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Kalton's Problem 2: does a bi-Lipschitz copy of c0c_0 force a linear copy of c0c_0?

Godefroy, Kalton and Lancien (2000) proved that a Banach space bi-Lipschitz equivalent to c0c_0, or to a linear subspace of c0c_0, is linearly isomorphic to such a space. Those theorems concern an equivalence of whole spaces and say nothing about a space that merely contains a bi-Lipschitz image of c0c_0. Kalton recorded that question as Problem 2 in his 2008 survey of the nonlinear geometry of Banach spaces, and Hajek, Johanis and Schlumprecht described it as apparently open in 2024. If a Banach space contains a bi-Lipschitz image of c0c_0, must it contain a closed linear subspace isomorphic to c0c_0?

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Construction
Field
Nonlinear geometry of Banach spaces
Posed by
Nigel J. Kalton, The nonlinear geometry of Banach spaces, Rev. Mat. Complut. 21 (2008), Problem 2
Year posed
2008
Years open
18y
Solved
2026-09-26
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
28 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: there is a separable real Banach space ZZ with no closed subspace isomorphic to c0c_0 and an onto bi-Lipschitz map F:Z⊕∞c0→ZF:Z\oplus_\infty c_0\to Z. Restricting FF to the c0c_0 summand embeds c0c_0 bi-Lipschitzly in ZZ, so Problem 2 has a negative answer. With Aharoni's theorem, ZZ is bi-Lipschitz universal for separable metric spaces, and ZZ, Z⊕∞c0Z\oplus_\infty c_0 form a second separable Lipschitz-isomorphic, linearly non-isomorphic pair. Real scalars only; it does not address analogous questions for other classical spaces.

What the AI did

The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result, across roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. This result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The README also cautions that unformalized results could have issues.

Verification

No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the TeX source, read against Kalton's Problem 2 as the manuscript quotes it; Kalton's survey itself was not opened here. Lean-checked on the Comparator challenge C0Absorption (OAI.C0Absorption.main_result, OAI/Analysis/C0Absorption/Main.lean), listed in the release's formalization catalogue. Its statement, read here, asserts a complete separable real normed space ZZ with no bounded-below linear map from c0c_0, a surjective bi-Lipschitz map Z×c0→ZZ\times c_0\to Z (sup-norm product), a bi-Lipschitz map c0→Zc_0\to Z, bi-Lipschitz universality for separable metric spaces, and no continuous linear equivalence Z×c0≃ZZ\times c_0\simeq Z: the headline claim in full. Permitted axioms: propext, Quot.sound, Classical.choice. The statement was not independently audited and the development was not rebuilt here.

Sources

Changelog1 change

Discussion