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arXiv · 2512.01436

Statistical Properties of the Rooted-Tree Encoding of $\mathbb{N}$

Pierluigi Contucci·Claudio Giberti·Godwin Osabutey·Cecilia Vernia
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Abstract

We prime-encode the natural numbers via recursive factorisation, iterated to the exponents, generating a corpus of planar rooted trees equivalently represented as Dyck words. This forms a deterministic text endowed with internal rules. Statistical analysis of the corpus reveals that the dictionary and the entropy grow sublinearly, compression shows non-monotonic trend, and the rank-frequency curves assume a stable parabolic form deviating from Zipf's law. Correlation analysis using mean-squared displacement reveals a transition from normal diffusion to superdiffusion in the associated walk. These findings characterise the tree-encoded sequence as a statistically structured text with long-range correlations grounded in its generative arithmetic law, providing an empirical basis for subsequent theoretical and learnability

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Pierluigi Contucci, Claudio Giberti, Godwin Osabutey, Cecilia Vernia. 2025-12-01. Statistical Properties of the Rooted-Tree Encoding of $\mathbb{N}$. https://arxiv.org/abs/2512.01436

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