Bi-Lipschitz equivalence without linear isomorphism

Source and verification

This article explains family 324 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

Overview

Family 324 gives two claimed separable counterexamples to recovering linear structure from a bijection with bounded metric distortion. The first separates the presence of $c_0({\ell }_2)$; the second constructs a space that absorbs an entire $c_0$ summand by a bi-Lipschitz map while containing no linear copy of $c_0$. Both constructions make a nonlinear change of variables compatible with a carefully chosen linear obstruction. (OpenAI, 2026) (OpenAI, 2026a)

🏷️ Preliminaries: three notions of equivalence

A bounded linear isomorphism $L:X\to Y$ is a linear bijection with bounded inverse. A bi-Lipschitz equivalence is a possibly nonlinear bijection $\Psi$ such that

$$\begin{array}{c}c\Vert x-x^{\prime }{\Vert }_X\le \Vert \Psi (x)-\Psi (x^{\prime }){\Vert }_Y\\ \le C\Vert x-x^{\prime }{\Vert }_X.\end{array}$$

for fixed $0<c\le C<\infty$. A bi-Lipschitz embedding satisfies the same inequalities without being onto. Onto-ness is a genuine additional requirement in this family.

The space $c_0$ consists of scalar sequences tending to zero, with the supremum norm. Its block-valued version is

$$\begin{array}{c}{c}_0({\ell }_2)\\ =\{(v_i):v_i\in {\ell }_2,\Vert v_i{\Vert }_2\to 0\},\\ \Vert (v_i)\Vert =\underset{i}{\sup }\Vert v_i{\Vert }_2.\end{array}$$

Excluding the block-valued space does not exclude ordinary $c_0$.

The two manuscript claims

The first paper constructs separable real Banach spaces $X,Y$ and an onto map with lower Lipschitz bound $4/21$ and upper bound $76/25$. The space $X$ contains $c_0({\ell }_2)$ linearly isometrically, whereas $Y$ contains no closed linear subspace isomorphic to it. (OpenAI, 2026b)

The second constructs a separable real Banach space $Z$ with no linear copy of $c_0$ and an onto bi-Lipschitz map $Z{\oplus }_{\infty }{c}_0\to Z$. (OpenAI, 2026a)

Neither statement asserts reflexivity. The first obstruction alone should not be strengthened to the second.

🏷️ Lipschitz-free linearization

For a pointed metric space $(M,o)$, let ${\mathrm{Lip}}_0(M)$ be the real Lipschitz functions vanishing at $o$, normed by their Lipschitz constant. The Lipschitz-free space is

$$\begin{array}{c}\mathcal{F}(M)\\ =\overline{\mathrm{span}}\{{\delta }_x:x\in M\}\subset {\mathrm{Lip}}_0(M{)}^{\ast }.\end{array}$$

Here ${\delta }_x(f)=f(x)$. In particular,

$${\delta }_o=0.$$

The identity $\Vert {\delta }_x-{\delta }_y\Vert =d(x,y)$ follows from the Lipschitz inequality and the test $z\mapsto d(z,y)-d(o,y)$.

A Lipschitz map $D:M\to U$ fixing zero has a bounded linearization $q:\mathcal{F}(M)\to U$ with $q{\delta }_x=D(x)$. This converts a nonlinear correction into a linear operator on a larger space. It does not make the original map linear.

In the first construction, $M=U{\oplus }_{2}V$, $V={\ell }_2$, and an onto bi-Lipschitz map $h:M\to U$ gives

$$\begin{array}{c}D(u,v)=h(u,v)-u,\\ q{\delta }_{(u,v)}=D(u,v).\end{array}$$

The source arranges local orthogonality: near finitely many distinct inputs, sufficiently high correction coordinates occupy mutually orthogonal output subspaces. Its localization argument then proves that $q$ is completely continuous: weakly null sequences have norm-null images.

