Plectis Atlas
Search

↑ and ↓ move, Enter opens, Esc closes.

Lean statement

Midpoint-lens tail estimate

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.SegmentLenses.midpoint_lens_main

Lean statement file: MidpointLenses.leanShow on the mapFamily 331. 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 midpoint_lens_main :    (∀ (H : Finset ForestVertex) (_ : Ancestral ForestAncestor H)      (x y : X) (_ : x = head H x) (R : ℝ) (_ : 0 ≤ R)      (_ : ‖x + y‖ ≤ R) (_ : ‖x - y‖ ≤ R),      ‖tail H y‖ ≤ 2 * Real.sqrt (R ^ 2 - ‖x‖ ^ 2)) ∧    (∀ (H : Finset InfiniteVertex) (_ : Ancestral List.IsPrefix H)      (x y : BInfinite) (_ : x.val = head H x.val) (R : ℝ) (_ : 0 ≤ R)      (_ : ‖x + y‖ ≤ R) (_ : ‖x - y‖ ≤ R),      ‖tail H y.val‖ ≤ 2 * Real.sqrt (R ^ 2 - ‖x‖ ^ 2)) :=  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.

  • abbreviation OAI.SegmentLenses.InfiniteVertex line 18

    abbrev InfiniteVertex := List ℕ

  • definition OAI.SegmentLenses.ForestVertex line 20

    def ForestVertex := {p : ℕ × List ℕ // 1 ≤ p.1 ∧ p.2.length ≤ p.1}

  • definition OAI.SegmentLenses.ForestAncestor line 22

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

  • definition OAI.SegmentLenses.Ancestral line 28

    def Ancestral {V : Type*} (ancestor : V → V → Prop) (H : Finset V) : Prop :=  ∀ ⦃a b⦄, b ∈ H → ancestor a b → a ∈ H

  • class OAI.SegmentLenses.SegmentSystem line 31

    class SegmentSystem (V : Type*) where  isSegment : Finset V → Prop  singleton : ∀ v, isSegment {v}

  • structure OAI.SegmentLenses.SegmentFamily line 35

    structure SegmentFamily (V : Type*) [SegmentSystem V] where  sets : Finset (Finset V)  segment : ∀ S ∈ sets, SegmentSystem.isSegment S  disjoint : (sets : Set (Finset V)).PairwiseDisjoint id

  • definition OAI.SegmentLenses.SegmentFamily.column line 44

    def column (F : SegmentFamily V) (v : V) : EuclideanSpace ℝ F.sets :=  by classical exact WithLp.toLp 2 (fun S => if v ∈ S.val then 1 else 0)

  • definition OAI.SegmentLenses.SegmentFamily.test line 47

    def test (F : SegmentFamily V) : (V →₀ ℝ) →ₗ[ℝ] EuclideanSpace ℝ F.sets :=  Finsupp.linearCombination ℝ F.column

  • definition OAI.SegmentLenses.SegmentFamily.seminorm line 50

    def seminorm (F : SegmentFamily V) : Seminorm ℝ (V →₀ ℝ) :=  (normSeminorm ℝ (EuclideanSpace ℝ F.sets)).comp F.test

  • definition OAI.SegmentLenses.SegmentFamily.singleton line 53

    def singleton (v : V) : SegmentFamily V where
    Show the remaining 10 lines
      sets := {{v}}  segment := by    intro S h    have he : S = {v} := Finset.mem_singleton.mp h    rw [he]    exact SegmentSystem.singleton v  disjoint := by    intro S hS T hT hne    simp only [Finset.mem_coe, Finset.mem_singleton] at hS hT    exact False.elim (hne (hS.trans hT.symm))

  • definition OAI.SegmentLenses.segmentSeminorm line 67

    def segmentSeminorm (V : Type*) [SegmentSystem V] : Seminorm ℝ (V →₀ ℝ) :=  ⨆ F : SegmentFamily V, F.seminorm

