Bi-Lipschitz coordinates at regular RCD points: controlling distortion across scales

Source and verification

This article explains family 357 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.

🏷️ Tangent spaces and open coordinate charts

A metric space can look increasingly Euclidean under magnification without a chosen coordinate system having uniformly bounded distortion. Family 357 concerns the gap between these two statements. Its manuscript claims that every regular point of a noncollapsed RCD space has an open bi-Lipschitz coordinate neighborhood, with distortion bounded in terms of the dimension alone. The proof constructs successive corrections of harmonic coordinates and controls their cumulative effect on distances. (OpenAI, 2026)

For a pointed metric space $(X,d,p)$, a tangent at $p$ is a pointed Gromov–Hausdorff limit of rescaled spaces $(X,r_j^{-1}d,p)$ with $r_j\downarrow 0$. Gromov–Hausdorff convergence means, roughly, that bounded regions can be matched with distance errors tending to zero. A point is $n$-regular when every such tangent is Euclidean ${\mathbb{R}}^n$ with its usual distance and distinguished origin.

A map $F:U\to V\subset {\mathbb{R}}^n$ is an $L$-bi-Lipschitz chart if it is a homeomorphism and

$$\begin{array}{c}{L}^{-1}d(x,z)\le |F(x)-F(z)|,\\ |F(x)-F(z)|\le Ld(x,z),\\ x,z\in U.\end{array}$$

Here both $U$ and $V$ must be open. The distance on $U$ is the original ambient distance restricted to pairs in $U$, not the infimum of lengths of curves constrained to stay in $U$. This distinction matters when proving a bound for every pair of points.

The background condition $\mathrm{RCD}(K,n)$ consists of the full curvature-dimension condition $\mathrm{CD}(K,n)$, formulated through optimal transport, together with quadratic Sobolev energy. The quadratic energy gives a Hilbert-space calculus of weak gradients. On an unweighted smooth $n$-manifold, this is the framework corresponding to a lower Ricci bound $K$.

In this paper noncollapsed means that the reference measure is exactly the $n$-dimensional Hausdorff measure ${\mathcal{H}}_d^n$. The spaces are complete, separable, and geodesic, and their measure has full support and is finite on bounded sets. The exact measure, integer dimension, and finite-dimensional condition remain part of the theorem.

Open bi-Lipschitz charts, as stated in the manuscript

For each integer $n\ge 2$, there is a constant $L_n$ such that every $n$-regular point of every noncollapsed $\mathrm{RCD}(K,n)$ space has an open neighborhood bi-Lipschitz homeomorphic to an open subset of ${\mathbb{R}}^n$, with constant $L_n$. The neighborhood can depend on the space and point. The distortion constant depends only on $n$, while $K$ may be any real number. (OpenAI, 2026)

This is stronger than countable rectifiability, which covers almost every point by measurable pieces with bi-Lipschitz parametrizations. It is also stronger than an open bi-Hölder chart. It does not claim that the metric tensor in the resulting coordinates is Hölder continuous, or that the coordinate functions have bounded Laplacians.

🏷️ Why small errors at every scale are insufficient

Suppose a construction changes a length by a factor $1+{\eta }_i$ at dyadic scale $2^{-i}$. Even if ${\eta }_i\to 0$, the cumulative product can diverge. For example,

$$\prod _{i=1}^N\left(1+\frac{1}{i}\right)=N+1.$$

Qualitative Euclidean approximation at small scales therefore does not automatically give a uniform bi-Lipschitz constant. A second obstacle appears when the available errors are energies $E_i$: summability of $E_i$ does not imply summability of $\sqrt{E_i}$. The sequence $E_i=(i+1{)}^{-2}$ demonstrates the difference.

The manuscript permits the matrix describing the required coordinate stretching to grow without bound as the scale decreases. It instead proves a directional estimate for actual differences of coordinate values, and uses that estimate without summing square roots of the energy defects. This is the main connection between the geometric and analytic parts of the proof.

🏷️ Harmonic functions, Gram matrices, and initial open coordinates

A weakly harmonic function satisfies $\Delta u=0$, with the Laplacian defined by integration by parts against test functions. Harmonic functions on the spaces in question have local Lipschitz and second-order Sobolev regularity. After sufficiently strong magnification at the regular point, the paper constructs a harmonic column

$$u=(u_1,\dots ,u_n{)}^T$$

approximating ordinary Euclidean coordinates on a large region. The curvature lower bound rescales from $K$ to $K{r}_{\ast }^2$ when distances are divided by $r_{\ast }$. Thus its negative part can be made arbitrarily small on a fixed working region.

