Statistical patterns in the equations of physics and the emergence of a meta-law of nature

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@Article{Constantin:2026:RSTA,

• author = "Andrei Constantin and Pedro Ferreira and Harry Desmond and Deaglan Bartlett",
• title = "Statistical patterns in the equations of physics and the emergence of a meta-law of nature",
• journal = "Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences",
• year = "2026",
• volume = "384",
• number = "2317",
• pages = "20250091",
• month = "9 " # apr,
• keywords = "genetic algorithms, genetic programming, physics formulae, statistical patterns, symbolic regression, mathematical physics",
• ISSN = "1364-503X",
• URL = "https://royalsocietypublishing.org/rsta/article-pdf/doi/10.1098/rsta.2025.0091/6131125/rsta.2025.0091.pdf",
• DOI = "10.1098/rsta.2025.0091",
• size = "16 pages",
• abstract = "Physics seeks to uncover the laws of Nature and express them through mathematical equations . Despite the vast diversity of natural phenomena, physical equations exhibit structural regularities that set them apart from arbitrary mathematical expressions. While principles such as dimensional analysis have long guided the formulation of physical models, the exploration of more subtle statistical patterns within the equations of physics remains an open question. Here, by analysing four corpora of physics equations and applying advanced implicit-likelihood techniques, we find that the frequency of mathematical operators follows an exponential decay law, in contrast to Zipf power law for word frequencies in natural languages. This reveals a statistical meta-law of physics, possibly reflecting a combination of communication efficiency and constraints imposed by Nature itself. The meta-law offers practical benefits for symbolic regression by drastically narrowing down the space of physically plausible expressions. More broadly, it may inform the development of language models that can generate coherent mathematical representations, advancing the automation of physical law discovery.",
• notes = "part of the discussion meeting issue Symbolic regression in the physical sciences \cite{Bartlett:2026:RSTAintro}.",

}

Genetic Programming entries for Andrei Constantin Pedro G Ferreira Harry Desmond Deaglan J Bartlett

Citations