  • definition OAI.SegmentLenses.Test line 70

    def Test (V : Type*) := V →₀ ℝ

  • definition OAI.SegmentLenses.Test.coeff line 78

    def coeff (u : Test V) : V →₀ ℝ := u

  • definition OAI.SegmentLenses.Test.ofFinsupp line 80

    def ofFinsupp (u : V →₀ ℝ) : Test V := u

  • definition OAI.SegmentLenses.Test.evaluation line 185

    def evaluation (v : V) : Test V →L[ℝ] ℝ := by
    Show the remaining 88 lines
      have norm_column_le {V : Type _} [SegmentSystem V] (F : SegmentFamily V) (v : V) : ‖F.column v‖ ≤ 1 := by    classical    by_cases hv : ∃ S ∈ F.sets, v ∈ S    · obtain ⟨S, hS, hvS⟩ := hv      have he : F.column v = PiLp.single 2 (⟨S, hS⟩ : F.sets) (1 : ℝ) := by        ext T        change (if v ∈ T.val then (1 : ℝ) else 0) = _        by_cases hTS : T = ⟨S, hS⟩        · subst T          simp [hvS]        · have hn : v ∉ T.val := by            intro hvT            have hne : T.val ≠ S := fun h => hTS (Subtype.ext h)            exact Finset.disjoint_left.mp (F.disjoint T.property hS hne) hvT hvS          simp [hTS, hn]      rw [he, PiLp.norm_single]      norm_num    · have he : F.column v = 0 := by        ext S        have hn : v ∉ S.val := fun h => hv ⟨S.val, S.property, h⟩        simp [SegmentFamily.column, hn]      rw [he, norm_zero]      norm_num  have test_apply {V : Type _} [SegmentSystem V] (F : SegmentFamily V) (u : V →₀ ℝ) (S : F.sets) :      F.test u S = ∑ v ∈ S.val, u v := by    classical    simp only [SegmentFamily.test, Finsupp.linearCombination_apply, Finsupp.sum,      WithLp.ofLp_sum, Finset.sum_apply, PiLp.smul_apply, SegmentFamily.column,      smul_eq_mul]    simp only [mul_ite, mul_one, mul_zero]    rw [← Finset.sum_filter]    apply Finset.sum_subset    · intro v hv      exact (Finset.mem_filter.mp hv).2    · intro v hv hnv      have hnot : v ∉ u.support := by        intro hvs        exact hnv (Finset.mem_filter.mpr ⟨hvs, hv⟩)      exact Finsupp.notMem_support_iff.mp hnot  have norm_test_le {V : Type _} [SegmentSystem V] (F : SegmentFamily V) (u : V →₀ ℝ) :      ‖F.test u‖ ≤ ∑ v ∈ u.support, |u v| := by    classical    rw [SegmentFamily.test, Finsupp.linearCombination_apply, Finsupp.sum]    calc      ‖∑ v ∈ u.support, u v • F.column v‖ ≤          ∑ v ∈ u.support, ‖u v • F.column v‖ := norm_sum_le _ _      _ ≤ ∑ v ∈ u.support, |u v| := by        apply Finset.sum_le_sum        intro v hv        rw [norm_smul, Real.norm_eq_abs]        exact mul_le_of_le_one_right (abs_nonneg _) (norm_column_le F v)  have seminorm_sq {V : Type _} [SegmentSystem V] (F : SegmentFamily V) (u : V →₀ ℝ) :      F.seminorm u ^ 2 = ∑ S : F.sets, (∑ v ∈ S.val, u v) ^ 2 := by    change ‖F.test u‖ ^ 2 = _    rw [EuclideanSpace.real_norm_sq_eq]    apply Finset.sum_congr rfl    intro S _    rw [test_apply]  have singleton_seminorm {V : Type _} [SegmentSystem V] (v : V) (u : V →₀ ℝ) :      (SegmentFamily.singleton v).seminorm u = |u v| := by    have h := seminorm_sq (SegmentFamily.singleton v) u    have hh : (∑ S : (SegmentFamily.singleton v).sets, (∑ w ∈ S.val, u w) ^ 2) = u v ^ 2 := by      exact (Finset.sum_coe_sort (SegmentFamily.singleton v).sets        (fun S : Finset V => (∑ w ∈ S, u w) ^ 2)).trans (by simp [SegmentFamily.singleton])    rw [hh] at h    have hn := apply_nonneg (SegmentFamily.singleton v).seminorm u    nlinarith [sq_abs (u v), abs_nonneg (u v)]  have segmentTests_bdd (V : Type _) [SegmentSystem V] :      BddAbove (Set.range (SegmentFamily.seminorm (V := V))) := by    rw [Seminorm.bddAbove_iff]    refine ⟨fun u => ∑ v ∈ u.support, |u v|, ?_⟩    rintro _ ⟨p, ⟨F, rfl⟩, rfl⟩ u    exact norm_test_le F u  have le_segmentSeminorm {V : Type _} [SegmentSystem V]      (F : SegmentFamily V) (u : V →₀ ℝ) : F.seminorm u ≤ segmentSeminorm V u := by    have h := le_ciSup (f := fun G : SegmentFamily V => G.seminorm) (segmentTests_bdd V) F    exact Seminorm.le_def.mp h u  have abs_coord_le {V : Type _} [SegmentSystem V] (u : V →₀ ℝ) (v : V) :      |u v| ≤ segmentSeminorm V u := by    rw [← singleton_seminorm v u]    exact le_segmentSeminorm _ _  exact let f : Test V →ₗ[ℝ] ℝ :=      { toFun := fun u => u.coeff v        map_add' := fun _ _ => rfl        map_smul' := fun _ _ => rfl }    f.mkContinuous 1 (fun u => by      change |u.coeff v| ≤ 1 * segmentSeminorm V u.coeff      simpa only [one_mul] using abs_coord_le u.coeff v)