Define the Gram matrix of the weak gradients and its ball averages by

$$\begin{array}{rl}{G}_{ab}& =⟨\nabla u_a,\nabla u_b⟩,\\ R_i(y)& =\frac{1}{m(B_{r_i}(y))}{\int }_{B_{r_i}(y)}Gdm,\\ r_i& =2^{-i}.\end{array}$$

In Euclidean coordinates, $G=I$. In the manuscript’s construction, each $R_i(y)$ is positive definite, and the normalized map

$$U_{i,y}=\frac{R_i(y{)}^{-1/2}(u-u(y))}{r_i}$$

is close to a Euclidean coordinate map on the corresponding rescaled ball. The approximation is uniform over nearby centers and all dyadic scales once the initial magnification is sufficiently strong. Nearby average matrices are relatively close. Across many scales their variation has an arbitrarily small power bound, but no uniform positive lower bound independent of scale is assumed.

The normalized maps already give an open homeomorphism $u$ on a working ball. Injectivity follows by testing each distinct pair at a scale comparable to its distance. For openness, consider a small ellipsoid around $u(x)$ measured in the quadratic form $R_i(x{)}^{-1}$. Approximate Euclidean surjectivity produces successive points whose coordinate residuals shrink at radii $r_i,r_i/4,r_i/16,\dots$. Their displacements form a geometric series. Completeness gives a limit point with the desired coordinate value, and a strict interior margin keeps it inside the ball.

This supplies an open Euclidean domain $\Omega =u(B_1(p))$ on which ordinary smooth corrections can be defined. It does not yet give the final uniform distance bound. (OpenAI, 2026)

🏷️ Positive Bochner mass and accumulated stretching

On a smooth manifold with $\mathrm{Ric}\ge -\kappa$, the Bochner formula for harmonic $w$ gives

$$\frac{1}{2}\Delta |\nabla w{|}^2+\kappa |\nabla w{|}^2\ge |\mathrm{Hess}w{|}^2.$$

The corresponding RCD inequality is measure-valued. Applied to every linear combination of the coordinates, it defines a positive semidefinite matrix-valued measure

$$M=\frac{1}{2}\Delta G+\kappa Gm.$$

For a vector $b$, the scalar measure $b^{T}Mb$ is nonnegative and controls the squared Hessian of $b\cdot u$. The measure may have a singular part, which remains part of the estimates.

Fix an enlargement factor $L$. The scale-normalized mass matrix and its expression on coordinate differences are

$$\begin{array}{rl}{C}_{i,y}& =r_i^{2-n}M(B_{L{r}_i}(y)),\\ D_{i,y}& =R_i(y{)}^{-1}{C}_{i,y}{R}_i(y{)}^{-1}.\end{array}$$

For a positive matrix $A$, write $A[\xi ]={\xi }^{T}A\xi$ and $|\xi {|}_A=\sqrt{A[\xi ]}$. The two inverse factors in $D_{i,y}$ are essential: it is a quadratic form on differences in the space of coordinate values.

The paper compares these local masses with heat averages of $G$. If $P_t$ is the heat semigroup and $\chi$ a cutoff on a much larger region, put

$$\begin{array}{rl}{Q}_i(x)& =P_{r_i^2}(\chi G)(x)+\lambda r_i^{2}I,\\ H_i(x)& =Q_i(x{)}^{-1}.\end{array}$$

The small correction $\lambda >0$ absorbs cutoff and curvature errors. Positivity of the resulting time derivative makes $Q_i$ decrease as $i$ increases; hence $H_i$ increases. Relative comparison with $R_i$ also gives

$$H_i\le H_{i+1}\le (1+\epsilon )H_i.$$

The forms $H_i$ describe how much stretching is needed at each scale. They need not converge to a finite matrix.

For boundedly many nearby patches, let $Z_i$ be the sum of their $D_{i,y}$ and let $S_i$ be a weighted sum with weights in $[0,1]$. Require one weight-one patch whose enlarged mass ball contains the testing ball. The manuscript proves, with a fixed constant $C_{\ast }$,

$$\begin{array}{rl}{Z}_i& \le \epsilon H_i,\\ I+\sum _{i<k}{Z}_i& \le C_{\ast }{H}_k,\\ H_k& \le C_{\ast }\left(I+\sum _{i<k}{S}_i\right).\end{array}$$

These cumulative estimates allow unbounded stretching while measuring all accumulated mass against the appropriate final-scale form. Matrices at different scales need not commute. The inverse-increment argument conjugates a single positive decrement by $Q_i^{-1/2}$, applies scalar spectral inequalities to that one matrix, and then sums quadratic forms. Simultaneous diagonalization is not used. (OpenAI, 2026)

