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Lean statement

Convex injectivity domains under weak MTW

Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.

Theorem selected for verification: OAI.WeakMTWGlobalSupport.current_main_convexity

Lean statement file: WeakMTWGlobalSupport.leanShow on the mapFamily 360. OpenAI lists this as a main result in its Lean catalogue.

See the candidate theoremWho ran the proof-verification programReport a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

theorem current_main_convexity    {n : ℕ} (hn : 2 ≤ n) {M : Type*}    [MetricSpace M] [ChartedSpace (WeakMTW.Model n) M]    [IsManifold (WeakMTW.model n) ∞ M]    [CompactSpace M] [ConnectedSpace M]    [RiemannianBundle (fun x : M => TangentSpace (WeakMTW.model n) x)]    [IsContMDiffRiemannianBundle (WeakMTW.model n) ∞ (WeakMTW.Model n)      (fun x : M => TangentSpace (WeakMTW.model n) x)]    [IsRiemannianManifold (WeakMTW.model n) M]    (hmtw : WeakMTW.HasWeakMTW (n := n) (M := M)) :    WeakMTW.MainConvexity (n := n) (M := M) :=

A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.

Definitions needed to read it

Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.

  • abbreviation OAI.WeakMTWGlobalSupport.WeakMTW.Model line 14

    abbrev Model (n : ℕ) := EuclideanSpace ℝ (Fin n)

  • abbreviation OAI.WeakMTWGlobalSupport.WeakMTW.model line 15

    abbrev model (n : ℕ) := 𝓘(ℝ, Model n)

  • definition OAI.WeakMTWGlobalSupport.WeakMTW.CompleteGeodesicWithInitialData line 24

    def CompleteGeodesicWithInitialData (x : M) (v : TangentSpace (model n) x)    (γ : ℝ → M) : Prop :=  ContMDiff 𝓘(ℝ, ℝ) (model n) ∞ γ ∧  ∃ h : γ 0 = x,    (h ▸ mfderiv 𝓘(ℝ, ℝ) (model n) γ 0 (1 : ℝ)) = v ∧    ∀ t : ℝ, ∃ ε : ℝ, 0 < ε ∧      ∀ s ∈ Metric.ball t ε, ∀ r ∈ Metric.ball t ε,        dist (γ s) (γ r) = ‖v‖ * |s - r|

  • definition OAI.WeakMTWGlobalSupport.WeakMTW.ExpEndpoint line 33

    def ExpEndpoint (x : M) (v : TangentSpace (model n) x) (y : M) : Prop :=  ∃ γ : ℝ → M, CompleteGeodesicWithInitialData x v γ ∧ γ 1 = y

  • definition OAI.WeakMTWGlobalSupport.WeakMTW.exp line 36

    noncomputable def exp (x : M) (v : TangentSpace (model n) x) : M :=  by    classical    exact if h : ∃ y, ExpEndpoint x v y then Classical.choose h else x

  • definition OAI.WeakMTWGlobalSupport.WeakMTW.injectivityDomain line 41

    def injectivityDomain (x : M) : Set (TangentSpace (model n) x) :=  {v | ∃ a : ℝ, 1 < a ∧ dist x (exp x (a • v)) = a * ‖v‖}

  • definition OAI.WeakMTWGlobalSupport.WeakMTW.cost line 44

    def cost (x y : M) : ℝ := dist x y ^ 2 / 2

  • definition OAI.WeakMTWGlobalSupport.WeakMTW.mtw line 46

    noncomputable def mtw (x : M) (v ξ η : TangentSpace (model n) x) : ℝ :=  -(3 / 2 : ℝ) * deriv (fun s : ℝ =>    deriv (fun s' : ℝ => deriv (fun t : ℝ =>      deriv (fun t' : ℝ => cost (exp x (t' • ξ)) (exp x (v + s' • η))) t) 0) s) 0

  • definition OAI.WeakMTWGlobalSupport.WeakMTW.HasWeakMTW line 51

    def HasWeakMTW : Prop :=  ∀ (x : M) (v : TangentSpace (model n) x), v ∈ injectivityDomain x →    ∀ ξ η : TangentSpace (model n) x, inner ℝ ξ η = 0 → 0 ≤ mtw x v ξ η

Possible paper match, with no recorded review

Atlas selected this possible match by comparing words in the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.

Words of the Lean statement, its names split into words, that also appear in the theorem: convexity, injectivity, domains, weak and mtw. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: convex.

