What was already known
The separable quotient problem remained open for arbitrary Banach spaces. Positive cases included separable spaces and many dual spaces, while earlier set-theoretic results did not provide this CH counterexample in both scalar fields.
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
Establishes failure of the separable quotient assertion under the continuum hypothesis.
Limitations
This is the negative direction under CH. The linked Lean note does not check the manuscript’s positive consistency direction or full relative-independence conclusion.
Technical verification
OpenAI reports a Lean proof matching the selected claim.
Formal checking
The pinned Lean note proves CH-conditional counterexamples for real and complex Banach spaces.
- Checker
- OpenAI-reported Lean 4 / Comparator
- Scope
- Assuming the continuum hypothesis, there are infinite-dimensional real and complex Banach spaces with no bounded linear surjection onto any infinite-dimensional separable Banach space.
- 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