Uniformly discrete free spaces with AP but no BAP

Source and verification

This article explains family 330 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

Uniformly separated metric points give a Lipschitz-free space with the approximation property. Family 330 claims that they need not give bounded approximation. Its construction places an obstruction to each proposed norm bound in a separate bounded block, then joins all blocks at one basepoint. (OpenAI, 2026)

🏷️ Preliminaries: free spaces and approximation bounds

For a pointed metric space $(M,o)$, ${\mathrm{Lip}}_0(M)$ is the space of real Lipschitz functions vanishing at $o$. Its norm is the Lipschitz constant. The associated 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 map $x\mapsto {\delta }_x$ preserves distances. A pointed Lipschitz map induces a bounded linear map of free spaces with the same Lipschitz bound.

AP means uniform approximation of the identity on each compact set by bounded finite-rank linear operators. The property $\lambda$-BAP requires all needed operators to have norm at most the same $\lambda$. BAP means $\lambda$-BAP for some finite $\lambda$; MAP is $1$-BAP.

Uniform discreteness means that all distinct points have distance at least one fixed positive number. Properness means that closed bounded balls are compact. A bounded infinite uniformly discrete set is not compact, so these two metric conditions are quite different.

Manuscript claim

There is a countable, unbounded, nonproper pointed metric space with $d(x,y)\ge 1$ for distinct points whose real free space has AP but fails $\lambda$-BAP for every finite $\lambda$. (OpenAI, 2026)

The result deliberately lies outside the countable proper case. It also does not say that an individual bounded uniformly discrete free space fails BAP.

🏷️ A bounded obstruction for each norm budget

For each integer $p\ge 1$, the source constructs a bounded countably infinite block $M_p$, a finite anchor set $A_p$ containing the basepoint, and

$$B_p=300p^2+150p+5$$

such that $\mathrm{diam}{M}_p\le B_p$, distinct points are at distance at least one, and every finite-rank $T$ with $\Vert T\Vert \le p$ satisfies

$$\underset{a\in A_p}{\max }\Vert T{\delta }_a-{\delta }_a\Vert \ge \frac{1}{2}.$$

The quantifiers matter: the same finite anchors obstruct every finite-rank range, not just projections onto one prescribed subspace.

A finite metric space alone could never provide this obstruction: its entire free space is finite-dimensional, and the identity has norm one. Each $M_p$ must be infinite.

For coefficients on a bounded uniformly discrete block, the source proves

$$\frac{1}{2}\Vert c{\Vert }_1\le \Vert \sum _{x\ne o}{c}_x{\delta }_x\Vert \le B_p\Vert c{\Vert }_1.$$

The upper bound follows by the triangle inequality and $d(x,o)\le B_p$. For the lower bound, prescribe $f(o)=0$ and $f(x)=\frac{1}{2}\mathrm{sign}(c_x)$. Differences of values are at most one, so uniform separation makes $f$ $1$-Lipschitz.

Thus each block is linearly isomorphic to ${\ell }_1$, and coordinate truncation gives $2B_p$-BAP. It can fail $p$-BAP while retaining a larger finite bound. At $p=1$, this also gives an equivalent norm on real ${\ell }_1$ failing MAP.

🏷️ Wedges separate AP from BAP

Identify the basepoints of all blocks. For points in different blocks, use

$$d(x,y)=d(x,o)+d(y,o).$$

Collapsing every other block to $o$ is a pointed $1$-Lipschitz retraction onto $M_p$. Its linearization and the inclusion show that $\mathcal{F}(M_p)$ is contractively complemented in $\mathcal{F}(M)$.

If the whole free space had $\lambda$-BAP, composition with these two contractions would give $\lambda$-BAP in every block. Choosing an integer $p\ge \lambda$ contradicts the anchor obstruction.

Why does AP survive? Free spaces over this metric wedge form an ${\ell }_1$ sum of the block free spaces. Truncation to finitely many blocks converges uniformly on a compact set: pointwise tail convergence is upgraded to uniform convergence by a finite compact net and the contractive truncation bound. Within each of the finitely many retained blocks, approximate by finite-rank operators. Only finitely many norm bounds occur, so the resulting operator is bounded.

As the compact set and accuracy change, more blocks may be needed and the operator norms may grow. This is precisely what AP permits and BAP forbids.

🏷️ The source of the block obstruction

The manuscript attaches large finite colored graphs to finitely many anchors, obtaining a bounded infinite block. Labels record capped distances to color classes. Small metric balls inside the source graphs remain uniformly bounded in size.

Given a finite-rank operator, a small perturbation lets its range be represented on one finite support set. Its values become signed measures ${\mu }_x$ of total mass one. Positivity is not assumed. Accurate anchor images force substantial pairing with localized nonnegative bump functions.

A long nonbacktracking walk would be matched if nearby support points could follow it with compatible labels. Many independent colors make such matches rare. At a good vertex, the maximal matching length in each direction singles out at most one exceptional direction per support point.

The source then combines a bump at one endpoint with the negative bump at the other to make a single Lipschitz test on an edge. Averaging over a high-degree graph divides the loss from exceptional directions by the degree; rare bad vertices contribute little. The resulting pairing is too large for the assumed operator norm.

