What was already known
The strict converse was known; Nam, Sly, and Zhang conjectured the positive direction, including the boundary. Nonreconstruction alone does not construct the required IID factor.
AI involvement
OpenAI credits itself as manuscript author and states that its internal frontier model produced the results in this release. Individual human input is not detailed enough to establish task-only autonomy.
OpenAI is named as author on the manuscript, and its release attributes the results to an internal frontier model. The individual human input is not detailed enough to establish autonomy.
Importance
Settles the exact IID-factor threshold for free Ising spins on regular trees.
Limitations
The selected Lean scope gives almost-sure equivariance for each fixed automorphism, not the manuscript abstract’s pointwise every-label convention; no finitary coding bound is included.
Technical verification
OpenAI reports a Lean proof matching the selected claim.
Formal checking
The pinned OpenAI Lean scope note identifies this selected claim.
- Checker
- OpenAI-reported Lean 4 / Comparator
- Scope
- For every d≥3 and β≥0, the free zero-field ferromagnetic Ising law on the infinite d-regular tree is a factor of IID exactly when tanh β≤1/√(d−1), including equality.
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Few is the default when evidence for a higher level is insufficient. No named person is documented as explaining the central argument for this assessed contribution.
Correction and publication history
Primary manuscript, attribution, prior-work, and scoped checking review for non-notifying local inclusion.
Read this published revision