Arithmetic Trace and Primality in Homogeneous Descending Processes

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Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-14
DOI
https://doi.org/10.5281/zenodo.22744162
Primary Topic
Random Matrices and Applications
Type
preprint
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preprint

Arithmetic Trace and Primality in Homogeneous Descending Processes

Sylvain Gefffroy
Zenodo (CERN European Organization for Nuclear Research)
Random Matrices and Applications
preprint

Arithmetic Trace and Primality in Homogeneous Descending Processes

Sylvain Gefffroy
preprint en

Abstract

This paper comes from the the simple idea : " What happens if, instead of building numbers by successive ascent, we start from a Whole and only allow repeated withdrawal of equal operational portions, without ever seeing the next step in advance and without assigning any cardinality to the starting content?" This paper explores that descending process. The successive withdrawals generate their own grading. Finite formal traces form the free monoid on one generator; the traces that are actually realized form an initial segment of this monoid. Addition appears as concatenation, multiplication as repeated equal blocks, and divisibility as exact tiling of a trace. At each stage the earlier states form a graded memory. A step length signals precisely the addresses that it divides: composite addresses admit nontrivial closures, whereas prime addresses remain silent. Descent in the Whole thereby produces an ascending trace coordinate. Elapsed age becomes measurable through the history of the process, while remaining capacity still requires a terminal anchor. An elementary extension argument isolates the information that cannot be recovered before termination is observed. For a separate direct divisibility protocol under uniformly random presentation order, the exact expected immediate-certification rate is derived and classical Ramanujan–Wilson and Selberg–Delange estimates are applied to its deficit, retaining an explicit prime correction. The global rate converges to one in probability, although the conditional rate for primes is asymptotic to two divided by the square root of the cutoff. Persistent shared witnesses are then shown to prevent a logarithmic fluctuation limit: the concentration function of the centered rate tends to zero under every normalization of the form log N over (log log N) to the power b, for any fixed non-negative b. The argument is first given as an abstract criterion for systems of independent arrival coordinates and witness sets, then specialized to divisor families. The fluctuation analysis uses the classical Hoeffding–ANOVA decomposition and its ordered Doob grouping; the Efron–Stein inequality supplies a new upper variance bound. A univariate generating function isolates the remaining analytic problem. The construction deliberately keeps three questions distinct: how many withdrawals have been performed, how much content has actually been removed, and how much content remains. Only the first becomes accessible through the history of the process. The aim is therefore to separate counting acts from measuring content and to specify exactly which observations each arithmetic conclusion uses. Version notes (R8): adds an exact cyclic-counter realization of closure signals, a small illustrative example, a sufficient (not optimal) counter-state memory bound, and the connection to witness availability, without changing the random-order protocol. Reproducibility scripts for finite-sum verification and coupling diagnostics are included.

Zenodo (CERN European Organization for Nuclear Research)
Random Matrices and Applications
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