catalysts use them. The philosophy of the BTS and fishhook bridge fishhook seems to be to, at the first provocation, expand outwards with an expendable "shield" whose role is to crash into the chaos in order to protect the core so that the core can recover properly. In addition, there is a middle ground: The fishhook, block, and boat each typically recover from recovery states of comparable size (although the first two occasionally adopt an approach that somewhat resembles fishhook bridge fishhook and BTS but that typically does not involve intermittent disconnection from the chaos).

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x = 27, y = 38, rule = B3/S23
9bo16bo2$6b2obo13b4o$5bo2bo13bo$5b4o11b5o$21b3o$2b2o$bobo$bo20b2o$2o20b
2o6$7b2o$7bo2$5b2o13b2o3bo$7bo2bo11b4o$2b2o5b2o$bobo$bo20b2o$2o20b2o3$
9bo$6bo$6b3o$8bo$8b2o$8bo$5b4o12b3o$6b2o13b5o$2b2o$bobo$bo20b2o$2o20b2o!

Of course, for any type of recovery, the same principle holds: Any recovery state that a catalyst can recover from in the course of a catalysis is a recovery state that that catalyst can recover from when forming from a soup, and still-lives that can easily form from recovery states tend to be able to form easily in general; those points (especially the first one) are merely most pertinent for shrink-then-grow-back catalysts. (There is an exception for rocks, which don't lose their motifs and therefore don't need them to re-form. Catalysts with uncommon motifs, such as the snake, hook with tail, and long hook with tail, tend to be rocks.)

[Kazyan]

The observation that natural motifs are more likely to be catalysts is kind of profound, and IMO, could be the basis of another complex catalyst searcher.

[MathAndCode]

The observation that natural motifs are more likely to be catalysts is kind of profound, and IMO, could be the basis of another complex catalyst searcher.

For all I know, it may have already been employed in a successful catalyst search. I don't know how AbhpzTa found the R49 catalyst, including how many previous possibilities were tried or what they were, but it's possible that AbhpzTa was selectively trying motifs that looked more likely to reform. A worm/big S-type motif certainly seems more likely to reform than a snake motif.