  • abbreviation OAI.SegmentLenses.Primal line 277

    abbrev Primal (V : Type*) [SegmentSystem V] := UniformSpace.Completion (Test V)

  • abbreviation OAI.SegmentLenses.FullDual line 279

    abbrev FullDual (V : Type*) [SegmentSystem V] := StrongDual ℝ (Primal V)

  • definition OAI.SegmentLenses.unitVector line 281

    def unitVector {V : Type*} [SegmentSystem V] (v : V) : Primal V :=  (Test.ofFinsupp (Finsupp.single v 1) : Test V)

  • definition OAI.SegmentLenses.coordinate line 284

    def coordinate {V : Type*} [SegmentSystem V] (v : V) : FullDual V :=  (Test.evaluation v).extend UniformSpace.Completion.toComplL

  • definition OAI.SegmentLenses.coordinateSpan line 287

    def coordinateSpan (V : Type*) [SegmentSystem V] : Submodule ℝ (FullDual V) :=  (Submodule.span ℝ (Set.range (coordinate (V := V)))).topologicalClosure

  • abbreviation OAI.SegmentLenses.CoordinatePredual line 290

    abbrev CoordinatePredual (V : Type*) [SegmentSystem V] : Type _ := ↥(coordinateSpan V)

  • definition OAI.SegmentLenses.head line 292

    def head {V : Type*} [SegmentSystem V] (H : Finset V) (x : FullDual V) : FullDual V :=  ∑ v ∈ H, x (unitVector v) • coordinate v

  • definition OAI.SegmentLenses.tail line 295

    def tail {V : Type*} [SegmentSystem V] (H : Finset V) (x : FullDual V) : FullDual V :=  x - head H x

  • abbreviation OAI.SegmentLenses.X line 322

    abbrev X := FullDual ForestVertex

1 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: predual, dual, forest, coordinate, ancestral, infinite and sqrt. Found in the paper’s title, which its main theorem states: segment, midpoint and spaces. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: height and finite.

Statements in Midpoint lenses in segment spaces come first because the Lean file and module names share words with that paper’s title.

