OpenAI · 323-01 · 2026-09-23 · Functional analysis
Relative independence of the separable quotient problem
preprints/Relative-independence-of-the-separable-quotient-problem-September-23-2026
Family 323. Relative independence of the separable quotient problem.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Establishes, relative to the consistency of a measurable cardinal, that the separable quotient problem is independent of ZFC. The assertion that every infinite-dimensional Banach space has a separable infinite-dimensional quotient can hold for all real and complex Banach spaces, whereas the continuum hypothesis yields counterexamples over both fields.
The separable quotient problem asks whether every infinite-dimensional Banach space has a separable infinite-dimensional quotient. We prove that this statement is independent of ZFC, relative to the consistency of ZFC with a measurable cardinal. This holds over both the real and complex fields. For spaces of norm density ℵ1, we obtain independence relative to the consistency of ZFC alone.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 323: Independence of the separable quotient problem. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.