Lean statement

All-pairs diamond distortion bound

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.DiamondDistortion.headline_all_pairs

Lean statement file: DiamondDistortion.leanShow on the mapFamily 331. OpenAI lists this as a main result in its Lean catalogue.

See the candidate theorem Who ran the proof-verification program Report 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 headline_all_pairs :    (∀ (k : ℕ) (f : Diamond.Vertex k → ForestDual) (D s : ℝ),      0 < s → (∀ u v,s*Diamond.distance k u v ≤ ‖f u-f v‖ ∧        ‖f u-f v‖ ≤ D*s*Diamond.distance k u v) → 1+(k:ℝ)/4 ≤ D^2) ∧    (∀ (k : ℕ) (f : Diamond.Vertex k → InfinitePredual) (D s : ℝ),      0 < s → (∀ u v,s*Diamond.distance k u v ≤ ‖f u-f v‖ ∧        ‖f u-f v‖ ≤ D*s*Diamond.distance k u v) → 1+(k:ℝ)/4 ≤ D^2) := by  sorry

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.

structure OAI.DiamondDistortion.SegmentSystem line 15

structure SegmentSystem (ι : Type*) where  IsSegment : Finset ι → Prop  singleton : ∀ v, IsSegment {v}

structure OAI.DiamondDistortion.SegmentSystem.Family line 24

structure Family where  n : ℕ  seg : Fin n → Finset ι  allowed : ∀ i, T.IsSegment (seg i)  disjoint : Pairwise (fun i j => Disjoint (seg i) (seg j))

definition OAI.DiamondDistortion.SegmentSystem.Family.test line 39

def Family.test (F : T.Family) : (ι →₀ ℝ) →ₗ[ℝ] EuclideanSpace ℝ (Fin F.n) where  toFun u := WithLp.toLp 2 (fun i => ∑ v ∈ F.seg i, u v)  map_add' u v := by    ext i    simp [Finset.sum_add_distrib]  map_smul' c u := by    ext i    simp [Finset.mul_sum]

definition OAI.DiamondDistortion.SegmentSystem.Family.seminorm line 48

def Family.seminorm (F : T.Family) : Seminorm ℝ (ι →₀ ℝ) :=  (normSeminorm ℝ (EuclideanSpace ℝ (Fin F.n))).comp F.test

definition OAI.DiamondDistortion.SegmentSystem.primalSeminorm line 53

def primalSeminorm : Seminorm ℝ (ι →₀ ℝ) := ⨆ F : T.Family, F.seminorm

definition OAI.DiamondDistortion.SegmentSystem.singletonFamily line 56

def singletonFamily (v : ι) : T.Family where  n := 1  seg _ := {v}  allowed _ := T.singleton v  disjoint := by intro i j h; exact False.elim (h (Subsingleton.elim _ _))

definition OAI.DiamondDistortion.SegmentSystem.Primal line 62
```
def Primal (_T : SegmentSystem ι) := ι →₀ ℝ
```
abbreviation OAI.DiamondDistortion.SegmentSystem.JamesSpace line 130
```
abbrev JamesSpace := UniformSpace.Completion T.Primal
```
abbreviation OAI.DiamondDistortion.SegmentSystem.Dual line 133
```
abbrev Dual := StrongDual ℝ T.JamesSpace
```

definition OAI.DiamondDistortion.SegmentSystem.coordinateFinite line 136

def coordinateFinite (v : ι) : T.Primal →L[ℝ] ℝ := by

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  have Family.test_apply (F : T.Family) (u : ι →₀ ℝ) (i : Fin F.n) :      F.test u i = ∑ v ∈ F.seg i, u v := rfl  have Family.seminorm_apply (F : T.Family) (u : ι →₀ ℝ) :      F.seminorm u = ‖F.test u‖ := rfl  have Family.test_single_le (F : T.Family) (v : ι) (r : ℝ) :      ‖F.test (Finsupp.single v r)‖ ≤ |r| := by    classical    rw [EuclideanSpace.norm_eq]    apply (Real.sqrt_le_left (abs_nonneg r)).2    by_cases h : ∃ i, v ∈ F.seg i    · obtain ⟨i, hi⟩ := h      have hne : ∀ j, j ≠ i → v ∉ F.seg j := by        intro j hji hj        exact Finset.disjoint_left.mp (F.disjoint hji) hj hi      rw [Finset.sum_eq_single i]      · simp [Family.test, Finsupp.single_apply, hi]      · intro j _ hji        simp [Family.test, Finsupp.single_apply, hne j hji]      · simp    · have hn : ∀ i, v ∉ F.seg i := by simpa using h      simp [Family.test, Finsupp.single_apply, hn, sq_nonneg]  have Family.test_le_l1 (F : T.Family) (u : ι →₀ ℝ) :      ‖F.test u‖ ≤ u.sum (fun _ r => |r|) := by    classical    calc      ‖F.test u‖ = ‖∑ v ∈ u.support, F.test (Finsupp.single v (u v))‖ := by        rw [← map_sum]        congr 2        exact (Finsupp.sum_single u).symm      _ ≤ ∑ v ∈ u.support, ‖F.test (Finsupp.single v (u v))‖ := norm_sum_le _ _      _ ≤ ∑ v ∈ u.support, |u v| := Finset.sum_le_sum fun v _ => Family.test_single_le F v (u v)      _ = u.sum (fun _ r => |r|) := rfl  have families_bddAbove : BddAbove (Set.range (Family.seminorm (T := T))) := by    rw [Seminorm.bddAbove_range_iff]    intro u    exact ⟨u.sum (fun _ r => |r|), by rintro _ ⟨F, rfl⟩; exact Family.test_le_l1 F u⟩  have test_le_primal (F : T.Family) (u : ι →₀ ℝ) :      ‖F.test u‖ ≤ T.primalSeminorm u := by    rw [primalSeminorm, Seminorm.iSup_apply families_bddAbove]    exact le_ciSup (⟨u.sum (fun _ r => |r|), by      rintro _ ⟨G, rfl⟩; exact Family.test_le_l1 G u⟩ :      BddAbove (Set.range (fun G : T.Family => G.seminorm u))) F  have coordinate_le_primal (u : ι →₀ ℝ) (v : ι) : |u v| ≤ T.primalSeminorm u := by    have h := test_le_primal (T.singletonFamily v) u    rw [EuclideanSpace.norm_eq] at h    change Real.sqrt (∑ _ : Fin 1, ‖∑ w ∈ ({v} : Finset ι), u w‖ ^ 2) ≤ _ at h    simpa [Real.sqrt_sq_eq_abs] using h  exact (show T.Primal →ₗ[ℝ] ℝ from Finsupp.lapply v).mkContinuous 1 (by      intro u      change ‖(show ι →₀ ℝ from u) v‖ ≤ 1 * T.primalSeminorm u      simpa only [one_mul, Real.norm_eq_abs] using coordinate_le_primal u v)