Theorem 1.1 of 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}

Other possible paper matches, closest wording first

  • Corollary 10.3 of Midpoint lenses in segment spaces. Words of the Lean statement, its names split into words, that also appear in the theorem: predual, forest, coordinate, infinite, space and sqrt. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: height, estimate and finite.
  • Proposition 19.3 of Midpoint lenses in segment spaces. Words of the Lean statement, its names split into words, that also appear in the theorem: eq, ancestor, segment, forest, coordinate, space and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: height and finite.
  • Proposition 22.1 of Midpoint lenses in segment spaces. Words of the Lean statement, its names split into words, that also appear in the theorem: tail, unit, vectors, head, supported, ancestral and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: finite.
  • 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, natural, disjoint, completion, integers, real and eight more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: tree, estimate and finite.
  • 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, unit, forest, coordinate, vectors and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: height and finite.
  • Corollary 23.5 of Midpoint lenses in segment spaces. Words of the Lean statement, its names split into words, that also appear in the theorem: vertex, forest and sqrt.

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 229 s; peak memory 6.2 GiB).

    Reported 10 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 controls midpoint lenses in Banach spaces defined by tree-segment norms. The formalization proves the following estimate in both the full dual of the finite-height forest space and the coordinate predual of the infinite-tree space. If $x$ is supported on a finite ancestral set $H$, $R\ge0$, and $\|x+y\|,\|x-y\|\le R$, then the part of $y$ outside $H$ has norm at most $2\sqrt{R^2-\|x\|^2}$.”

