Mathematically clean and tidy would be a better description. This system is particularly close to the natural harmonics that either occur when planets form, or that planetary systems tend towards.

It's akin to watching a group of metronomes move into sync. Mathematically, it's not that special. To us monkeys, that particular pattern feels a lot better and cleaner.

Scientifically, it's quite useful. By judging the age of the system, we can tell a lot about early formation and planetary movements. As the article mentions, it's likely that planets form in such a resonance, then get knocked into chaotic orbits. However that's not been tested. This system would let us test some of our models against reality.

Mathematics doesn't try anything. It's abstract. Humans try to fit mathematics to real world phenomena, often with great success, but we don't know why things seem to work out, only that they do.

Whether it means the universe is fundamentally mathematical is a bigger question than for both science and philosophy combined, and so mathematics is merely a useful modelling tool until we know for certain. (And we might never know.)

The "perfect" mathematics in this case is that this planetary system has all planets in what's called resonance, that is, the orbit time of the planets is in some relatively simple integer ratio. e.g. the innermost planet might go around twice for every once of the next furthest out (2:1), or three times for every two (3:2), etc.

The article doesn't give the ratios, but there is a tendency for planetary resonances to self-reinforce, that is, once nearby planets in non-resonant motion get close to an orbital resonance, there is a tendency that their orbit speeds will shift towards that resonance over time, and outside interference is usually needed to break a resonance once it has been set up.

There was no grand mathematical plan, only an emergent phenomenon.

Maybe this system has ended up with 32:16:8:4:2:1. That would be neat.

You're thinking of physics.