definition OAI.DiamondDistortion.SegmentSystem.coordinate line 189

def coordinate (v : ι) : T.Dual :=  (T.coordinateFinite v).extend UniformSpace.Completion.toComplL

definition OAI.DiamondDistortion.SegmentSystem.CoordinatePredual line 201

def CoordinatePredual : Submodule ℝ T.Dual :=  (Submodule.span ℝ (Set.range T.coordinate)).topologicalClosure

definition OAI.DiamondDistortion.intervalSystem line 207

def intervalSystem {ι : Type*} (r : ι → ι → Prop)    (hrefl : ∀ a, r a a) (hanti : ∀ a b, r a b → r b a → a = b) : SegmentSystem ι where

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  IsSegment s := ∃ a b, r a b ∧ ∀ v, v ∈ s ↔ r a v ∧ r v b  singleton v := by    refine ⟨v, v, hrefl v, ?_⟩    intro w    simp only [Finset.mem_singleton]    constructor    · rintro rfl      exact ⟨hrefl _, hrefl _⟩    · intro h      exact hanti w v h.2 h.1

class OAI.DiamondDistortion.Ancestry line 221

class Ancestry (ι : Type*) extends PartialOrder ι where  predecessors_total : ∀ {a b c : ι}, le a c → le b c → le a b ∨ le b a

definition OAI.DiamondDistortion.Ancestry.segments line 231

def segments : SegmentSystem ι :=  intervalSystem (· ≤ ·) (fun _ => le_rfl) (fun _ _ => le_antisymm)

definition OAI.DiamondDistortion.ForestNode line 236

def ForestNode := {hw : ℕ × List ℕ // 1 ≤ hw.1 ∧ hw.2.length ≤ hw.1}

definition OAI.DiamondDistortion.ForestAncestor line 238

def ForestAncestor (a b : ForestNode) : Prop :=  a.val.1 = b.val.1 ∧ a.val.2.IsPrefix b.val.2

definition OAI.DiamondDistortion.forestAncestry line 241

@[instance_reducible] def forestAncestry : Ancestry ForestNode where

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  le := ForestAncestor  lt a b := ForestAncestor a b ∧ ¬ ForestAncestor b a  le_refl a := ⟨rfl, List.prefix_refl _⟩  le_trans _ _ _ hab hbc := ⟨hab.1.trans hbc.1, hab.2.trans hbc.2⟩  le_antisymm a b hab hba := by    apply Subtype.ext    exact Prod.ext hab.1 (hab.2.sublist.antisymm hba.2.sublist)  predecessors_total hab hac := by    rcases List.prefix_or_prefix_of_prefix hab.2 hac.2 with h | h    · exact Or.inl ⟨hab.1.trans hac.1.symm, h⟩    · exact Or.inr ⟨hac.1.trans hab.1.symm, h⟩

definition OAI.DiamondDistortion.forestSegments line 254

def forestSegments : SegmentSystem ForestNode :=  Ancestry.segments (A := forestAncestry)

definition OAI.DiamondDistortion.infiniteAncestry line 258

@[instance_reducible] def infiniteAncestry : Ancestry (List ℕ) where  le := List.IsPrefix  lt a b := a.IsPrefix b ∧ ¬ b.IsPrefix a  le_refl := List.prefix_refl  le_trans _ _ _ := List.IsPrefix.trans  le_antisymm _ _ hab hba := hab.sublist.antisymm hba.sublist  predecessors_total := List.prefix_or_prefix_of_prefix

definition OAI.DiamondDistortion.infiniteSegments line 267

def infiniteSegments : SegmentSystem (List ℕ) :=  Ancestry.segments (A := infiniteAncestry)

abbreviation OAI.DiamondDistortion.ForestDual line 270
```
abbrev ForestDual := forestSegments.Dual
```

abbreviation OAI.DiamondDistortion.InfinitePredual line 271

abbrev InfinitePredual := infiniteSegments.CoordinatePredual

definition OAI.DiamondDistortion.Diamond.Edge line 276

@[reducible] def Edge : ℕ → Type  | 0 => Unit  | n + 1 => Edge n × ℕ × Bool

4 more definitions are reached the same way; the full file lists them.

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: eq, integer, segment, infinite, space, norm and sqrt. Found in the paper’s title, which its main theorem states: uniform, tree and diamond. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: distortion and embedding.