Statements in Global Support and Convex Injectivity Domains under Weak MTW come first because the release’s name for this Lean statement shares words with that paper’s title.

Theorem 1.1 of Global Support and Convex Injectivity Domains under Weak MTW

Theorem (Convexity of injectivity domains)

If (M,g) satisfies (eq:weak-mtw), then I(x) is convex for every x\in M. More precisely, for every v_0,v_1\in I(x) and \theta\in[0,1],

(1-\theta)v_0+\theta v_1\in I(x).
Section 1, Introduction Read it in the paper TeX source, line 47
Show the TeX
If $(M,g)$ satisfies \eqref{eq:weak-mtw}, then $I(x)$ is convex for
every $x\in M$. More precisely, for every $v_0,v_1\in I(x)$ and
$\theta\in[0,1]$,
\[
 (1-\theta)v_0+\theta v_1\in I(x).
\]

Other possible paper matches, closest wording first

  • Corollary 4.9 of Global Support and Convex Injectivity Domains under Weak MTW. Words of the Lean statement, its names split into words, that also appear in the theorem: injectivity and domains. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: convex.
  • Corollary 4.10 of Global Support and Convex Injectivity Domains under Weak MTW. Words of the Lean statement, its names split into words, that also appear in the theorem: 2/2, endpoints, compact and exp. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: convex.
  • Theorem 1.2 of Global Support and Convex Injectivity Domains under Weak MTW. Words of the Lean statement, its names split into words, that also appear in the theorem: 2/2, weak, mtw, compact and exp. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: convex.
  • Corollary 3.6 of Uniform Bi-Holder Transport from Weak MTW. Words of the Lean statement, its names split into words, that also appear in the theorem: convexity, weak and mtw. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: assume.
  • Corollary 2.6 of Uniform Bi-Holder Transport from Weak MTW. Words of the Lean statement, its names split into words, that also appear in the theorem: injectivity and domains. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: convex.
  • Proposition 4.11 of Global Support and Convex Injectivity Domains under Weak MTW. Words of the Lean statement, its names split into words, that also appear in the theorem: geodesic, compact and exp.

How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.

Reported Comparator runs

two independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.

For a configured theorem, Comparator tests whether OpenAI’s proof proves the exact Lean statement. Some files instead configure definitions for comparison by name and type. Acceptance does not establish that the statement means the same thing as the paper’s theorem.

Carlos Toledo’s cert-machine

How they ran it, in their words

Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners (.github/workflows/openai-math-kernel.yml), one j…

  • Accepted Their term: “CERTIFIED”.

    Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.

    Reported 10 October 2026, at the release version shown here. Original verification report

MathVet

How they ran it, in their words

An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.

  • Accepted Their term: “comparator-sandboxed-pass”.

    Re-run through Comparator in a sandboxed virtual machine: accepted.

    Reported 8 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.

This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.

Allowed axioms: propext, Quot.sound, Classical.choice

Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.

Do family 360’s Lean statements express the advertised result?

These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.

MathVet and cert-machine agree on how much of the advertised result the Lean statements express.

  • MathVet

    Finds the whole advertised result stated or implied Their term: “full”.

    The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record

    Their note: current_main_convexity: on a compact connected smooth Riemannian manifold of dimension n ≥ 2, weak MTW (-3/2·∂⁴c ≥ 0 for orthogonal ξ, η at interior velocities) implies every open tangent injectivity domain is convex. uniform_biHolder_transport: for squared-distance cost and densities between fixed bounds there is one (α, C) such that for all pairs of densities the optimal map exists, is a homeomorphism, is a.e. unique, and T and T⁻¹ are α-Hölder with constant C.

    Definitions the group asks readers to compare with the paper: Riemannian exponential is defined by if h : ∃ γ, complete geodesic with initial data then (Classical.choose h) 1 else x (geodesics characterized metrically as locally constant-speed distance realizers), injectivityDomain x := {v | ∃ a > 1, dist x (exp x (a•v)) = a‖v‖}, and mtw is an iterated ordinary derivative of the cost along exp-curves; calibrated Hausdorff measure metricVolume as volume

  • Carlos Toledo’s cert-machine

    Finds the whole advertised result stated or implied Their term: “MATCHES”.