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

343 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: 134 to its own declarations, 141 to Mathlib’s documentation. 31 declared names also link to the same declaration in OpenAI’s library. 4 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 section/-!# Segment tests, their completion, and coordinate functionals-/open scoped BigOperatorsopen Finsetnamespace SegmentLensesabbrev InfiniteVertex := List ℕdef ForestVertex := {p : ℕ × List ℕ // 1 ≤ p.1 ∧ p.2.length ≤ p.1}def ForestAncestor (a b : ForestVertex) : Prop :=  a.val.1 = b.val.1 ∧ a.val.2.IsPrefix b.val.2def IsInterval {V : Type*} (ancestor : V → V → Prop) (S : Finset V) : Prop :=  ∃ a b, ancestor a b ∧ ∀ v, v ∈ S ↔ ancestor a v ∧ ancestor v bdef Ancestral {V : Type*} (ancestor : V → V → Prop) (H : Finset V) : Prop :=  ∀ ⦃a b⦄, b ∈ H → ancestor a b → a ∈ Hclass SegmentSystem (V : Type*) where  isSegment : Finset V → Prop  singleton : ∀ v, isSegment {v}structure SegmentFamily (V : Type*) [SegmentSystem V] where  sets : Finset (Finset V)  segment : ∀ S ∈ sets, SegmentSystem.isSegment S  disjoint : (sets : Set (Finset V)).PairwiseDisjoint idnamespace SegmentFamilyvariable {V : Type*} [SegmentSystem V]def column (F : SegmentFamily V) (v : V) : EuclideanSpace ℝ F.sets :=  by classical exact WithLp.toLp 2 (fun S => if v ∈ S.val then 1 else 0)def test (F : SegmentFamily V) : (V →₀ ℝ) →ₗ[ℝ] EuclideanSpace ℝ F.sets :=  Finsupp.linearCombination ℝ F.columndef seminorm (F : SegmentFamily V) : Seminorm ℝ (V →₀ ℝ) :=  (normSeminorm ℝ (EuclideanSpace ℝ F.sets)).comp F.testdef singleton (v : V) : SegmentFamily V where  sets := {{v}}  segment := by    intro S h    have he : S = {v} := Finset.mem_singleton.mp h    rw [he]    exact SegmentSystem.singleton v  disjoint := by    intro S hS T hT hne    simp only [Finset.mem_coe, Finset.mem_singleton] at hS hT    exact False.elim (hne (hS.trans hT.symm))end SegmentFamilydef segmentSeminorm (V : Type*) [SegmentSystem V] : Seminorm ℝ (V →₀ ℝ) :=  ⨆ F : SegmentFamily V, F.seminormdef Test (V : Type*) := V →₀ ℝnamespace Testvariable {V : Type*}instance : AddCommGroup (Test V) := inferInstanceAs (AddCommGroup (V →₀ ℝ))instance : Module ℝ (Test V) := inferInstanceAs (Module ℝ (V →₀ ℝ))def coeff (u : Test V) : V →₀ ℝ := udef ofFinsupp (u : V →₀ ℝ) : Test V := uvariable [SegmentSystem V]instance : Norm (Test V) where norm u := segmentSeminorm V u.coeffinstance : NormedAddCommGroup (Test V) := by  have norm_column_le {V : Type _} [SegmentSystem V] (F : SegmentFamily V) (v : V) : ‖F.column v‖ ≤ 1 := by    classical    by_cases hv : ∃ S ∈ F.sets, v ∈ S    · obtain ⟨S, hS, hvS⟩ := hv      have he : F.column v = PiLp.single 2 (⟨S, hS⟩ : F.sets) (1 : ℝ) := by        ext T        change (if v ∈ T.val then (1 : ℝ) else 0) = _        by_cases hTS : T = ⟨S, hS⟩        · subst T          simp [hvS]        · have hn : v ∉ T.val := by            intro hvT            have hne : T.val ≠ S := fun h => hTS (Subtype.ext h)            exact Finset.disjoint_left.mp (F.disjoint T.property hS hne) hvT hvS          simp [hTS, hn]      rw [he, PiLp.norm_single]      norm_num    · have he : F.column v = 0 := by        ext S        have hn : v ∉ S.val := fun h => hv ⟨S.val, S.property, h⟩        simp [SegmentFamily.column, hn]      rw [he, norm_zero]      norm_num  have test_apply {V : Type _} [SegmentSystem V] (F : SegmentFamily V) (u : V →₀ ℝ) (S : F.sets) :      F.test u S = ∑ v ∈ S.val, u v := by    classical    simp only [SegmentFamily.test, Finsupp.linearCombination_apply, Finsupp.sum,      WithLp.ofLp_sum, Finset.sum_apply, PiLp.smul_apply, SegmentFamily.column,      