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Math-2696 · mathematics · Analysis

Linear rigidity of bi-Lipschitz equivalent separable Banach spaces

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
If two separable real Banach spaces are joined by a bi-Lipschitz bijection, must they be linearly isomorphic?

Short English statement

If two separable real Banach spaces are joined by a bi-Lipschitz bijection, must they be linearly isomorphic?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2696-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2696

The displayed question has a negative answer under the manuscript's stated construction. Its counterexample meets the conditions in this problem and defeats the universal assertion, so this is a full disproof claim. The supporting result is manuscript 1 (family 324; 2026-09-24).

Linear rigidity of bi-Lipschitz equivalent separable Banach spaces

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 324; 2026-09-24), the asserted result is that a global bi-Lipschitz bijection can exist between separable real Banach spaces for which no linear isomorphism exists.

Mathematical statement of the claimed result

Manuscript 1.
The claimed counterexample consists of separable real Banach spaces $X,Y$ and a bijection $\Psi:X\to Y$ with \[ \tfrac4{21}\|s-t\|_X\le\|\Psi(s)-\Psi(t)\|_Y\le\tfrac{76}{25}\|s-t\|_X \quad(s,t\in X). \] Yet $X$ contains a linearly isometric copy of $c_0(\ell_2)$, the space of $\ell_2$-valued null sequences with supremum norm, whereas $Y$ contains no closed subspace linearly isomorphic to it. Thus the metric equivalence does not yield a linear Banach-space isomorphism.

Outline of the argument

For manuscript 1 (family 324; 2026-09-24), weighted graph spaces give a linear obstruction detected by complete continuity. A triangular change of variables is localized into moving Hilbert slots with common motion budgets and orthogonality; finite inverses converge to an onto nonlinear equivalence.

Formal verification

For manuscript 1 (family 324; 2026-09-24), OpenAI lists Lean statements for this assertion: a global bi-Lipschitz bijection can exist between separable real Banach spaces for which no linear isomorphism exists.

The central assertion is within the documented formal scope; the additional applications retain their own scope limits.

The linked family Comparator checks are C0Absorption.lean, LipschitzEquivalence.lean.

Their named declarations include OAI.C0Absorption.main_result, OAI.LipschitzCounterexample.main.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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