    The challenge statements state the headline as OpenAI wrote it. Their record

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“On a compact connected smooth Riemannian manifold of dimension at least two, the formalized result proves that weak MTW curvature implies convexity of every open tangent injectivity domain. Prior convexity or nonfocality is not assumed.”

An AI agent drafted an interpretation of that note, classifying this statement as “states a paper’s theorem as the paper does”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

79 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.

Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.

Links in the file: 35 to its own declarations, 42 to Mathlib’s documentation. 11 declared names also link to the same declaration in OpenAI’s library. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

import Mathlibnamespace OAInamespace WeakMTWGlobalSupportopen scoped Manifold ContDiff Topologyopen Set Filter Manifold Bundlenamespace WeakMTWnoncomputable sectionabbrev Model (n : ℕ) := EuclideanSpace ℝ (Fin n)abbrev model (n : ℕ) := 𝓘(ℝ, Model n)variable {n : ℕ} {M : Type*} [MetricSpace M] [ChartedSpace (Model n) M]  [IsManifold (model n) ∞ M]  [RiemannianBundle (fun x : M => TangentSpace (model n) x)]  [IsContMDiffRiemannianBundle (model n) ∞ (Model n)    (fun x : M => TangentSpace (model n) x)]  [IsRiemannianManifold (model n) M]def CompleteGeodesicWithInitialData (x : M) (v : TangentSpace (model n) x)    (γ : ℝ → M) : Prop :=  ContMDiff 𝓘(ℝ, ℝ) (model n) ∞ γ ∧  ∃ h : γ 0 = x,    (h ▸ mfderiv 𝓘(ℝ, ℝ) (model n) γ 0 (1 : ℝ)) = v ∧    ∀ t : ℝ, ∃ ε : ℝ, 0 < ε ∧      ∀ s ∈ Metric.ball t ε, ∀ r ∈ Metric.ball t ε,        dist (γ s) (γ r) = ‖v‖ * |s - r|def ExpEndpoint (x : M) (v : TangentSpace (model n) x) (y : M) : Prop :=  ∃ γ : ℝ → M, CompleteGeodesicWithInitialData x v γ ∧ γ 1 = ynoncomputable def exp (x : M) (v : TangentSpace (model n) x) : M :=  by    classical    exact if h : ∃ y, ExpEndpoint x v y then Classical.choose h else xdef injectivityDomain (x : M) : Set (TangentSpace (model n) x) :=  {v | ∃ a : ℝ, 1 < a ∧ dist x (exp x (a • v)) = a * ‖v‖}def cost (x y : M) : ℝ := dist x y ^ 2 / 2noncomputable def mtw (x : M) (v ξ η : TangentSpace (model n) x) : ℝ :=  -(3 / 2 : ℝ) * deriv (fun s : ℝ =>    deriv (fun s' : ℝ => deriv (fun t : ℝ =>      deriv (fun t' : ℝ => cost (exp x (t' • ξ)) (exp x (v + s' • η))) t) 0) s) 0def HasWeakMTW : Prop :=  ∀ (x : M) (v : TangentSpace (model n) x), v ∈ injectivityDomain x →    ∀ ξ η : TangentSpace (model n) x, inner ℝ ξ η = 0 → 0 ≤ mtw x v ξ ηdef MainConvexity : Prop :=  ∀ x : M, Convex ℝ (injectivityDomain (n := n) x)endend WeakMTW/-- Weak MTW implies linear convexity of every open tangent injectivity domain. -/theorem current_main_convexity    {n : ℕ} (hn : 2 ≤ n) {M : Type*}    [MetricSpace M] [ChartedSpace (WeakMTW.Model n) M]    [IsManifold (WeakMTW.model n) ∞ M]    [CompactSpace M] [ConnectedSpace M]    [RiemannianBundle (fun x : M => TangentSpace (WeakMTW.model n) x)]    [IsContMDiffRiemannianBundle (WeakMTW.model n) ∞ (WeakMTW.Model n)      (fun x : M => TangentSpace (WeakMTW.model n) x)]    [IsRiemannianManifold (WeakMTW.model n) M]    (hmtw : WeakMTW.HasWeakMTW (n := n) (M := M)) :    WeakMTW.MainConvexity (n := n) (M := M) :=  by    sorryend WeakMTWGlobalSupportend OAI
Every extracted statement of the accompanying papers

The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.

