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The Lipschitz isomorphism problem for separable Banach spaces

Banach spaces X,YX,Y are Lipschitz isomorphic if there is a bijection Ψ:X→Y\Psi:X\to Y with c∥s−t∥≤∥Ψs−Ψt∥≤C∥s−t∥c\|s-t\|\le\|\Psi s-\Psi t\|\le C\|s-t\| for constants 0<c≤C0<c\le C. Linear isomorphism implies this. Aharoni and Lindenstrauss (1978) constructed nonseparable Banach spaces that are Lipschitz isomorphic but not linearly isomorphic and asked for separable examples. Ribe and Johnson-Lindenstrauss-Schechtman found separable spaces that are uniformly homeomorphic but not isomorphic, Godefroy-Kalton-Lancien showed c0c_0 is determined by its Lipschitz structure, and Albiac-Kalton gave separable quasi-Banach counterexamples for 0<p<10<p<1. Kalton listed the question as Problem 3 in his 2008 survey. Must two separable Banach spaces that are Lipschitz isomorphic be linearly isomorphic?

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Construction
Field
Nonlinear geometry of Banach spaces
Posed by
Israel Aharoni and Joram Lindenstrauss, Uniform equivalence between Banach spaces, Bull. Amer. Math. Soc. 84 (1978), pp. 281-282; recorded as Problem 3 by Nigel Kalton (2008)
Year posed
1978
Years open
48y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
52 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: there are separable real Banach spaces X,YX,Y and a bijection Ψ:X→Y\Psi:X\to Y with 421∥s−t∥≤∥Ψ(s)−Ψ(t)∥≤7625∥s−t∥\frac{4}{21}\|s-t\|\le\|\Psi(s)-\Psi(t)\|\le\frac{76}{25}\|s-t\|, where XX contains a linearly isometric copy of c0(ℓ2)c_0(\ell_2) and YY contains no subspace isomorphic to c0(ℓ2)c_0(\ell_2), so XX and YY are not linearly isomorphic. The construction uses a near-isometric bi-Lipschitz change of variables between Hilbert spaces with a local orthogonality property, Lipschitz-free spaces, and a triangular graph-space construction. Real scalars only; it asserts no reflexive or complex analogue and does not say YY lacks ordinary c0c_0. The companion on c0c_0 absorption gives a second, independent counterexample and is a separate entry.

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 the question as Aharoni-Lindenstrauss and Kalton posed it according to the manuscript; the proof was not refereed. Lean-checked on the Comparator challenge LipschitzEquivalence (OAI.LipschitzCounterexample.main, OAI/Analysis/LipschitzEquivalence/Main.lean), listed in the release's formalization catalogue. Its statement, read here, asserts separable complete real normed spaces X,YX,Y, a bijection with Lipschitz bounds 4/214/21 and 76/2576/25, no continuous linear equivalence between them, a linear isometric copy of c0(ℓ2)c_0(\ell_2) in XX, and no bounded-below linear map from c0(ℓ2)c_0(\ell_2) into YY: the headline claim in full, with its constants. Permitted axioms: propext, Quot.sound, Classical.choice. The statement was not independently audited and the development was not rebuilt here.

Sources

Changelog1 change

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