Statements in Asymptotic midpoint uniform convexity and unbounded diamond distortion in a reflexive tree space come first because the AI draft reading of OpenAI’s note shares words with that paper’s title.

Theorem 1.1 of Asymptotic midpoint uniform convexity and unbounded diamond distortion in a reflexive tree space

Theorem

The space X=J^* defined by (eq:segment-norm) is infinite dimensional, separable, and reflexive, and has no equivalent AUC norm. Its given norm satisfies

\widehat\delta_X(t)\ge\sqrt{1+t^2/12}-1\qquad(t>0).

For every integer k\ge0, every embedding of D_k into X has distortion at least \sqrt{1+k/12}.

Section 1, Introduction Read it in the paper TeX source, line 99

Show the TeX

The space $X=J^*$ defined by \eqref{eq:segment-norm} is infinite
dimensional, separable, and reflexive, and has no equivalent AUC norm.
Its given norm satisfies
\[
 \widehat\delta_X(t)\ge\sqrt{1+t^2/12}-1\qquad(t>0).
\]
For every integer $k\ge0$, every embedding of $D_k$ into $X$ has
distortion at least $\sqrt{1+k/12}$.

Other possible paper matches, closest wording first

Theorem 1.1 of Midpoint lenses in segment spaces. Words of the Lean statement, its names split into words, that also appear in the theorem: predual, height, dual, forest, coordinate, ancestral and three more. Found in the paper’s title, which its main theorem states: segment and spaces.
Proposition 8.2 of Midpoint lenses in segment spaces. Words of the Lean statement, its names split into words, that also appear in the theorem: single, including, natural, tree, disjoint, path and twelve more.
Proposition 5.1 of Midpoint convexity from bounded tree potentials and path costs. Words of the Lean statement, its names split into words, that also appear in the theorem: continuous, linear, dual, pairing, test, coordinate and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: complete.
Proposition 9.1 of Midpoint lenses in segment spaces. Words of the Lean statement, its names split into words, that also appear in the theorem: using, vertex, distinct, child, path, root and eight more.
Theorem 1.1 of Distortion of countably branching diamonds from midpoint and tree energies. Words of the Lean statement, its names split into words, that also appear in the theorem: integer. Found in the paper’s title, which its main theorem states: tree, countably and diamonds. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: distortion and embedding.
Corollary A.1 of Distortion of countably branching diamonds from midpoint and tree energies. Words of the Lean statement, its names split into words, that also appear in the theorem: map, pairs, segment, forest, bound and space.

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

three independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet; Neal McBurnett’s openai-math-lean-tooling. 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

Neal McBurnett’s openai-math-lean-tooling

How they ran it, in their words

Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory; the run was still in progress at this snapshot (192 accepted, 224 pending).

Accepted Their term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 73 s; peak memory 6.2 GiB).
Reported 11 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 331’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: ForestSpace.main_theorem (unconditional) states for FullDual Vertex (continuous dual of the completed disjoint-segment-norm space over a forest): infinite-dimensional, separable, canonically reflexive, no equivalent norm is AUC, averaged midpoint modulus ≥ √(1+t²/12) - 1 for all t > 0 (this implies positivity of the usual max-version midpoint modulus, since max ≥ average: the one routine step), and every embedding of the depth-k countably branching diamond has distortion ≥ √(1+k/12). The other six challenges are companion results of the seven papers (lens tail bound, all-pairs diamond bound 1+k/4 ≤ D², exact Daugavet-subspace moduli, tree-potential, independent-product and recursive-potential spaces), not needed for the headline.
Definitions the group asks readers to compare with the paper: aucModulus, averageModulus and Cofinite are hand-built (inf_{N x=1} sup_{F cofinite closed} inf_{y∈F, N y=1} (N(x+ty)-1), resp. with the average of N(x±ty) minus 1), matching the standard asymptotic moduli; IsAUC N := ∀ t > 0, 0 < aucModulus N t and EquivalentNorm as a seminorm with α‖x‖ ≤ N x ≤ β‖x‖ ; Diamond.diamond k := base.iterate k (each edge replaced by ℕ-many 2-edge paths, graph distance via SimpleGraph.dist) and EmbeddingBound (∃ scale s > 0, s·d ≤ ‖f u - f v‖ ≤ C·s·d): standard distortion; docs note 22 companion inputs are assumed in some graph deductions, not in the headline theorem
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

“The paper studies embeddings of countably branching diamond graphs into Banach spaces built from tree segments. The formalization proves that every embedding of the depth-$k$ diamond with distortion $D$, at any positive scale, satisfies $1+k/4\le D^2$ in either of two spaces: the full dual of the completed finite-height forest space and the closed coordinate span in the dual of the infinite-tree space. The bound applies to all pairs of vertices and is unconditional.”
Quoted from OpenAI’s Lean note for this family

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

316 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: 53 to its own declarations, 142 to Mathlib’s documentation. 32 declared names also link to the same declaration in OpenAI’s library. 6 identifiers stay plain because they resolve to no single declaration: variables Lean binds automatically, notation, or names the scan of the pinned sources did not find. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