smul_eq_mul]    simp only [mul_ite, mul_one, mul_zero]    rw [← Finset.sum_filter]    apply Finset.sum_subset    · intro v hv      exact (Finset.mem_filter.mp hv).2    · intro v hv hnv      have hnot : v ∉ u.support := by        intro hvs        exact hnv (Finset.mem_filter.mpr ⟨hvs, hv⟩)      exact Finsupp.notMem_support_iff.mp hnot  have norm_test_le {V : Type _} [SegmentSystem V] (F : SegmentFamily V) (u : V →₀ ℝ) :      ‖F.test u‖ ≤ ∑ v ∈ u.support, |u v| := by    classical    rw [SegmentFamily.test, Finsupp.linearCombination_apply, Finsupp.sum]    calc      ‖∑ v ∈ u.support, u v • F.column v‖ ≤          ∑ v ∈ u.support, ‖u v • F.column v‖ := norm_sum_le _ _      _ ≤ ∑ v ∈ u.support, |u v| := by        apply Finset.sum_le_sum        intro v hv        rw [norm_smul, Real.norm_eq_abs]        exact mul_le_of_le_one_right (abs_nonneg _) (norm_column_le F v)  have seminorm_sq {V : Type _} [SegmentSystem V] (F : SegmentFamily V) (u : V →₀ ℝ) :      F.seminorm u ^ 2 = ∑ S : F.sets, (∑ v ∈ S.val, u v) ^ 2 := by    change ‖F.test u‖ ^ 2 = _    rw [EuclideanSpace.real_norm_sq_eq]    apply Finset.sum_congr rfl    intro S _    rw [test_apply]  have singleton_seminorm {V : Type _} [SegmentSystem V] (v : V) (u : V →₀ ℝ) :      (SegmentFamily.singleton v).seminorm u = |u v| := by    have h := seminorm_sq (SegmentFamily.singleton v) u    have hh : (∑ S : (SegmentFamily.singleton v).sets, (∑ w ∈ S.val, u w) ^ 2) = u v ^ 2 := by      exact (Finset.sum_coe_sort (SegmentFamily.singleton v).sets        (fun S : Finset V => (∑ w ∈ S, u w) ^ 2)).trans (by simp [SegmentFamily.singleton])    rw [hh] at h    have hn := apply_nonneg (SegmentFamily.singleton v).seminorm u    nlinarith [sq_abs (u v), abs_nonneg (u v)]  have segmentTests_bdd (V : Type _) [SegmentSystem V] :      BddAbove (Set.range (SegmentFamily.seminorm (V := V))) := by    rw [Seminorm.bddAbove_iff]    refine ⟨fun u => ∑ v ∈ u.support, |u v|, ?_⟩    rintro _ ⟨p, ⟨F, rfl⟩, rfl⟩ u    exact norm_test_le F u  have le_segmentSeminorm {V : Type _} [SegmentSystem V]      (F : SegmentFamily V) (u : V →₀ ℝ) : F.seminorm u ≤ segmentSeminorm V u := by    have h := le_ciSup (f := fun G : SegmentFamily V => G.seminorm) (segmentTests_bdd V) F    exact Seminorm.le_def.mp h u  have abs_coord_le {V : Type _} [SegmentSystem V] (u : V →₀ ℝ) (v : V) :      |u v| ≤ segmentSeminorm V u := by    rw [← singleton_seminorm v u]    exact le_segmentSeminorm _ _  have segmentSeminorm_eq_zero {V : Type _} [SegmentSystem V] (u : V →₀ ℝ) :      segmentSeminorm V u = 0 ↔ u = 0 := by    constructor    · intro h      ext v      have hh := abs_coord_le u v      rw [h] at hh      simpa using hh    · rintro rfl      exact map_zero _  exact NormedAddCommGroup.ofCore (𝕜 := ℝ) {    norm_nonneg u := apply_nonneg (segmentSeminorm V) u.coeff    norm_smul c u := map_smul_eq_mul (segmentSeminorm V) c u.coeff    norm_triangle u v := map_add_le_add (segmentSeminorm V) u.coeff v.coeff    norm_eq_zero_iff u := segmentSeminorm_eq_zero u.coeff }instance : NormedSpace ℝ (Test V) where  norm_smul_le c u := le_of_eq (map_smul_eq_mul (segmentSeminorm V) c u.coeff)def evaluation (v : V) : Test V →L[ℝ] ℝ := by  have norm_column_le {V : Type _} [SegmentSystem V] (F : SegmentFamily V) (v : V) : ‖F.column v‖ ≤ 1 := by    classical    by_cases hv : ∃ S ∈ F.sets, v ∈ S    · obtain ⟨S, hS, hvS⟩ := hv      have he : F.column v = PiLp.single 2 (⟨S, hS⟩ : F.sets) (1 : ℝ) := by        ext T        change (if v ∈ T.val then (1 : ℝ) else 0) = _        by_cases hTS : T = ⟨S, hS⟩        · subst T          simp [hvS]        · have hn : v ∉ T.val := by            intro hvT            have hne : T.val ≠ S := fun h => hTS (Subtype.ext h)            exact Finset.disjoint_left.mp (F.disjoint T.property hS hne) hvT hvS          simp [hTS, hn]      rw [he, PiLp.norm_single]      norm_num    · have he : F.column v = 0 := by        ext S        have hn : v ∉ S.val := fun h => hv ⟨S.val, S.property, h⟩        simp [SegmentFamily.column, hn]      rw [he, norm_zero]      norm_num  have test_apply {V : Type _} [SegmentSystem V] (F : SegmentFamily V) (u : V →₀ ℝ) (S : F.sets) :      F.test u S = ∑ v ∈ S.val, u v := by    classical    simp only [SegmentFamily.test, Finsupp.linearCombination_apply, Finsupp.sum,      WithLp.ofLp_sum, Finset.sum_apply, PiLp.smul_apply, SegmentFamily.column,      smul_eq_mul]    simp only [mul_ite, mul_one, mul_zero]    rw [← Finset.sum_filter]    apply Finset.sum_subset    · intro v hv      exact (Finset.mem_filter.mp hv).2    · intro v hv hnv      have hnot : v ∉ u.support := by        intro hvs        exact hnv (Finset.mem_filter.mpr ⟨hvs, hv⟩)      exact Finsupp.notMem_support_iff.mp hnot  have norm_test_le {V : Type _} [SegmentSystem V] (F : SegmentFamily V) (u : V →₀ ℝ) :      ‖F.test u‖ ≤ ∑ v ∈ u.support, |u v| := by    classical    rw [SegmentFamily.test, Finsupp.linearCombination_apply, Finsupp.sum]    calc      ‖∑ v ∈ u.support, u v • F.column v‖ ≤          ∑ v ∈ u.support, ‖u v • F.column v‖ := norm_sum_le _ _      _ ≤ ∑ v ∈ u.support, |u v| := by        apply Finset.sum_le_sum        intro v hv        rw [norm_smul, Real.norm_eq_abs]        exact mul_le_of_le_one_right (abs_nonneg _) (norm_column_le F v)  have seminorm_sq {V : Type _} [SegmentSystem V] (F : SegmentFamily V) (u : V →₀ ℝ) :      F.seminorm u ^ 2 = ∑ S : F.sets, (∑ v ∈ S.val, u v) ^ 2 := by    change ‖F.test u‖ ^ 2 = _    rw [EuclideanSpace.real_norm_sq_eq]    apply Finset.sum_congr rfl    intro S _    rw [test_apply]  have singleton_seminorm {V : Type _} [SegmentSystem V] (v : V) (u : V →₀ ℝ) :      (SegmentFamily.singleton v).seminorm u = |u v| := by    have h := seminorm_sq (SegmentFamily.singleton v) u    have hh : (∑ S : (SegmentFamily.singleton v).sets, (∑ w ∈ S.val, u w) ^ 2) = u v ^ 2 := by      exact (Finset.sum_coe_sort (SegmentFamily.singleton v).sets        (fun S : Finset V => (∑ w ∈ S, u w) ^ 2)).trans (by simp [SegmentFamily.singleton])    rw [hh] at h    have hn := apply_nonneg (SegmentFamily.singleton v).seminorm u    nlinarith [sq_abs (u v), abs_nonneg (u v)]  have segmentTests_bdd (V : Type _) [SegmentSystem V] :      BddAbove (Set.range (SegmentFamily.seminorm (V := V))) := by    rw [Seminorm.bddAbove_iff]    refine ⟨fun u => ∑ v ∈ u.support, |u v|, ?_⟩    rintro _ ⟨p, ⟨F, rfl⟩, rfl⟩ u    exact norm_test_le F u  have le_segmentSeminorm {V : Type _} [SegmentSystem V]      (F : SegmentFamily V) (u : V →₀ ℝ) : F.seminorm u ≤ segmentSeminorm V u := by    have h := le_ciSup (f := fun G : SegmentFamily V => G.seminorm) (segmentTests_bdd V) F    exact Seminorm.le_def.mp h u  have abs_coord_le {V : Type _} [SegmentSystem V] (u : V →₀ ℝ) (v : V) :      |u v| ≤ segmentSeminorm V u := by    rw [← singleton_seminorm v u]    exact le_segmentSeminorm _ _  exact let f : Test V →ₗ[ℝ] ℝ :=      { toFun := fun u => u.coeff v        map_add' := fun _ _ => rfl        map_smul' := fun _ _ => rfl }    f.mkContinuous 1 (fun u => by      change |u.coeff v| ≤ 1 * segmentSeminorm V u.coeff      simpa only [one_mul] using abs_coord_le u.coeff v)end Testabbrev Primal (V : Type*) [SegmentSystem V] := UniformSpace.Completion (Test V)abbrev FullDual (V : Type*) [SegmentSystem V] := StrongDual ℝ (Primal V)def unitVector {V : Type*} [SegmentSystem V] (v : V) : Primal V :=  (Test.ofFinsupp (Finsupp.single v 1) : Test V)def coordinate {V : Type*} [SegmentSystem V] (v : V) : FullDual V :=  (Test.evaluation v).extend UniformSpace.Completion.toComplLdef coordinateSpan (V : Type*) [SegmentSystem V] : Submodule ℝ (FullDual V) :=  (Submodule.span ℝ (Set.range (coordinate (V := V)))).topologicalClosureabbrev CoordinatePredual (V : Type*) [SegmentSystem V] : Type _ := ↥(coordinateSpan V)def head {V : Type*} [SegmentSystem V] (H : Finset V) (x : FullDual V) : FullDual V :=  ∑ v ∈ H, x (unitVector v) • coordinate vdef tail {V : Type*} [SegmentSystem V] (H : Finset V) (x : FullDual V) : FullDual V :=  x - head H xinstance infiniteSegments : SegmentSystem InfiniteVertex where  isSegment := IsInterval List.IsPrefix  singleton v := by    refine ⟨v, v, List.prefix_rfl, ?_⟩    intro w    simp only [Finset.mem_singleton]    constructor    · rintro rfl; exact ⟨List.prefix_rfl, List.prefix_rfl⟩    · rintro ⟨hvw, hwv⟩      exact hwv.eq_of_length (Nat.le_antisymm hwv.length_le hvw.length_le)instance forestSegments : SegmentSystem ForestVertex where  isSegment := IsInterval ForestAncestor  singleton v := by    refine ⟨v, v, ⟨rfl, List.prefix_rfl⟩, ?_⟩    intro w    simp only [Finset.mem_singleton]    constructor    · rintro rfl; exact ⟨⟨rfl, List.prefix_rfl⟩, ⟨rfl, List.prefix_rfl⟩⟩    · rintro ⟨hvw, hwv⟩      exact Subtype.ext (Prod.ext hwv.1 (hwv.2.eq_of_length (Nat.le_antisymm hwv.2.length_le hvw.2.length_le)))abbrev J := Primal ForestVertexabbrev X := FullDual ForestVertexabbrev JInfinite := Primal InfiniteVertexabbrev BInfinite := CoordinatePredual InfiniteVertextheorem midpoint_lens_main :    (∀ (H : Finset ForestVertex) (_ : Ancestral ForestAncestor H)      (x y : X) (_ : x = head H x) (R : ℝ) (_ : 0 ≤ R)      (_ : ‖x + y‖ ≤ R) (_ : ‖x - y‖ ≤ R),      ‖tail H y‖ ≤ 2 * Real.sqrt (R ^ 2 - ‖x‖ ^ 2)) ∧    (∀ (H : Finset InfiniteVertex) (_ : Ancestral List.IsPrefix H)      (x y : BInfinite) (_ : x.val = head H x.val) (R : ℝ) (_ : 0 ≤ R)      (_ : ‖x + y‖ ≤ R) (_ : ‖x - y‖ ≤ R),      ‖tail H y.val‖ ≤ 2 * Real.sqrt (R ^ 2 - ‖x‖ ^ 2)) :=  sorryend SegmentLensesendend 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
Show the TeX
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
Show the TeX
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
Show the TeX
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
Show the TeX
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
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
35 local vocabulary declarations
Lines of Lean, without comments or blank lines
275
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

Midpoint-lens tail estimate

Family 331: Reflexive midpoint convexity and diamond distortion

Candidate pairing

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}.

Midpoint lenses in segment spaces

TeX label thm:main-tail, build/sections/introduction.tex, line 54

The Lean statement says

theorem midpoint_lens_main :    (∀ (H : Finset ForestVertex) (_ : Ancestral ForestAncestor H)      (x y : X) (_ : x = head H x) (R : ℝ) (_ : 0 ≤ R)      (_ : ‖x + y‖ ≤ R) (_ : ‖x - y‖ ≤ R),      ‖tail H y‖ ≤ 2 * Real.sqrt (R ^ 2 - ‖x‖ ^ 2)) ∧    (∀ (H : Finset InfiniteVertex) (_ : Ancestral List.IsPrefix H)      (x y : BInfinite) (_ : x.val = head H x.val) (R : ℝ) (_ : 0 ≤ R)      (_ : ‖x + y‖ ≤ R) (_ : ‖x - y‖ ≤ R),      ‖tail H y.val‖ ≤ 2 * Real.sqrt (R ^ 2 - ‖x‖ ^ 2)) :=  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.

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

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

Compare this statement with the paperShare a Comparator run reportSuggest a change