report · Zenodo (CERN European Organization for Nuclear Research)
This work presents a collection of mathematical research notes complemented by experimental computational measurements. The volume is structured into two main sections, reviewing foundational concepts in number theory alongside original experimental findings. These experimental results address four empty positions within an arithmetic classification framework, extend the Sato-Tate spectrometer to complex multiplication involving specific algebraic rings, and analyse data spanning the historical Return of Coppersmith's Attack window from 2016 to 2017. All reported numerical values are provided to exact digits and verified via self-contained, dependency-free Python scripts designed to ensure full computational reproducibility. Rather than presenting formal proofs for unsolved mathematical conjectures, the records serve as an empirical laboratory log documenting predicted arithmetic behaviours against direct computational verification.
Pure mathematics and computational number theory often depend on empirical validation to test theoretical expectations. By pairing theoretical arithmetic frameworks with fully reproducible, dependency-free code, this laboratory notebook allows researchers to inspect, verify, and build upon precise calculations without needing proprietary software environments or unsubstantiated theoretical leaps.
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This volume gathers ten notes written between 16 and 31 August 2026, extended by experimental measurements performed on 4–5 September 2026. It is organised in two parts: classical background (Brun's and Selberg's sieves, the Hardy–Littlewood singular series, Brahmagupta's identity, the LTE lemma), cited to their authors; and an original measurement notebook filling four empty cells of the arithmetic Mendeleev table, extending the Sato–Tate spectrometer to complex multiplication by Z[ω], and traversing the historical ROCA window (2016–2017). Every measurement is reproducible: dependency-free pure-Python source code is published alongside, and numerical constants are reported to the digit. We claim no proof of any open conjecture; we publish an honest laboratory notebook, in which the map predicts and the console confirms. Source code (reproducible): https://github.com/fouad-bensmail/congruential-audit Francais Ce recueil rassemble dix notes rédigées entre le 16 et le 31 août 2026, augmentées de mesuresexpérimentales réalisées entre le 4 et le 5 septembre 2026. Il est organisé en deux parties : desrappels classiques (crible de Brun et Selberg, série singulière de Hardy–Littlewood, identité deBrahmagupta, lemme LTE) cités à leurs auteurs, et un cahier de mesures originales portantsur quatre cases vides de la table de Mendeleïev arithmétique, sur l’extension du spectromètre deSato–Tate à la multiplication complexe par Z[ω], et sur la traversée de la fenêtre historique ROCA(2016–2017). Chaque mesure est reproductible : le code source Python pur (sans dépendance) estpublié en regard, et les constantes numériques sont rapportées au chiffre près. Nous ne prétendonsdémontrer aucune conjecture ouverte; nous publions un cahier de laboratoire honnête, où la carteprédit et la console confirme
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DOI: 10.5281/zenodo.22314058
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