import Mathlibnamespace OAInoncomputable sectionuniverse u vopen scoped BigOperators Classical Topologyopen Setnamespace DiamondDistortionstructure SegmentSystem (ι : Type*) where  IsSegment : Finset ι → Prop  singleton : ∀ v, IsSegment {v}namespace SegmentSystemvariable {ι : Type*} (T : SegmentSystem ι)/-- The represented disjoint families in the definition of the primal segment norm. -/structure Family where  n : ℕ  seg : Fin n → Finset ι  allowed : ∀ i, T.IsSegment (seg i)  disjoint : Pairwise (fun i j => Disjoint (seg i) (seg j))instance : Nonempty T.Family := ⟨{  n := 0  seg := Fin.elim0  allowed := fun i => Fin.elim0 i  disjoint := fun {i} => Fin.elim0 i }⟩variable {T}/-- Segment testing as a Euclidean-valued linear map. -/def Family.test (F : T.Family) : (ι →₀ ℝ) →ₗ[ℝ] EuclideanSpace ℝ (Fin F.n) where  toFun u := WithLp.toLp 2 (fun i => ∑ v ∈ F.seg i, u v)  map_add' u v := by    ext i    simp [Finset.sum_add_distrib]  map_smul' c u := by    ext i    simp [Finset.mul_sum]def Family.seminorm (F : T.Family) : Seminorm ℝ (ι →₀ ℝ) :=  (normSeminorm ℝ (EuclideanSpace ℝ (Fin F.n))).comp F.testvariable (T)def primalSeminorm : Seminorm ℝ (ι →₀ ℝ) := ⨆ F : T.Family, F.seminorm/-- Singleton families ensure that the segment seminorm separates points. -/def singletonFamily (v : ι) : T.Family where  n := 1  seg _ := {v}  allowed _ := T.singleton v  disjoint := by intro i j h; exact False.elim (h (Subsingleton.elim _ _))def Primal (_T : SegmentSystem ι) := ι →₀ ℝinstance : AddCommGroup T.Primal := inferInstanceAs (AddCommGroup (ι →₀ ℝ))instance : Module ℝ T.Primal := inferInstanceAs (Module ℝ (ι →₀ ℝ))instance : NormedAddCommGroup T.Primal := by  have Family.test_apply (F : T.Family) (u : ι →₀ ℝ) (i : Fin F.n) :      F.test u i = ∑ v ∈ F.seg i, u v := rfl  have Family.seminorm_apply (F : T.Family) (u : ι →₀ ℝ) :      F.seminorm u = ‖F.test u‖ := rfl  have Family.test_single_le (F : T.Family) (v : ι) (r : ℝ) :      ‖F.test (Finsupp.single v r)‖ ≤ |r| := by    classical    rw [EuclideanSpace.norm_eq]    apply (Real.sqrt_le_left (abs_nonneg r)).2    by_cases h : ∃ i, v ∈ F.seg i    · obtain ⟨i, hi⟩ := h      have hne : ∀ j, j ≠ i → v ∉ F.seg j := by        intro j hji hj        exact Finset.disjoint_left.mp (F.disjoint hji) hj hi      rw [Finset.sum_eq_single i]      · simp [Family.test, Finsupp.single_apply, hi]      · intro j _ hji        simp [Family.test, Finsupp.single_apply, hne j hji]      · simp    · have hn : ∀ i, v ∉ F.seg i := by simpa using h      simp [Family.test, Finsupp.single_apply, hn, sq_nonneg]  have Family.test_le_l1 (F : T.Family) (u : ι →₀ ℝ) :      ‖F.test u‖ ≤ u.sum (fun _ r => |r|) := by    classical    calc      ‖F.test u‖ = ‖∑ v ∈ u.support, F.test (Finsupp.single v (u v))‖ := by        rw [← map_sum]        congr 2        exact (Finsupp.sum_single u).symm      _ ≤ ∑ v ∈ u.support, ‖F.test (Finsupp.single v (u v))‖ := norm_sum_le _ _      _ ≤ ∑ v ∈ u.support, |u v| := Finset.sum_le_sum fun v _ => Family.test_single_le F v (u v)      _ = u.sum (fun _ r => |r|) := rfl  have families_bddAbove : BddAbove (Set.range (Family.seminorm (T := T))) := by    rw [Seminorm.bddAbove_range_iff]    intro u    exact ⟨u.sum (fun _ r => |r|), by rintro _ ⟨F, rfl⟩; exact Family.test_le_l1 F u⟩  have test_le_primal (F : T.Family) (u : ι →₀ ℝ) :      ‖F.test u‖ ≤ T.primalSeminorm u := by    rw [primalSeminorm, Seminorm.iSup_apply families_bddAbove]    exact le_ciSup (⟨u.sum (fun _ r => |r|), by      rintro _ ⟨G, rfl⟩; exact Family.test_le_l1 G u⟩ :      BddAbove (Set.range (fun G : T.Family => G.seminorm u))) F  have coordinate_le_primal (u : ι →₀ ℝ) (v : ι) : |u v| ≤ T.primalSeminorm u := by    have h := test_le_primal (T.singletonFamily v) u    rw [EuclideanSpace.norm_eq] at h    change Real.sqrt (∑ _ : Fin 1, ‖∑ w ∈ ({v} : Finset ι), u w‖ ^ 2) ≤ _ at h    simpa [Real.sqrt_sq_eq_abs] using h  have primal_eq_zero_iff (u : ι →₀ ℝ) : T.primalSeminorm u = 0 ↔ u = 0 := by    constructor    · intro h      ext v      have := coordinate_le_primal u v      simpa [h] using this    · rintro rfl      exact map_zero _  exact AddGroupNorm.toNormedAddCommGroup    { T.primalSeminorm.toAddGroupSeminorm with      eq_zero_of_map_eq_zero' := fun u h => (primal_eq_zero_iff u).mp h }instance : NormedSpace ℝ T.Primal where  norm_smul_le c u := le_of_eq (T.primalSeminorm.smul' c u)abbrev JamesSpace := UniformSpace.Completion T.Primal/-- Its full real continuous dual. -/abbrev Dual := StrongDual ℝ T.JamesSpace/-- Coordinate evaluation on finite arrays, bounded by the singleton test. -/def coordinateFinite (v : ι) : T.Primal →L[ℝ] ℝ := by  have Family.test_apply (F : T.Family) (u : ι →₀ ℝ) (i : Fin F.n) :      F.test u i = ∑ v ∈ F.seg i, u v := rfl  have Family.seminorm_apply (F : T.Family) (u : ι →₀ ℝ) :      F.seminorm u = ‖F.test u‖ := rfl  have Family.test_single_le (F : T.Family) (v : ι) (r : ℝ) :      ‖F.test (Finsupp.single v r)‖ ≤ |r| := by    classical    rw [EuclideanSpace.norm_eq]    apply (Real.sqrt_le_left (abs_nonneg r)).2    by_cases h : ∃ i, v ∈ F.seg i    · obtain ⟨i, hi⟩ := h      have hne : ∀ j, j ≠ i → v ∉ F.seg j := by        intro j hji hj        exact Finset.disjoint_left.mp (F.disjoint hji) hj hi      rw [Finset.sum_eq_single i]      · simp [Family.test, Finsupp.single_apply, hi]      · intro j _ hji        simp [Family.test, Finsupp.single_apply, hne j hji]      · simp    · have hn : ∀ i, v ∉ F.seg i := by simpa using h      simp [Family.test, Finsupp.single_apply, hn, sq_nonneg]  have Family.test_le_l1 (F : T.Family) (u : ι →₀ ℝ) :      ‖F.test u‖ ≤ u.sum (fun _ r => |r|) := by    classical    calc      ‖F.test u‖ = ‖∑ v ∈ u.support, F.test (Finsupp.single v (u v))‖ := by        rw [← map_sum]        congr 2        exact (Finsupp.sum_single u).symm      _ ≤ ∑ v ∈ u.support, ‖F.test (Finsupp.single