Global Support and Convex Injectivity Domains under Weak MTW

Theorem (Convexity of injectivity domains)

If (M,g) satisfies (eq:weak-mtw), then I(x) is convex for every x\in M. More precisely, for every v_0,v_1\in I(x) and \theta\in[0,1],

(1-\theta)v_0+\theta v_1\in I(x).
Section 1, Introduction Read it in the paper TeX source, line 47
Show the TeX
If $(M,g)$ satisfies \eqref{eq:weak-mtw}, then $I(x)$ is convex for
every $x\in M$. More precisely, for every $v_0,v_1\in I(x)$ and
$\theta\in[0,1]$,
\[
 (1-\theta)v_0+\theta v_1\in I(x).
\]
Theorem (Global support and intermediate geometry)

Under (eq:weak-mtw), every potential u, with its dual potential v, has the following properties.

  1. (i)

    Every p\in\partial u(x) belongs to G_x and satisfies

    u(x')\geq u(x)+c(x,\exp_xp)-c(x',\exp_xp) \qquad(x'\in M).

    For every 0<t<1, the map F_t:\Gamma_u\to M, F_t(x,p)=\exp_x(tp), is a homeomorphism; tp\in I(x) and x is the unique global minimizing pole of Q_tu at F_t(x,p).

  2. (ii)

    For every dual pair (u,v), every x and every r, the lifted section

    \mathcal S_x(r)=\left\{p\in G_x: u(x)+\frac{|p|^2}{2}+v(\exp_xp)\leq r\right\}

    is compact and convex. If it is nonempty and r\geq0, then

    \operatorname{diam}\mathcal S_x(r)^2 \leq8\left(r+\osc_{p\in\mathcal S_x(r)}v(\exp_xp)\right).

    • For 0<t<1, one has Q_tu=-Q_{1-t}v\in C^{1,1}(M). Its C^{1,1} seminorm and the Lipschitz constants of the pole and momentum maps F_t^{-1} depend only on M,g,t. There are also r_t,b_t>0, depending only on M,g,t, such that, for z=F_t(x,p),

    u(x')+\frac{c(x',z)}t \geq u(x)+\frac{c(x,z)}t+b_t d(x,x')^2 \quad\text{if }d(x,x')<r_t.

    These constants can be chosen uniformly on compact subintervals of (0,1).

No density hypothesis is required.

Section 1, Introduction Read it in the paper TeX source, line 94
Show the TeX
Under \eqref{eq:weak-mtw}, every potential $u$, with its dual potential $v$,
has the following properties.
\begin{enumerate}[label=\textup{(\roman*)}]
\item Every $p\in\partial u(x)$ belongs to $G_x$ and satisfies
\[
 u(x')\geq u(x)+c(x,\exp_xp)-c(x',\exp_xp)
 \qquad(x'\in M).
\]
For every $0<t<1$, the map $F_t:\Gamma_u\to M$,
$F_t(x,p)=\exp_x(tp)$, is a homeomorphism; $tp\in I(x)$ and $x$
is the unique global minimizing pole of $Q_tu$ at $F_t(x,p)$.
\item For every dual pair $(u,v)$, every $x$ and every $r$, the lifted section
\[
 \mathcal S_x(r)=\left\{p\in G_x:
 u(x)+\frac{|p|^2}{2}+v(\exp_xp)\leq r\right\}
\]
is compact and convex. If it is nonempty and $r\geq0$, then
\[
 \operatorname{diam}\mathcal S_x(r)^2
 \leq8\left(r+\osc_{p\in\mathcal S_x(r)}v(\exp_xp)\right).
\]
\item For $0<t<1$, one has $Q_tu=-Q_{1-t}v\in C^{1,1}(M)$.
Its $C^{1,1}$ seminorm and the Lipschitz constants of the pole and momentum
maps $F_t^{-1}$ depend only on $M,g,t$. There are also $r_t,b_t>0$,
depending only on $M,g,t$, such that, for $z=F_t(x,p)$,
\[
 u(x')+\frac{c(x',z)}t
 \geq u(x)+\frac{c(x,z)}t+b_t d(x,x')^2
 \quad\text{if }d(x,x')<r_t.
\]
These constants can be chosen uniformly on compact subintervals of $(0,1)$.
\end{enumerate}
No density hypothesis is required.

