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The separable quotient problem for Banach spaces

Every infinite-dimensional Banach space has a separable infinite-dimensional closed subspace. The separable quotient problem asks for the dual statement: does every infinite-dimensional Banach space XX (real or complex) admit a bounded linear surjection onto a separable infinite-dimensional Banach space, equivalently a closed subspace MM with X/MX/M separable and infinite dimensional? Positive answers were known for separable spaces (Johnson-Rosenthal), dual spaces (Argyros-Dodos-Kanellopoulos), spaces of density below b\mathfrak b (Saxon-Sanchez Ruiz), and, consistently, densities at least ℵω\aleph_\omega under large cardinals (Dodos-Lopez-Abad-Todorcevic). Does every infinite-dimensional Banach space have a separable infinite-dimensional quotient?

Result
Independent(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Banach space theory; set-theoretic functional analysis
Posed by
Traditionally attributed to Stefan Banach and Stanislaw Mazur; early written formulation by H. P. Rosenthal
Year posed
1969
Years open
57y
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
50 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for F∈{R,C}\mathbb F\in\{\mathbb R,\mathbb C\}, ZFC proves that if 2ℵ02^{\aleph_0} is real-valued measurable then every infinite-dimensional Banach space over F\mathbb F has a separable infinite-dimensional quotient, and that under CH there is a counterexample of density ℵ1\aleph_1. So the problem is independent of ZFC if ZFC plus a measurable cardinal is consistent; restricted to density ℵ1\aleph_1 it is independent relative to ZFC alone (Corollary 1.2, with Martin's axiom). It does not give a ZFC answer, does not remove the measurable-cardinal strength for the unrestricted statement, and leaves open which other axioms decide it.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family is a single manuscript (September 23, 2026).

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the problem. ZFC proves (i) if the continuum is real-valued measurable, every infinite-dimensional Banach space over R\mathbb R or C\mathbb C has a separable infinite-dimensional quotient, and (ii) under CH some space has none; hence independence relative to a measurable cardinal, and full independence for density ℵ1\aleph_1 relative to ZFC. The challenge lean/ComparatorChallenges/SeparableQuotientNegative.json and its solution module OAI.Analysis.SeparableQuotients.Main exist at the pinned commit (found via lean/docs/323.md) but are not in the formalization catalogue. The statement was read here: CH implies the separable-quotient assertion fails over both R\mathbb R and C\mathbb C. That is narrower than the headline: the positive direction and the independence conclusion are not formalised. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice. The independence result is relative to the consistency of a measurable cardinal. Listed as Unreviewed rather than Lean-checked because its formal statement covers only one direction of the independence claim (CH implies a counterexample).

Sources

Changelog1 change

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