Moving prime environments and arithmetic cluster limits

The following declaration describes the research and the published source edition: The research was organized and directed by the author with substantial assistance from GPT-6 Astra (OpenAI), the only language-model system used for this article. Assistance included exploring proof routes, drafting and reorganizing mathematical arguments, writing finite diagnostic programs, checking references, producing the manuscript and conducting adversarial reviews through separate instances. The author determines the claims and scope and accepts responsibility for the arguments, references and remaining errors. AI-assisted checks are not independent human peer review. The mathematical claims rest on the printed arguments and specified external results, not on model output or the success of finite tests. For this Hexagon deposit, OpenAI Codex assisted source packaging, compilation checks and the preparation of submission metadata. This preparation does not constitute independent mathematical peer review. The published edition used for this deposit, including its supplementary source material, is available at https://doi.org/10.5281/zenodo.22878806. At the time of preparation for Hexagon, the submitter declares the text primarily AI-generated and declares that no part of the contribution has been checked and understood by a human.