v (u v))‖ := norm_sum_le _ _      _ ≤ ∑ v ∈ u.support, |u v| := Finset.sum_le_sum fun v _ => Family.test_single_le F v (u v)      _ = u.sum (fun _ r => |r|) := rfl  have families_bddAbove : BddAbove (Set.range (Family.seminorm (T := T))) := by    rw [Seminorm.bddAbove_range_iff]    intro u    exact ⟨u.sum (fun _ r => |r|), by rintro _ ⟨F, rfl⟩; exact Family.test_le_l1 F u⟩  have test_le_primal (F : T.Family) (u : ι →₀ ℝ) :      ‖F.test u‖ ≤ T.primalSeminorm u := by    rw [primalSeminorm, Seminorm.iSup_apply families_bddAbove]    exact le_ciSup (⟨u.sum (fun _ r => |r|), by      rintro _ ⟨G, rfl⟩; exact Family.test_le_l1 G u⟩ :      BddAbove (Set.range (fun G : T.Family => G.seminorm u))) F  have coordinate_le_primal (u : ι →₀ ℝ) (v : ι) : |u v| ≤ T.primalSeminorm u := by    have h := test_le_primal (T.singletonFamily v) u    rw [EuclideanSpace.norm_eq] at h    change Real.sqrt (∑ _ : Fin 1, ‖∑ w ∈ ({v} : Finset ι), u w‖ ^ 2) ≤ _ at h    simpa [Real.sqrt_sq_eq_abs] using h  exact (show T.Primal →ₗ[ℝ] ℝ from Finsupp.lapply v).mkContinuous 1 (by      intro u      change ‖(show ι →₀ ℝ from u) v‖ ≤ 1 * T.primalSeminorm u      simpa only [one_mul, Real.norm_eq_abs] using coordinate_le_primal u v)def coordinate (v : ι) : T.Dual :=  (T.coordinateFinite v).extend UniformSpace.Completion.toComplL/-- Primal unit coordinate and its canonical image in the completion. -/def unitFinite (v : ι) : T.Primal := Finsupp.single v 1def unitVector (v : ι) : T.JamesSpace := (T.unitFinite v : T.JamesSpace)def headDual (H : Finset ι) : T.Dual →L[ℝ] T.Dual :=  ∑ v ∈ H, (ContinuousLinearMap.apply ℝ ℝ (T.unitVector v)).smulRight (T.coordinate v)/-- The norm-closed coordinate span, not the full infinite-tree dual. -/def CoordinatePredual : Submodule ℝ T.Dual :=  (Submodule.span ℝ (Set.range T.coordinate)).topologicalClosureend SegmentSystem/-- A finite interval in an ancestral order.  This keeps the root in every tree. -/def intervalSystem {ι : Type*} (r : ι → ι → Prop)    (hrefl : ∀ a, r a a) (hanti : ∀ a b, r a b → r b a → a = b) : SegmentSystem ι where  IsSegment s := ∃ a b, r a b ∧ ∀ v, v ∈ s ↔ r a v ∧ r v b  singleton v := by    refine ⟨v, v, hrefl v, ?_⟩    intro w    simp only [Finset.mem_singleton]    constructor    · rintro rfl      exact ⟨hrefl _, hrefl _⟩    · intro h      exact hanti w v h.2 h.1/-- An ancestral partial order: the predecessors of any vertex form a chain. -/class Ancestry (ι : Type*) extends PartialOrder ι where  predecessors_total : ∀ {a b c : ι}, le a c → le b c → le a b ∨ le b anamespace Ancestryopen scoped Classicalvariable {ι : Type*} [A : Ancestry ι]local instance : PartialOrder ι := A.toPartialOrder/-- Nonempty finite intervals for an ancestral order. -/def segments : SegmentSystem ι :=  intervalSystem (· ≤ ·) (fun _ => le_rfl) (fun _ _ => le_antisymm)end Ancestrydef ForestNode := {hw : ℕ × List ℕ // 1 ≤ hw.1 ∧ hw.2.length ≤ hw.1}def ForestAncestor (a b : ForestNode) : Prop :=  a.val.1 = b.val.1 ∧ a.val.2.IsPrefix b.val.2@[instance_reducible] def forestAncestry : Ancestry ForestNode where  le := ForestAncestor  lt a b := ForestAncestor a b ∧ ¬ ForestAncestor b a  le_refl a := ⟨rfl, List.prefix_refl _⟩  le_trans _ _ _ hab hbc := ⟨hab.1.trans hbc.1, hab.2.trans hbc.2⟩  le_antisymm a b hab hba := by    apply Subtype.ext    exact Prod.ext hab.1 (hab.2.sublist.antisymm hba.2.sublist)  predecessors_total hab hac := by    rcases List.prefix_or_prefix_of_prefix hab.2 hac.2 with h | h    · exact Or.inl ⟨hab.1.trans hac.1.symm, h⟩    · exact Or.inr ⟨hac.1.trans hab.1.symm, h⟩def forestSegments : SegmentSystem ForestNode :=  Ancestry.segments (A := forestAncestry)/-- Prefix order on all finite words, explicitly distinct from their lexicographic order. -/@[instance_reducible] def infiniteAncestry : Ancestry (List ℕ) where  le := List.IsPrefix  lt a b := a.IsPrefix b ∧ ¬ b.IsPrefix a  le_refl := List.prefix_refl  le_trans _ _ _ := List.IsPrefix.trans  le_antisymm _ _ hab hba := hab.sublist.antisymm hba.sublist  predecessors_total := List.prefix_or_prefix_of_prefix/-- The infinite-height tree of all finite words; this includes the empty root. -/def infiniteSegments : SegmentSystem (List ℕ) :=  Ancestry.segments (A := infiniteAncestry)abbrev ForestDual := forestSegments.Dualabbrev InfinitePredual := infiniteSegments.CoordinatePredualnamespace Diamond/-- Stage edges, retaining their parent, child number, and side. -/@[reducible] def Edge : ℕ → Type  | 0 => Unit  | n + 1 => Edge n × ℕ × Bool/-- At every replacement every parent edge receives countably many distinct new vertices. -/@[reducible] def Vertex : ℕ → Type  | 0 => Bool  | n + 1 => Vertex n ⊕ (Edge n × ℕ)def endpoint : (n : ℕ) → Edge n → Bool → Vertex n  | 0, _, b => b  | n + 1, (e, _, false), false => Sum.inl (endpoint n e false)  | _ + 1, (e, i, false), true => Sum.inr (e, i)  | _ + 1, (e, i, true), false => Sum.inr (e, i)  | n + 1, (e, _, true), true => Sum.inl (endpoint n e true)/-- The undirected replacement graph. -/def graph (n : ℕ) : SimpleGraph (Vertex n) :=  SimpleGraph.fromRel (fun u v => ∃ e : Edge n, u = endpoint n e false ∧ v = endpoint n e true)/-- Final edges have length 2⁻ᵏ: this is the shortest-path metric of the normalized graph. -/def distance (k : ℕ) (u v : Vertex k) : ℝ :=  ((graph k).dist u v : ℝ) / (Nat.cast 2 : ℝ) ^ kend Diamondtheorem headline_all_pairs :    (∀ (k : ℕ) (f : Diamond.Vertex k → ForestDual) (D s : ℝ),      0 < s → (∀ u v,s*Diamond.distance k u v ≤ ‖f u-f v‖ ∧        ‖f u-f v‖ ≤ D*s*Diamond.distance k u v) → 1+(k:ℝ)/4 ≤ D^2) ∧    (∀ (k : ℕ) (f : Diamond.Vertex k → InfinitePredual) (D s : ℝ),      0 < s → (∀ u v,s*Diamond.distance k u v ≤ ‖f u-f v‖ ∧        ‖f u-f v‖ ≤ D*s*Diamond.distance k u v) → 1+(k:ℝ)/4 ≤ D^2) := by  sorryend DiamondDistortionendend 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.

