What was already known
Alpöge’s earlier three-variable Jacobian counterexample, credited in contemporary accounts to Claude Fable, yields a Poisson counterexample on three canonical pairs. This construction reduces the count to two pairs.
AI involvement
Christopher D. Long credits an interactive ChatGPT 5.6 Sol session with the new four-variable map and Hamiltonian correction. Long directed and edited the manuscript; Claude Fable supplied separate algebraic audits.
The original provenance expressly attributes the new four-variable map and Hamiltonian correction to an interactive ChatGPT research session, beyond drafting or routine computation.
- Supporting source (arxiv.org) — Exact computational audit and its stated coverage
Importance
If the full argument is correct, it disproves the rank-two conjecture and extends the counterexample to every higher canonical rank.
Limitations
Two independent exact-arithmetic programs check the displayed brackets, coordinate inverse and fiber calculations. A completed human or machine check of every deduction from those identities to the theorem is not documented.
Technical verification
The paper reports two exact, independent programs checking the six canonical brackets, coordinate inverse, coefficient identities, collision and fiber calculations. These cover the computation-heavy construction; no documented completed human or machine review of every logical step tying the computations to the full conjecture statement was found. The programs were not run in this review.
Independent checking
Exact symbolic polynomial and rational-arithmetic checks
- Checker
- The paper's SymPy and independent sparse-polynomial audit programs
- Scope
- Six Poisson bracket identities, source inverse, coefficient identity and noninjective three-point fiber for the displayed four-variable map
- Basis
- Reported by the linked checker
Human understanding
The manuscript explains the coordinate and Hamiltonian mechanism, but it does not identify a human explanation or account of understanding separate from model-assisted organization. Few is a missing-evidence default.
Correction and publication history
If the full argument is correct, it disproves the rank-two conjecture and extends the counterexample to every higher canonical rank.
Read this published revision