Controlling total variation, rather than pretending the signed measures are positive, is essential to this estimate. It supplies the finite anchor obstruction used in the wedge argument.

🏷️ Nonlinear approximation

The paper also applies Kalton’s net theorem to deduce that this free space is not approximable by an equi-uniformly continuous sequence of maps with relatively compact ranges converging pointwise to the identity. This is a separate nonlinear approximation notion; AP alone does not provide the common modulus of continuity it requires.

🏷️ Key steps and proof outlines

Finite rank first becomes finite support

A finite-rank operator on a Lipschitz-free space can have range vectors with infinite support. The proof therefore writes $T_{0}u={\sum }_{i=1}^r{\varphi }_i(u)v_i$ and approximates each $v_i$ by a finite combination $w_i$ of evaluation vectors. Choosing the errors so that ${\sum }_i\Vert {\varphi }_i\Vert \Vert v_i-w_i\Vert <\eta$ gives an operator $T$ with one common finite support and $\Vert T-T_0\Vert <\eta$. A sufficiently small $\eta$ preserves the required anchor approximation while allowing the convenient bound $\Vert T\Vert \le 2p$.

This reduction is essential because the subsequent graph argument is allowed to depend on the support of the proposed approximating operator. The metric block and its finite anchor set, however, were fixed before that operator was presented. (OpenAI, 2026)

Signed measures encode an arbitrary linear operator

Write $T{\delta }_x={\sum }_{j\ne o}{c}_x(j){\delta }_j$ and add mass $1-{\sum }_{j\ne o}{c}_x(j)$ at the basepoint $o$. This defines a signed measure ${\mu }_x$ of total mass one, satisfying

$$\int gd{\mu }_x=(T{\delta }_x)(g-g(o))+g(o).$$

Separation of the metric by one lets a sign function on the finite support have Lipschitz constant at most two. Testing against it bounds total variation by $V=1+4pB$, where $B$ bounds the block diameter. Positivity of the measures is neither available nor needed.

The label map $h$ supplies a slowly varying bump

$$b_x(t)=L{\left(1-\frac{\Vert h(x)-h(t){\Vert }_{\infty }}{K}\right)}_{+}.$$

The nearby anchor has the same label as $x$, so the bump equals $L$ there. Its small Lipschitz constant $L/K$ converts anchor approximation into $\int b_{x}d{\mu }_x\ge 3L/4$, even though the operator norm may be as large as $2p$.

One scalar test compares two neighboring measures

For a graph incidence $e:x\to y$, the continuation construction associates exceptional state sets $E_e,E_{\bar{e}}$. Off these sets, one can prescribe the values $b_x$ and $-b_y$ simultaneously as one 1-Lipschitz function. Pairs of states at distance at least $2L$ are harmless because the value difference is at most $2L$. For closer pairs, the exceptional-direction lemma rules out incompatible positive prescriptions.

The finite prescribed function extends to the whole block by the formula $F(z)={\min }_j(v(j)+d(z,j))$, followed by clipping to $[-L,L]$. Denote the result by $f_e$. Then

$$\int f_{e}d({\mu }_x-{\mu }_y)\ge \frac{3L}{2}-2L(|{\mu }_x|(E_e)+|{\mu }_y|(E_{\bar{e}})).$$

The same integral is at most $2p$ in absolute value, since $d(x,y)\le 1$ and $\Vert T\Vert \le 2p$. Total variation makes the lower estimate valid for arbitrary signs of the coefficients.

Averaging removes dependence on support size

At a good vertex, each state is exceptional in at most one of the $\Delta$ incidence directions. Averaging therefore costs at most $V/\Delta$, independent of how many states lie in the common support. At a bad vertex the crude cost is $V$; the rare-walk construction makes their proportion $\beta$ small. Thus the expected exceptional mass is at most $V/\Delta +V\beta <1/10$ at each endpoint. Reversal preserves uniform incidence sampling, including the prescribed loop convention.

Taking expectations in the scalar-test inequality and using $L=10p$ forces $2p>11L/10=11p$, a contradiction. The graph stage is chosen after the operator support, but all blocks and anchors were fixed earlier. This establishes the uniform finite-anchor obstruction.

Finally, bounded separated blocks have free spaces equivalent to ${\ell }_1$ and hence possess bounded approximation individually. Their wedge gives an ${\ell }_1$ sum, whose approximation property follows by finite-block truncation and finite-rank approximation on those blocks. A global uniform bound would restrict, through the canonical block inclusion and retraction, to a forbidden operator on a sufficiently large block. This explains how AP survives while BAP fails.

See Also

• The cotype–cotype problem uses ordinary AP while carefully avoiding BAP.
• Nonlinear equivalence uses free-space linearization to construct a different separation between metric and linear properties.

🏷️ Manuscripts and verification scope

• A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The page selects the uniformly discrete AP-without-BAP example and the quantitative renormings of real ell1. These are separate from the nonlinear approximability consequence. 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 2026. A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces.