Statistical Properties of the Rooted-Tree Encoding of $\mathbb{N}$
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914176005308416 |
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| author | Contucci, Pierluigi Giberti, Claudio Osabutey, Godwin Vernia, Cecilia |
| author_facet | Contucci, Pierluigi Giberti, Claudio Osabutey, Godwin Vernia, Cecilia |
| contents | 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 |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_01436 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Statistical Properties of the Rooted-Tree Encoding of $\mathbb{N}$ Contucci, Pierluigi Giberti, Claudio Osabutey, Godwin Vernia, Cecilia Mathematical Physics Number Theory 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 |
| title | Statistical Properties of the Rooted-Tree Encoding of $\mathbb{N}$ |
| topic | Mathematical Physics Number Theory |
| url | https://arxiv.org/abs/2512.01436 |