Asymptotic midpoint uniform convexity and unbounded diamond distortion in a reflexive tree space

Theorem

The space X=J^* defined by (eq:segment-norm) is infinite dimensional, separable, and reflexive, and has no equivalent AUC norm. Its given norm satisfies

\widehat\delta_X(t)\ge\sqrt{1+t^2/12}-1\qquad(t>0).

For every integer k\ge0, every embedding of D_k into X has distortion at least \sqrt{1+k/12}.

Section 1, Introduction Read it in the paper TeX source, line 99

Show the TeX

The space $X=J^*$ defined by \eqref{eq:segment-norm} is infinite
dimensional, separable, and reflexive, and has no equivalent AUC norm.
Its given norm satisfies
\[
 \widehat\delta_X(t)\ge\sqrt{1+t^2/12}-1\qquad(t>0).
\]
For every integer $k\ge0$, every embedding of $D_k$ into $X$ has
distortion at least $\sqrt{1+k/12}$.

Midpoint lenses in segment spaces

Theorem

Let Y be either the finite-height forest dual X=J^* or the infinite-height coordinate predual B_\infty. For every finite ancestral H, every x,y\in Y with x=P_Hx, and every R\ge0 with \|x+y\|,\|x-y\|\le R,

\|Q_Hy\|\le2\sqrt{R^2-\|x\|^2}.