Uniform Bi-Holder Transport from Weak MTW

Theorem (Uniform bi-Hölder transport)

Fix (M,g) as above and 0<\lambda\leq\Lambda<\infty for which the following density class is nonempty. There are \alpha\in(0,1] and C<\infty, depending only on (M,g),\lambda,\Lambda, such that the following holds for every pair of measurable densities satisfying

\int_M\rho_i\,d\vol=1,\qquad \lambda\leq\rho_i\leq\Lambda\quad\vol\text{-almost everywhere}, \qquad i=0,1.

The almost-everywhere unique optimal map from \rho_0\vol to \rho_1\vol for the cost c has a homeomorphic representative \widetilde T:M\to M, and

d(\widetilde T(x),\widetilde T(x'))\leq C d(x,x')^\alpha, \qquad d(\widetilde T^{-1}(y),\widetilde T^{-1}(y')) \leq C d(y,y')^\alpha

for all x,x',y,y'\in M.

Section 1, Introduction Read it in the paper TeX source, line 32
Show the TeX
Fix $(M,g)$ as above and $0<\lambda\leq\Lambda<\infty$ for which the
following density class is nonempty. There are
$\alpha\in(0,1]$ and $C<\infty$, depending only on
$(M,g),\lambda,\Lambda$, such that the following holds for every pair
of measurable densities satisfying
\[
 \int_M\rho_i\,d\vol=1,\qquad
 \lambda\leq\rho_i\leq\Lambda\quad\vol\text{-almost everywhere},
 \qquad i=0,1.
\]
The almost-everywhere unique optimal map from $\rho_0\vol$ to
$\rho_1\vol$ for the cost $c$ has a homeomorphic representative
$\widetilde T:M\to M$, and
\[
 d(\widetilde T(x),\widetilde T(x'))\leq C d(x,x')^\alpha,
 \qquad
 d(\widetilde T^{-1}(y),\widetilde T^{-1}(y'))
       \leq C d(y,y')^\alpha
\]
for all $x,x',y,y'\in M$.
Configuration and file facts
OpenAI’s Lean catalogue
OpenAI lists this as a main result in its Lean catalogue.
Allowed axioms
propext, Quot.sound, Classical.choiceThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports
Mathlib
Imports from outside Mathlib
None
Local vocabulary
10 local vocabulary declarations
Lines of Lean, without comments or blank lines
46
Second proof-verification program (nanoda)
Off in OpenAI's configuration.

Statement comparison

A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.

Plectis Atlas

Convex injectivity domains under weak MTW

Family 360: Weak MTW curvature gives convexity and regular optimal transport

Candidate pairing

Theorem (Convexity of injectivity domains)

If (M,g) satisfies (eq:weak-mtw), then I(x) is convex for every x\in M. More precisely, for every v_0,v_1\in I(x) and \theta\in[0,1],

(1-\theta)v_0+\theta v_1\in I(x).

Global Support and Convex Injectivity Domains under Weak MTW

TeX label thm:main, build/sections/introduction.tex, line 47

The Lean statement says

theorem current_main_convexity    {n : ℕ} (hn : 2 ≤ n) {M : Type*}    [MetricSpace M] [ChartedSpace (WeakMTW.Model n) M]    [IsManifold (WeakMTW.model n) ∞ M]    [CompactSpace M] [ConnectedSpace M]    [RiemannianBundle (fun x : M => TangentSpace (WeakMTW.model n) x)]    [IsContMDiffRiemannianBundle (WeakMTW.model n) ∞ (WeakMTW.Model n)      (fun x : M => TangentSpace (WeakMTW.model n) x)]    [IsRiemannianManifold (WeakMTW.model n) M]    (hmtw : WeakMTW.HasWeakMTW (n := n) (M := M)) :    WeakMTW.MainConvexity (n := n) (M := M) :=

A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.

Atlas selected this possible match because the two statements share words. No review of whether they mean the same thing is recorded.two independent groups report that Comparator accepted runs for this file at the release version shown here.Statement review: none recorded

10 local vocabulary declarationsAllowed axioms: propext, Quot.sound, Classical.choiceopenai/math at fd4aeebwcook04.github.io/plectis-atlas/statements/WeakMTWGlobalSupport.html

Recorded comparisons with the paper

Atlas records no contributed comparison of this statement with the paper.

Contributed Comparator run reports

Atlas records no contributed report of running Comparator for this statement file.

More about this statement

  • Commentary The doc lists further results as formalised that no Comparator theorem name covers. Robert Joseph George, Do the Lean statements in openai/math match their papers?, 2026-10-07 Source

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