Where Do Theorems Live? Dirichlet–Euler Mass, Growth Functions, and Spectral Invariance of Gödel Numberings

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Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-16
DOI
https://doi.org/10.5281/zenodo.22794138
Primary Topic
Advanced Combinatorial Mathematics
Type
preprint
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preprint

Where Do Theorems Live? Dirichlet–Euler Mass, Growth Functions, and Spectral Invariance of Gödel Numberings

Francesco Masedu
Zenodo (CERN European Organization for Nuclear Research)
Advanced Combinatorial Mathematics
preprint

Where Do Theorems Live? Dirichlet–Euler Mass, Growth Functions, and Spectral Invariance of Gödel Numberings

Francesco Masedu
preprint en

Abstract

We study the distribution of Gödel numbers of theorems in formal systems using the Dirichlet–Euler (zeta) probability measure. We introduce a proof-length growth function weighted by this measure, which carries strictly more information than standard counting functions. We establish that under power renumberings, the normalized profile of the growth function is invariant up to a deterministic scaling of the parameter. Finally, we show that the Dirichlet–Euler measure serves as the maximum-entropy "null model" for random formulas, making the dependence on the encoding fully explicit and controllable.

Zenodo (CERN European Organization for Nuclear Research)
University of L'Aquila (IT)
Advanced Combinatorial Mathematics
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