🏷️ An explicit triangular inverse

Put $E=\mathcal{F}(M)$ and use the sum norm on $E\oplus V$. Define

$$\begin{array}{c}{Q}_1(e,v)=(qe,v),\\ B(e,v)=e+{\delta }_{(qe,v)}.\end{array}$$

Applying $q$ gives the crucial identity

$$qB(e,v)=h(qe,v).$$

Given $z\in E$, recover $(u,v)=h^{-1}(qz)$ and set $e=z-{\delta }_{(u,v)}$. Then

$$qe=qz-[h(u,v)-u]=u,$$

so $B(e,v)=z$. This proves the inverse identity directly; the quantitative Lipschitz estimates still require the bounds on $h$.

For a linear $Q:A\to H$, form the graph space

$$\begin{array}{c}\Vert (s_i){\Vert }_{Z_Q}\\ =\max \left\{\sum _i{2}^{-i}\Vert s_i\Vert ,\underset{i}{\sup }\Vert Q{s}_i\Vert \right\}.\end{array}$$

Its elements satisfy both

$$\begin{array}{c}\sum _i{2}^{-i}\Vert s_i\Vert <\infty ,\\ (Q{s}_i)\in c_0(H).\end{array}$$

Applying $B$ coordinatewise gives the proposed equivalence between $Z_{Q_1}$ and $Z_q$. The identity $qB=h{Q}_1$ controls the output coordinates as well as the weighted input norm.

The $V$ coordinates place $c_0(V)$ in the first space. In the second, complete continuity of $q$ forces Hilbert-block directions to disappear in fixed output coordinates. A disjoint-support selection would then force an ordinary $c_0$ copy into a weighted ${\ell }_1(E)$ space. The source excludes this using weak sequential completeness of its free-space input. Local orthogonality and that obstruction are essential analytic steps.

🏷️ Absorbing ordinary null sequences

The second paper uses a related identity with a different construction. Suppose

$$g:E{\oplus }_{\infty }U\to U$$

is an onto bi-Lipschitz map, $Q:Z\to U$ is linear, and $K:E{\oplus }_{\infty }U\to Z$ is Lipschitz with

$$QK(t,u)=g(t,u)-u.$$

Then $F(b,t)=b+K(t,Qb)$ satisfies $QF(b,t)=g(t,Qb)$.

For $z\in Z$, write $(t,u)=g^{-1}(Qz)$ and $b=z-K(t,u)$. The identity gives $Qb=u$, hence $F(b,t)=z$. This is why lifting the correction is enough. A Lipschitz right inverse of $Q$ is neither asserted nor needed.

For $E=c_0$, the manuscript builds $g$ by shifts through finite coordinate blocks, with input-dependent weights. Uniform inverse bounds and a terminal coordinate establish onto-ness. It constructs $Z$ from point differences tested against localized scalar functions. A hypothetical $c_0$ basis would have uniformly bounded signed sums; selected tests instead force those sums to grow. That contradiction excludes linear $c_0$.

Restricting $F$ to the $c_0$ summand gives a nonlinear copy of $c_0$ in $Z$. Combined with the classical embedding of every separable metric space into $c_0$, the source obtains a universal separable metric target without a linear $c_0$ subspace.

🏷️ Key steps and proof outlines

Local orthogonality is designed into the Hilbert-space rearrangement

For a frozen family of unit directions ${\xi }_n$ in finite-dimensional Hilbert blocks, decompose $u_n$ into its component along ${\xi }_n$ and its orthogonal complement. The source rearranges the scalar components together with the extra Hilbert input coordinates, leaving the orthogonal complements in place. The sum of squared norms is unchanged, so the frozen map is an isometry with an explicit inverse.

The directions are then allowed to depend on the input. Finite stages and separating coordinates ensure that, near finitely many distinct points, high correction coordinates occupy orthogonal output blocks. Quantitative bounds on how the directions move preserve the two-sided Lipschitz estimates. Onto-ness is proved through the explicit inverse construction, not inferred from a lower Lipschitz bound. (OpenAI, 2026b)

Localization turns orthogonal corrections into complete continuity

Let $D=h-{\pi }_U$ and let $q$ be its free-space linearization. To show $q$ sends weakly null sequences to norm-null sequences, the manuscript localizes free-space molecules near finitely many input centers. A cutoff ${\varphi }_{z,s}$ equals one on $B(z,s/2)$, vanishes outside $B(z,s)$, and has Lipschitz constant at most $2/s$.