Section 1, Introduction Read it in the paper TeX source, line 54

Show the TeX

Let $Y$ be either the finite-height forest dual $X=J^*$ or the
infinite-height coordinate predual $B_\infty$. For every finite
ancestral $H$, every $x,y\in Y$ with $x=P_Hx$, and every $R\ge0$
with $\|x+y\|,\|x-y\|\le R$,
\begin{equation}
 \|Q_Hy\|\le2\sqrt{R^2-\|x\|^2}.
\end{equation}

Distortion of countably branching diamonds from midpoint and tree energies

Theorem

For every integer k\ge0, every distortion-D embedding of \mathcal D_k into either X or B_\infty satisfies

D^2\ge1+k/4.

Section 1, Introduction Read it in the paper TeX source, line 53

Show the TeX

For every integer $k\ge0$, every distortion-$D$ embedding of
$\mathcal D_k$ into either $X$ or $B_\infty$ satisfies
\[
 D^2\ge1+k/4.
\]

Exact asymptotic moduli in a Daugavet subspace of L1

Theorem

Let E be an infinite-dimensional closed real subspace of L^1(\Omega,\mathbb P), where \mathbb P is a probability measure. Suppose its closed unit ball is totally bounded for convergence in measure and its inherited norm has the Daugavet property. Then every x\in S_E and t>0 satisfy

H_E(x,t)=\max\{t/2,t-1\},\qquad D_E(x,t)=\max\{0,t-2\}.

Consequently these are also the formulas for \widehat\delta_E(t) and \overline\delta_E(t). The space E admits no equivalent AUC norm.

Section 1, Introduction Read it in the paper TeX source, line 63

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Let $E$ be an infinite-dimensional closed real subspace of
$L^1(\Omega,\mathbb P)$, where $\mathbb P$ is a probability measure.
Suppose its closed unit ball is totally bounded for convergence in
measure and its inherited norm has the Daugavet property.  Then every
$x\in S_E$ and $t>0$ satisfy
\[
 H_E(x,t)=\max\{t/2,t-1\},\qquad
 D_E(x,t)=\max\{0,t-2\}.
\]
Consequently these are also the formulas for
$\widehat\delta_E(t)$ and $\overline\delta_E(t)$.  The space $E$ admits
no equivalent AUC norm.

Midpoint convexity from bounded tree potentials and path costs

Theorem

Each of the four completed spaces X_K is a separable infinite-dimensional real Banach space. For K\in\{K_{\rm s},K_{\rm p},K_{\rm a},K_{\rm g}\} and 0<t<1,

\widehat\delta_{X_K}(t)\ge\sqrt{1+t^2/4}-1.

None of these spaces admits an equivalent AUC norm.

Section 2, Four potential norms, one pair of inward tests Read it in the paper TeX source, line 63

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Each of the four completed spaces $X_K$ is a separable infinite-dimensional
real Banach space. For $K\in\{K_{\rm s},K_{\rm p},K_{\rm a},K_{\rm g}\}$
and $0<t<1$,
\begin{equation}
 \widehat\delta_{X_K}(t)\ge\sqrt{1+t^2/4}-1.
\end{equation}
None of these spaces admits an equivalent AUC norm.

Theorem

The real Banach space X_{\mathrm e} defined by (fr:tests) and (fr:norm) satisfies

\widehat\delta_{\|\cdot\|_{\mathrm e}}(t) \ge\frac{\sqrt{1+t^2/576}-1}{4}\qquad(0<t<1).

It admits no equivalent AUC norm. More precisely, if \alpha\|\cdot\|_{\mathrm e}\le N\le\beta\|\cdot\|_{\mathrm e}, where 0<\alpha\le\beta<\infty, then \overline\delta_N(\alpha/(4\beta))=0.

Section 4, A free root and a planar perturbation Read it in the paper TeX source, line 67

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The real Banach space $X_{\mathrm e}$ defined by \eqref{fr:tests} and
\eqref{fr:norm} satisfies
\begin{equation}
 \widehat\delta_{\|\cdot\|_{\mathrm e}}(t)
 \ge\frac{\sqrt{1+t^2/576}-1}{4}\qquad(0<t<1).
 
\end{equation}
It admits no equivalent AUC norm.  More precisely, if
$\alpha\|\cdot\|_{\mathrm e}\le N\le\beta\|\cdot\|_{\mathrm e}$,
where $0<\alpha\le\beta<\infty$, then
$\overline\delta_N(\alpha/(4\beta))=0$.

Theorem (Clipping estimate on a head and its complement)

If x,y\in X, P_Ax=x and P_Ay=0 for a finite head A, then, for every c>0,

\max\{\|x+y\|,\|x-y\|\} \ge\frac{\|x\|+c\|y\|/2}{\sqrt{1+4c^2}}.

In particular the left side is at least \sqrt{\|x\|^2+\|y\|^2/16}.

Section 5, Linear costs of root paths Read it in the paper TeX source, line 287

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If $x,y\in X$, $P_Ax=x$ and $P_Ay=0$ for a finite head $A$, then,
for every $c>0$,
\begin{equation}
 \max\{\|x+y\|,\|x-y\|\}
 \ge\frac{\|x\|+c\|y\|/2}{\sqrt{1+4c^2}}.
 
\end{equation}
In particular the left side is at least
$\sqrt{\|x\|^2+\|y\|^2/16}$.

