From "More Is Different" to Algorithmic Emergence: Regularity, Compression, and the Limits of Discovery

Knowing the fundamental laws doesn't hand you the bigger picture for free — and this paper proves, with real theorems, exactly why that gap can be permanent.

The intuitive hook is Philip Anderson's old slogan "more is different": reductionism can tell you the rules governing electrons, but it doesn't automatically tell you how to talk about a hurricane, a flock of birds, or a brain. The paper's move is to make this precise using compression. Science, in this framing, is the search for descriptions that are shorter than just listing every data point — a model captures what's shared, parameters capture what's specific, and if the model is reused often enough it pays for itself even if it's clunky. Reductionism works because simple systems are easier to compress. The open question is whether having the simple microscopic law in hand automatically gives you a good compressed description of the larger system built from those parts. The paper's answer, formalized with tools from algorithmic information theory (Kolmogorov complexity — the length of the shortest program that outputs a given string), is: no, not in general, and here's exactly where it breaks.

Three specific barriers get nailed down. First, even with dead-simple, fully known dynamics (their toy example is a cyclic shift register), the history an observer actually records can still carry almost all of its original information — nothing shorter exists, because the "boring" law just relays initial-condition detail forward in time instead of erasing it. Second, even when a shorter description does exist, there's no algorithm that's guaranteed to always find it and halt — open-ended search will eventually stumble on it, but you can't put a computable cap on how long "eventually" takes. Third, even algorithms that do successfully compress lots of things will always have blind spots: for every method, at every sufficiently large data length, there exist records with tiny logarithmic-length descriptions that the method fails to shorten at all.

Despite all that, people (and presumably other observers) obviously do find useful macroscopic models all the time — that's what science and perception are. The paper names this event "algorithmic emergence": acquiring a representation plus a reusable model that newly exposes compressible structure. Crucially, they make it testable rather than mystical — a candidate model passes if, counting its own cost, it beats a baseline on the data it was built on and beats it again on fresh data. The barriers mean no algorithm can be guaranteed to find a passing model whenever one exists, or to get close to the best possible one. But the same barriers point to what does help in practice: symmetry, restricted hypothesis classes, or physical constraints that shrink the search space — which is presumably why real science, unlike the general-purpose search this paper rules out, actually works.