The Monofractal Limit

Cities grow messy. Young urban areas have irregular population distributions — dense clusters here, voids there, no single pattern describing the whole. But analysis of 477 Dutch urban areas and major world cities from 1975 to 2023 reveals something unexpected: as cities mature, their spatial statistics converge. The variance of population counts across grid cells scales as a power law with the grid size, and the scaling exponent depends linearly on the mean-population exponent. Over time, cities drift toward a monofractal limit where a single correlation dimension captures the entire spatial structure.

Separately, a new analytical framework for polymer dynamics achieves something similar through mathematics rather than aging. By applying Gaussian closure — truncating higher-order correlations at the two-point level — one self-consistent diffusion equation handles coiled polymers, collapsed globules, and self-avoiding chains. Three fundamentally different physical states, unified under one description because the closure assumption makes the correlation structure tractable.

The pattern connecting these: mature systems converge toward simpler descriptions not because they become simple, but because their correlations become consistent enough to compress.

A young city has multifractal statistics — you need multiple scaling exponents to describe different neighborhoods, different densities, different growth patterns. The spatial correlations are heterogeneous. But as the city ages, these correlations align. Infrastructure creates pathways. Zoning creates regularities. Economic sorting creates predictable gradients. The city doesn't lose complexity — it gains coherence. And coherence is what makes a single number sufficient.

The Gaussian closure works by the same logic, applied deliberately rather than emergently. Real polymer chains have complicated many-body correlations — each monomer interacts with every other through excluded volume, hydrodynamics, activity. The full description requires the complete hierarchy of correlation functions. But the Gaussian truncation says: assume the correlations are pairwise-consistent, and the higher-order statistics will follow. This works — predicting even hyper-compacted chromatin states — because the pairwise level captures enough of the structure that higher orders are genuinely constrained.

The monofractal limit is not simplification. It is the state where the system's complexity has organized itself into a form that admits compression. The description gets shorter. The thing described stays rich.

This distinction matters everywhere correlation structures determine what you can know about a system. A stock market in its early phase — many independent sectors, heterogeneous returns — requires many parameters. A mature market with deeply correlated sectors might be described by fewer factors. Not because the market got simpler, but because its correlations aligned enough that a lower-dimensional description became faithful.

The deeper point: compressibility is not a property of the system. It is a property of the relationship between the system and the description. What changes during maturation is not the system's complexity but its coherence — and coherence is exactly what makes compression possible.