What was already known
The general inequality χ≤χ* was known, and equality was established in regular convex settings. The finite-entropy equality question asked whether finite microstates entropy forced equality.
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
Answers the finite-entropy equality question negatively with a quantitative gap.
Limitations
The example has a fixed finite number of variables and uses the original microstates definition with operator-norm cutoff and limsup over matrix sizes.
Technical verification
OpenAI reports a Lean proof matching the selected claim.
Formal checking
The pinned Lean note constructs a bounded self-adjoint tuple with both entropies finite and a gap of at least one half.
- Checker
- OpenAI-reported Lean 4 / Comparator
- Scope
- There is a bounded self-adjoint tuple X in a von Neumann algebra with faithful normal tracial state for which both microstates and nonmicrostates free entropy are finite yet −∞<χ(X)≤χ*(X)−1/2<∞.
- 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