Independent products in real L1: asymptotic midpoint convexity without AUC renormings

Theorem

For every multiplier W as above, the Banach space X_W is infinite dimensional and has no equivalent AUC norm. For the exponential and Gaussian-square laws, the inherited norms satisfy

\begin{aligned}\whdelta_{X_{\mathrm{exp}}}(t) &\ge \frac{7t}{30}c(10/t)>0 &&(t>0),\\ \whdelta_{X_{\mathrm{sq}}}(t) &\ge\frac{\PP\bigl(|G|\ge4\sqrt2(2+\sqrt2)/t\bigr)}{8}>0 &&(t>0),\\ \whdelta_{X_{\mathrm{sq}}}(t) &\ge\frac{\PP\bigl(|G|\ge8\sqrt6/t\bigr)}{4}>0 &&(t>0).\end{aligned}

Section 1, Introduction Read it in the paper TeX source, line 57

Show the TeX

For every multiplier $W$ as above, the Banach space $X_W$ is infinite
dimensional and has no equivalent AUC norm.
For the exponential and Gaussian-square laws, the inherited norms satisfy
\begin{align}
 \whdelta_{X_{\mathrm{exp}}}(t)
 &\ge \frac{7t}{30}c(10/t)>0 &&(t>0),\\
 \whdelta_{X_{\mathrm{sq}}}(t)
 &\ge\frac{\PP\bigl(|G|\ge4\sqrt2(2+\sqrt2)/t\bigr)}{8}>0
 &&(t>0),\\
 \whdelta_{X_{\mathrm{sq}}}(t)
 &\ge\frac{\PP\bigl(|G|\ge8\sqrt6/t\bigr)}{4}>0
 &&(t>0).
\end{align}

Midpoint convexity from two recursive potentials

Theorem

For N=N_\Sigma on X_\Sigma, or N=N_J on X_J, one has

\widehat\delta_N(t)\ge t^3/128\qquad(0<t<1).

For X_0, a finite initial set H, a vector x supported in H, and z\in F_H\setminus\{0\}, one has

\frac{N_0(x+z)+N_0(x-z)}2\ge N_0(x) +\frac{N_0(z)^3}{8(2N_0(x)+N_0(z))^2}.

Each of X_\Sigma,X_0,X_J admits no equivalent AUC norm.

Section 2, The fields, the roots, and the completed spaces Read it in the paper TeX source, line 65

Show the TeX

For $N=N_\Sigma$ on $X_\Sigma$, or $N=N_J$ on $X_J$, one has
\begin{equation}
 \widehat\delta_N(t)\ge t^3/128\qquad(0<t<1).
\end{equation}
For $X_0$, a finite initial set $H$, a vector $x$ supported in $H$, and
$z\in F_H\setminus\{0\}$, one has
\begin{equation}
 \frac{N_0(x+z)+N_0(x-z)}2\ge N_0(x)
 +\frac{N_0(z)^3}{8(2N_0(x)+N_0(z))^2}.
\end{equation}
Each of $X_\Sigma,X_0,X_J$ admits no equivalent AUC norm.

Theorem (Separated midpoint families)

For every \varepsilon>0 there is \eta(\varepsilon)\in(0,1) such that, for x,z_1,z_2,\ldots\in X_J,

\left. \begin{gathered} N_J(x+z_j),N_J(x-z_j)\le1\quad(j\ge1),\\ N_J(z_i-z_j)\ge\varepsilon\quad(i\ne j) \end{gathered} \right\} \quad\Longrightarrow\quad N_J(x)\le1-\eta(\varepsilon).

For 0<\varepsilon\le2, one can take \eta(\varepsilon)=\min\{1/2,\gamma(2,\varepsilon/16)\}.

Section 6, Logarithmic propagation on the joined tree Read it in the paper TeX source, line 480

Show the TeX

For every $\varepsilon>0$ there is $\eta(\varepsilon)\in(0,1)$ such
that, for $x,z_1,z_2,\ldots\in X_J$,
\[
 \left.
 \begin{gathered}
 N_J(x+z_j),N_J(x-z_j)\le1\quad(j\ge1),\\
 N_J(z_i-z_j)\ge\varepsilon\quad(i\ne j)
 \end{gathered}
 \right\}
 \quad\Longrightarrow\quad N_J(x)\le1-\eta(\varepsilon).
\]
For $0<\varepsilon\le2$, one can take
$\eta(\varepsilon)=\min\{1/2,\gamma(2,\varepsilon/16)\}$.

Theorem

Let x\in X_{\mathrm v} have finite total support. In each component where x^{(h)}\ne0, choose a finite ancestral set A_h containing that support. Suppose y\in X_{\mathrm v} vanishes on every such A_h. With S=\|x+y\|_{\mathrm v}^2+\|x-y\|_{\mathrm v}^2, one has

S-2\|x\|_{\mathrm v}^2 \ge c_6 S^{-2}\|y\|_{\mathrm v}^6, \qquad c_6=2\exp(-(\mathrm e-1))4^{-6}.

The right side is zero when S=0.

Section 7, Changing the child exponent with the height Read it in the paper TeX source, line 294

Show the TeX

Let $x\in X_{\mathrm v}$ have finite total support.  In each component
where $x^{(h)}\ne0$, choose a finite ancestral set $A_h$ containing
that support.  Suppose $y\in X_{\mathrm v}$ vanishes on every such
$A_h$.  With
$S=\|x+y\|_{\mathrm v}^2+\|x-y\|_{\mathrm v}^2$, one has
\begin{equation}
 S-2\|x\|_{\mathrm v}^2
 \ge c_6 S^{-2}\|y\|_{\mathrm v}^6,
 \qquad c_6=2\exp(-(\mathrm e-1))4^{-6}.
\end{equation}
The right side is zero when $S=0$.

Configuration and file facts

File: lean/ComparatorChallenges/DiamondDistortion.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Analysis/DiamondDistortion/Descent.lean
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: 37 local vocabulary declarations
Lines of Lean, without comments or blank lines: 224
Second proof-verification program (nanoda): Off in OpenAI's configuration.

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

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