Silver-rank zeros of the tensor-curl eta invariant on lens spaces: silver S-units, a Pell ladder, and a period law

Arithmetic results for silver-rank zeros of the specified tensor-curl eta invariant: S-unit and Pell constructions and residue-class period laws. Bounded computational searches are distinguished from classification theorems. AI-assisted research, computation, checking and editing used OpenAI Codex and Anthropic Claude. Luqman Mahmood directed the research and is responsible for its claims and final text. Internal checks are not external peer review or an establishment of priority. CC BY 4.0 covers scholarly documents and mathematical data. Software source and executable verification scripts are excluded; their rights remain reserved unless an individual file states otherwise. Proprietary engineering implementations are not included.