🏷️ Summable directional defects from squared distance

The second main input is more specific than a bound on the total size of $D_{i,y}$. For nearby points $x,z,y$, it states

$$\begin{array}{c}{D}_{i,y}[u(x)-u(z)]\\ \le C{r}_i^2(E_i(x)+E_i(z)),\\ \underset{x}{\sup }\sum _{i\ge 0}{E}_i(x)\le \epsilon .\end{array}$$

The same nonnegative defects work simultaneously for all permitted centers and scales. Their sum becomes arbitrarily small with the initial magnification. (OpenAI, 2026)

The starting point is the squared distance $q^a=d(a,\cdot {)}^2/2$. Laplacian comparison gives a positive defect measure

$$\begin{array}{c}{\mu }_i^a=a_{i}m-\Delta q^a,\\ a_i=n+C_0\kappa r_i^2.\end{array}$$

Solve a Dirichlet problem to replace $q^a$ by $f_i^a$ with constant Laplacian $a_i$ and the same boundary values on an enlarged ball. The normalized source mass, together with $\kappa r_i^2$, defines an energy defect $P_i(a)$.

Its summability comes from volume comparison. Write $A_a(t)=m(B_t(a))/t^n$. Radial integration by parts expresses the mass of ${\mu }_i^a$ in terms of a drop of $A_a$ between fixed multiples of $r_i$. Summing over dyadic radii telescopes. The small-scale volume density is at most the Euclidean one at every center, while the initial-scale volume ratio is uniformly nearly Euclidean. The curvature correction is summable because ${\sum }_i{r}_i^2<\infty$. Neighboring centers need not themselves be regular.

The Poisson replacements have controlled Hessian defect from the metric, including the full measure contribution. Radial derivatives of the harmonic coordinate span can then be approximated by affine combinations of those coordinates. An improvement recurrence across scales, with exponentially decaying weights, enlarges $P_i$ to summable defects $E_i$.

The sharp estimate compares replacements centered at two points. Their difference is harmonic because they have the same constant Laplacian. In normalized coordinates $U$, let this difference be $v$, and project it onto the affine span of $U$. If

$$s^2=E_i(x)+E_i(z),$$

the required residual estimate is $\Vert \nabla (v-b\cdot U){\Vert }_2\le Cs$, with the coefficient $b$ equal to $U(x)-U(z)$ up to $O(s)$.

A radial transport estimate preserves this squared order of error. It is applied to the unnormalized residual $w$ before dividing by its small norm $\alpha$. Its crucial error has the form $Cs\sqrt{\mathcal{A}}$, where $\mathcal{A}$ is an annular squared norm. Young’s inequality absorbs part of $\mathcal{A}$ and leaves $C{s}^2$.

If $\alpha /s$ were unbounded, the normalized residual would converge to a nonzero Euclidean harmonic function orthogonal to all affine functions and satisfying an Euler equation

$$X\cdot \nabla w_{\infty }-w_{\infty }=\ell \cdot X+k.$$

The operator on the left multiplies a homogeneous term of degree $q$ by $q-1$. Thus all terms of degree at least two vanish. The limit is affine, contradicting its orthogonality and unit norm. This yields the sharp projection estimate. Positivity and polarization of the Bochner measure turn the $O(s)$ coefficient error into an $O(s^2)$ mass error, producing the directional bound with the required powers of $r_i$.

🏷️ Correcting coordinates in their Euclidean value space

At scale $i$, choose a finite net of patch centers and smooth cutoffs ${\varphi }_{i,y}$ supported in normalized ellipsoids. Only boundedly many patches meet each point. In each patch choose one fixed coordinate value $c_{i,y}$ whose defect is within a factor two of the patch infimum. An arbitrarily small positive, summable floor in $E_i$ permits this choice without continuity of $E_i$ or a selection theorem.

Define a smooth covector field on value space by

$$v_i(P)=\sum _y{\varphi }_{i,y}(P)D_{i,y}(P-c_{i,y}).$$

Its derivative has leading part $S_i$, with cutoff errors controlled by directional pinching. The coordinate corrections are

$$\begin{array}{rl}{h}_0(P)& =P,J_i=D{h}_i,\\ h_{i+1}(P)& =h_i(P)+J_i(P{)}^{-T}{v}_i(P).\end{array}$$

All derivatives here are ordinary Euclidean derivatives on the fixed open set $\Omega$. The construction does not invert the weak differential of $u$ or differentiate the scale forms $H_i$.

