Research Notes on ζ(9): Constructions, Computations, and Open Problems

Abstract

Version 0.1 is a work-in-progress research note on the irrationality of ζ(9), documenting research conducted on 24–25 September 2026. This note does not claim a proof of the irrationality of ζ(9). The record documents nine research rounds, including rational-function constructions of linear forms A·ζ(9)+B, a Gram-determinant approach, local written arguments with explicit dependencies and caveats, archived finite computations, and a roadmap of unresolved mathematical questions. The source archive contains research notes, computational scripts, verification artifacts, and a limited abstract formalization layer. Statements are distinguished as written arguments, conditional statements, finite numerical evidence, heuristics, or open problems. In particular, the proposed sharper integrality multiplier is supported by a written proof sketch and finite checks, with unresolved proof-detail caveats. The abstract formal theorems do not constitute formal verification of the concrete ζ(9) constructions. No conclusion over an infinite sequence is inferred from finite tests. AI assistance: ChatGPT and DeepSeek were used for research exploration, drafting, coding, and proof-state auditing. The scope of the computational checks and the limitations of AI-assisted review are disclosed in Section 14 of the note. An AI assistant's assessment is not presented as independent human verification. Licensing: The research note and accompanying research documentation are licensed under CC BY 4.0. Original software files are licensed under the MIT License. Third-party materials retain their original licenses and notices. See the bundled README and LICENSES directory. Files: the research note (PDF), source archive (ZIP), README, and CHANGELOG.