2412.15561
Spirals, Tic-Tac-Toe Partition, and Deep Diagonal Maps
Zhengyu Zou
correctmedium confidence
- Category
- Not specified
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper rigorously proves Tk preserves both spiral classes (Theorem 1.1) and establishes precompactness for k=3 (both types) via four explicit T3-invariants and for k=2 (type-β) via Poisson Casimirs. The candidate solution instead appeals to a cluster-positivity/monodromy-invariant argument that is not established for planar Tk in this setting and contradicts the paper’s sign constraints for corner invariants of type-β 2-spirals. Key steps in the candidate’s proof require unproven positivity, subtraction-free dynamics, and properness claims.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} specialist/solid
\textbf{Justification:}
The work identifies robust geometric classes (type-α/β spirals) invariant under deep-diagonal maps and proves precompactness in central cases (k=3 and k=2 type-β). It contributes new birational and invariant structures for T3 and gives a clean geometric invariance proof. Exposition is clear and well illustrated. Minor clarifications (e.g., compactness via bounded corner invariants and explicit handling of backward orbits in the proof flow) would further strengthen the presentation.