The inverse transpose makes the main change in squared lengths equal to $2S_i[\xi ]$. Indeed, for $J=Dh$ and $W=J^{-T}v$, direct differentiation gives

$$\begin{array}{rl}& |(J+DW)\xi {|}^2-|J\xi {|}^2\\ & =2{\xi }^{T}Dv\xi \\ & -2⟨D^{2}h(\xi ,\xi ),W⟩\\ & +|DW\xi {|}^2.\end{array}$$

The second term is the delicate one. A crude bound followed by summation of $\sqrt{E_i}$ would lose the available summability. Instead fix a testing point and direction and set

$$q_i=|\xi {|}_{H_i},b_i=\frac{r_i|D^2{h}_i(\xi ,\xi )|}{q_i}.$$

The paper derives

$$\begin{array}{rl}{b}_{i+1}\le & \frac{1}{2}{b}_i+C_A\sqrt{Z_i[\xi ]}\\ & +C_A\sqrt{E_i}{q}_i,b_0=0.\end{array}$$

The factor $1/2$ comes from the next dyadic radius; monotonicity of $H_i$ helps rather than hurts. Convolution with the geometric sequence is bounded on ${\ell }^2$, so the cumulative matrix estimate implies

$$\sum _{i<k}{b}_i^2\le C_A{H}_k[\xi ].$$

Cauchy–Schwarz now bounds the troublesome sum by

$$\begin{array}{rl}& \sum _{i<k}{b}_i{q}_i\sqrt{E_i}\\ & \le {\left(\sum _{i<k}{b}_i^2\right)}^{1/2}{\left(\sum _{i<k}{E}_i{q}_i^2\right)}^{1/2}\\ & \le C_A\sqrt{\epsilon }{H}_k[\xi ].\end{array}$$

Only ${\sum }_i{E}_i$ enters. Combining the analogous cutoff and squared-increment estimates with the leading positive sum closes the metric bounds

$$A^{-1}{H}_i\le J_i^T{J}_i\le A{H}_i.$$

The update loses a derivative, so the proof also maintains bounds for every derivative order, using smooth cutoffs with controlled factorial growth. These are Gevrey-type bounds: the $m$th derivative is allowed to grow like a fixed exponential in $m$ times a fixed power of $m!$. Matrix inversion preserves the required exponential base. Changing to the next radius supplies a factor $2^{1-m}$, allowing one smallness choice to close all orders simultaneously.

The induction proves invertibility of each new Jacobian using estimates from earlier levels. It does not assume the next Jacobian is invertible in order to estimate it. Nor does it require each finite-stage map $h_i$ to be globally injective. (OpenAI, 2026)

🏷️ All-pairs distance bounds and openness of the limit

The increments satisfy a uniform summable value bound, giving a continuous limit

$$\begin{array}{c}F=\underset{i}{\lim }{h}_i\circ u,\\ |F(x)-h_i(u(x))|\le \tau r_i,\end{array}$$

where $\tau$ can be made arbitrarily small. Uniform convergence alone would not preserve a lower Lipschitz bound, so the proof selects a scale separately for each pair.

Take $t=d(x,z)>0$ in a sufficiently small final ball, and choose a dyadic radius with

$$Dt\le r_i<2Dt,$$

where the fixed large constant $D$ is chosen before the final magnification. For $\Delta =u(z)-u(x)$, normalized harmonic approximation gives $|\Delta {|}_{H_i(x)}$ between $t/2$ and $2t$.

The straight segment from $u(x)$ to $u(z)$ lies in a central ellipsoid whose entire preimage is inside the working region. Taylor’s theorem is therefore available along the whole segment. Its remainder is bounded by a constant times $t^2/r_i$, hence by $Ct/D$. The leading term $J_i(u(x))\Delta$ has size comparable to $t$, by the metric bound. The two limit tails contribute at most $2\tau r_i\le 4D\tau t$.

Choose $D$ first to control the Taylor error, then the initial magnification so that $\tau$ is small enough. The resulting estimates are

$$\begin{array}{c}\frac{d(x,z)}{4\sqrt{A}}\le |F(x)-F(z)|,\\ |F(x)-F(z)|\le 4\sqrt{A}d(x,z).\end{array}$$

They apply to every pair in the final ball, with the ambient distance. The map is consequently injective and has a continuous inverse onto its image. Since $u$ already gives an open coordinate domain, invariance of domain applied to $F\circ u^{-1}$ makes the image open.