If $f(z)=0$ and $\mathrm{Lip}(f)\le C$, then $|f|\le Cs$ on the cutoff support. The product rule for differences therefore gives $\mathrm{Lip}(f{\varphi }_{z,s})\le 3C$. The same bound holds for the complementary localized test. These estimates let the proof split a functional without losing control of its dual norm.

On separated pieces, high correction coordinates are orthogonal. Their norm cannot remain large on a weakly null sequence without producing a bounded Lipschitz test detecting it. Finite low-coordinate parts are handled by weak convergence. The localization lemma combines the two controls to obtain complete continuity of $q$.

The graph norm records both summability and a vanishing tail

The space $Z_Q$ sits as a closed graph inside a weighted ${\ell }_1(A)$ space times $c_0(H)$. Closedness follows because convergence in the weighted sum gives convergence of each input coordinate, while convergence in $c_0(H)$ gives convergence of its images; continuity of $Q$ identifies the two limits. This proves completeness and retains the condition $Q{s}_i\to 0$.

The triangular map $B$ controls both parts of this graph norm. Its Lipschitz estimate controls the weighted ${\ell }_1$ term, and $qB=h{Q}_1$ controls the output supremum and null-sequence condition. The explicit inverse verifies the same properties in reverse. Thus coordinatewise application really maps the full graph spaces onto one another, not just finitely supported sequences.

A hypothetical linear copy of $c_0({\ell }_2)$ in the second graph space is then tested on Hilbert block vectors. Complete continuity makes their fixed output coordinates small. A gliding-hump selection shifts the remaining norm into disjoint weighted-${\ell }_1$ coordinates, forcing a forbidden $c_0$ structure there. Weak sequential completeness of the free-space input is the external linear obstruction used in this step.

Ordinary c0 absorption uses a different detector construction

In the second manuscript, coordinate weights form local partitions of unity on finite cubes. Their denominator is bounded below because the grid includes boundary points and every point lies close to a grid vertex. These weights define state-dependent shifts through finite blocks. Uniform tail estimates show the output still belongs to the null-sequence space, and terminal scalar coordinates plus a finite-dimensional fixed-point argument establish surjectivity. (OpenAI, 2026a)

The space $Z$ is completed from point-difference molecules using seminorms built from selected localized tests. Their support and scale budgets make the correction lift $K$ Lipschitz. The identity $QK=g-u$ then gives the triangular inverse already calculated, so absorption follows once the construction estimates hold.

To exclude a linear $c_0$ copy, take a hypothetical normalized sequence $(m_i)$ with all finite signed sums bounded by $M$. Every fixed functional satisfies ${\sum }_i|u(m_i)|\le M\Vert u\Vert$, by choosing matching signs. The source selects individual detectors on a subsequence and combines compatible detectors whose supports and tails remain controlled. Their evaluations force a signed sum to exceed every fixed bound, contradicting the $c_0$ estimate. This detector argument excludes ordinary $c_0$, a stronger conclusion than the first construction’s Hilbert-block obstruction.

See Also

• Tingley’s problem treats exact sphere distances rather than bounded distortion of whole spaces.
• Free-space approximation introduces another use of linearized metric maps, this time to distinguish approximation properties.

🏷️ Manuscripts and verification scope

• Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic — pinned manuscript and companion source.
• Bi-Lipschitz Absorption of $c_0$ Without a Linear Copy of $c_0$ — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The page lists both the quantitative Hilbert-block counterexample and the ordinary c0 absorption result, with their distinct linear obstructions. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026a. Bi-Lipschitz Absorption of c₀ Without a Linear Copy of c₀.

🐻  OpenAI 2026b. Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic.