Nonconvergence of randomized subtraction games with odd moves

For every finite nonempty set of positive odd subtraction moves, with uniform choice among the legal moves, the probability of winning for the player to move does not converge as the heap grows. A positive finite-window invariant proves the result without a coprimality or root-simplicity assumption. This answers the separate all-odd nonconvergence assertion in Capitelli and Somma; the different MathDB root-simplicity conjecture is not resolved.

AI-assisted preprint by Myeongjin Kang, with no affiliation. Internal AI review is not external journal peer review. The accompanying Lean certificate covers the precise scoped main statement; inspect the formal endpoint, assumptions and provenance in the supporting package. Prior-art investigations are documented bounded searches, not worldwide-priority certification. Original prose is offered under CC BY 4.0 and original code/Lean under MIT; third-party rights and existing notices are preserved. The public supporting ZIP contains original proof sources and verification evidence, with bibliographic links and hashes instead of redistributed third-party source manuscripts.

Recorded prior-art/source-access limits: No earlier full resolution or directly subsuming theorem found in inspected primary literature; finite searches do not certify absolute priority.