All fixed constants depend only on $n$. Returning from the magnified metric to the original one scales the final coordinate values by the same factor as distances, preserving the distortion $L_n=4\sqrt{A(n)}$. Only the radius of the chart depends on the particular space and point. (OpenAI, 2026)

🏷️ The full-measure chart region and reading connections

Taking the union of the chart domains gives an open set containing every $n$-regular point. The known full-measure property of the regular set makes its complement null. Separability supplies a countable subcover. On overlaps, a transition map is a composition of an $L_n$-Lipschitz map with the inverse of another, so it is $L_n^2$-bi-Lipschitz.

The union may also contain points outside the exact regular locus. Thus the corollary does not assert that the exact regular locus itself is open.

The Alexandrov-curvature family introduces the RCD calculus and explains another passage from integrated estimates to a conclusion at every prescribed geometric object. Here the conclusion concerns every regular point and all pairs in its chart. The two articles use different curvature hypotheses and different analytic mechanisms.

The source review covers the theorem and introduction, selected harmonic and heat-matrix estimates, the telescoping volume defect, radial transport and sharp-projection arguments, the directional estimate, and the coordinate iteration and limiting-chart proof. The complete harmonic-normalization induction, heat-kernel comparison estimates, radial-invariance construction, and factorial inversion estimates were not independently verified. The claimed research theorem remains attributed to the manuscript.

🏷️ Key steps and proof outlines

Inverse increments are compared one matrix at a time

The heat Gram matrices decrease with scale, while their inverses $H_i$ increase. To estimate one inverse increment, conjugate its positive decrement by $Q_i^{-1/2}$. The resulting matrix is positive and can be diagonalized for that single step; scalar inequalities for $(1-s{)}^{-1}$ compare inverse growth with the normalized decrement. Undo the conjugation and sum inequalities of quadratic forms.

This does not require the matrices at different scales to commute. It also does not require $H_i$ to remain bounded: the cumulative inequalities compare all earlier corrections with the current $H_k$. The directional pinching estimate supplies the additional summability needed for actual coordinate differences. (OpenAI, 2026)

The inverse transpose creates the desired metric increment

Set $W=J^{-T}v$ for $J=Dh$. Differentiating $J^{-T}$ gives

$$D_{\xi }(J^{-T})=-J^{-T}(D_{\xi }J{)}^T{J}^{-T}.$$

Consequently the linear term in the change of $|J\xi {|}^2$ splits into the desired $2{\xi }^{T}Dv\xi$ and the error $-2⟨D^{2}h(\xi ,\xi ),W⟩$. The leading part of $Dv$ is the positive form $S_i$. Thus the update adds precisely the accumulated stretching prescribed by the Bochner mass, with all remaining terms explicitly identified.

Twice differentiating $W_i$ produces three terms:

$$D^2{W}_i(\xi ,\xi )=K_i{D}^2{v}_i(\xi ,\xi )+2(D_{\xi }{K}_i)(D{v}_i\xi )+(D_{\xi }^2{K}_i)v_i,$$

where $K_i=J_i^{-T}$. Frozen anisotropic normalization controls each term by the same directional forcing. Since the next radius is half as large and $H_{i+1}\ge H_i$, the normalized old second derivative contracts by at least one half. This gives the recurrence for $b_i$ stated above. (OpenAI, 2026)

Square summation avoids a false square-root summability assumption

The recurrence has the form $b_{i+1}\le b_i/2+C{f}_i$. Convolution by $(1,1/2,1/4,\dots )$ is bounded on ${\ell }^2$, so $\sum b_i^2\le C\sum f_i^2$. The forcing squares are controlled by accumulated Bochner mass plus $E_i{H}_i[\xi ]$, hence by $C{H}_k[\xi ]$ over any finite prefix.

The troublesome mixed error is now bounded by Cauchy–Schwarz against $\sum E_i{H}_i[\xi ]\le \epsilon H_k[\xi ]$. It costs only $C\sqrt{\epsilon }{H}_k[\xi ]$. Summing $\sqrt{E_i}$ directly would be invalid: a summable nonnegative sequence need not have summable square roots. This is the specific reason for tracking diagonal second derivatives and using a square-sum recurrence.

The resulting lower metric bound proves the next Jacobian invertible, closing the induction in the correct order. The final all-pairs estimates and openness argument concern the limiting map, rather than assuming global injectivity of every finite correction. Together they upgrade asymptotically Euclidean tangent information to an open coordinate neighborhood with uniform two-sided distance control.

🏷️ Manuscripts and verification scope

• Bi-Lipschitz Coordinates at Regular Points of Noncollapsed RCD Spaces — pinned manuscript and companion source.

No family scope page was present at lean/docs/357.md in the pinned snapshot. This does not establish that all related formalization code is absent. 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. Bi-Lipschitz Coordinates at Regular Points of Noncollapsed RCD Spaces.