Lean statement

Reflexive tree space, midpoint modulus, and diamond distortion

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.ForestSpace.main_theorem

Lean statement file: ForestSpace.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 main_theorem :    (¬FiniteDimensional ℝ (FullDual Vertex)) ∧    TopologicalSpace.SeparableSpace (FullDual Vertex) ∧    Reflexive (FullDual Vertex) ∧    (∀ (α β : ℝ) (R : EquivalentNorm (FullDual Vertex) α β), ¬IsAUC R.N) ∧    (∀ t : ℝ, 0 < t →      Real.sqrt (1+t^2/12)-1 ≤ averageModulus (FullDual Vertex) norm t) ∧    (∀ (k : ℕ) (f : (Diamond.diamond k).V → FullDual Vertex) (C : ℝ),      Diamond.EmbeddingBound k f C → Real.sqrt (1+(k:ℝ)/12) ≤ C) := 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.

definition OAI.ForestSpace.Cofinite line 19

def Cofinite : Type :=  {F : Submodule ℝ E // IsClosed (F : Set E) ∧ FiniteDimensional ℝ (E ⧸ F)}

definition OAI.ForestSpace.aucValues line 23

def aucValues (N : E → ℝ) (t : ℝ) (x : E) (F : Cofinite E) : Set ℝ :=  {a | ∃ y : E, y ∈ F.val ∧ N y = 1 ∧ a = N (x + t • y) - 1}

definition OAI.ForestSpace.aucInner line 26

noncomputable def aucInner (N : E → ℝ) (t : ℝ) (x : E) (F : Cofinite E) : ℝ :=  sInf (aucValues E N t x F)

definition OAI.ForestSpace.aucCenter line 29

noncomputable def aucCenter (N : E → ℝ) (t : ℝ) (x : E) : ℝ :=  sSup (Set.range (aucInner E N t x))

definition OAI.ForestSpace.aucModulus line 34

noncomputable def aucModulus (N : E → ℝ) (t : ℝ) : ℝ :=  sInf {a | ∃ x : E, N x = 1 ∧ a = aucCenter E N t x}

definition OAI.ForestSpace.averageValues line 38

def averageValues (N : E → ℝ) (t : ℝ) (x : E) (F : Cofinite E) : Set ℝ :=  {a | ∃ y : E, y ∈ F.val ∧ N y = 1 ∧    a = (N (x+t•y) + N (x-t•y))/((2 : ℕ) : ℝ) - 1}

definition OAI.ForestSpace.averageInner line 42

noncomputable def averageInner (N : E → ℝ) (t : ℝ) (x : E) (F : Cofinite E) : ℝ :=  sInf (averageValues E N t x F)

definition OAI.ForestSpace.averageCenter line 45

noncomputable def averageCenter (N : E → ℝ) (t : ℝ) (x : E) : ℝ :=  sSup (Set.range (averageInner E N t x))

definition OAI.ForestSpace.averageModulus line 49

noncomputable def averageModulus (N : E → ℝ) (t : ℝ) : ℝ :=  sInf {a | ∃ x : E, N x = 1 ∧ a = averageCenter E N t x}

structure OAI.ForestSpace.EquivalentNorm line 56

structure EquivalentNorm (E : Type) [NormedAddCommGroup E] [NormedSpace ℝ E]    (α β : ℝ) where  N : Seminorm ℝ E  alpha_pos : 0 < α  alpha_le_beta : α ≤ β  lower : ∀ x : E, α*‖x‖ ≤ N x  upper : ∀ x : E, N x ≤ β*‖x‖

definition OAI.ForestSpace.Reflexive line 65

def Reflexive (E : Type) [NormedAddCommGroup E] [NormedSpace ℝ E] : Prop :=  Function.Surjective (NormedSpace.inclusionInDoubleDual ℝ E)

definition OAI.ForestSpace.IsAUC line 69

def IsAUC {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] (N : E → ℝ) : Prop :=  ∀ t : ℝ, 0 < t → 0 < aucModulus E N t

definition OAI.ForestSpace.TestVector line 226

def TestVector (V : Type) := V →₀ ℝ

abbreviation OAI.ForestSpace.J line 332

abbrev J (V : Type) [PartialOrder V] := UniformSpace.Completion (TestVector V)

abbreviation OAI.ForestSpace.FullDual line 335

abbrev FullDual (V : Type) [PartialOrder V] := StrongDual ℝ (J V)

structure OAI.ForestSpace.Word line 347

structure Word where  letters : List ℕ  deriving DecidableEq

structure OAI.ForestSpace.Vertex line 364

structure Vertex where  height : ℕ  positive : 1 ≤ height  word : Word  bounded : word.letters.length ≤ height  deriving DecidableEq

structure OAI.ForestSpace.Diamond.Network line 388

structure Network where  V : Type  E : Type  left : E → V  right : E → V  distinct : ∀ e, left e ≠ right e

definition OAI.ForestSpace.Diamond.Network.graph line 399

def graph : SimpleGraph A.V where

Show the remaining 10 lines

  Adj u v := ∃ e : A.E, (A.left e = u ∧ A.right e = v) ∨    (A.right e = u ∧ A.left e = v)  symm := ⟨by    rintro u v ⟨e,h | h⟩    · exact ⟨e,Or.inr ⟨h.2,h.1⟩⟩    · exact ⟨e,Or.inl ⟨h.2,h.1⟩⟩⟩  loopless := ⟨by    rintro u ⟨e,h | h⟩    · exact A.distinct e (h.1.trans h.2.symm)    · exact A.distinct e (h.2.trans h.1.symm)⟩

definition OAI.ForestSpace.Diamond.Network.refine line 413

@[reducible] def refine : Network where  V := Sum A.V (A.E × ℕ)  E := A.E × ℕ × Bool  left e := if e.2.2 then Sum.inr (e.1,e.2.1) else Sum.inl (A.left e.1)  right e := if e.2.2 then Sum.inl (A.right e.1) else Sum.inr (e.1,e.2.1)  distinct := by rintro ⟨e,j,b⟩; cases b <;> simp

definition OAI.ForestSpace.Diamond.Network.iterate line 422

@[reducible] def iterate : Network → ℕ → Network  | A,0 => A  | A,n+1 => iterate A.refine n

definition OAI.ForestSpace.Diamond.base line 429

@[reducible] def base : Network where  V := Bool  E := Unit  left _ := false  right _ := true  distinct _ := by decide

definition OAI.ForestSpace.Diamond.diamond line 437

@[reducible] def diamond (k : ℕ) := base.iterate k

definition OAI.ForestSpace.Diamond.distance line 440

def distance (k : ℕ) (u v : (diamond k).V) : ℝ := ((diamond k).graph.dist u v : ℝ)

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: k/12, eq, 2/12, integer, separable, auc, dimensional, reflexive and seven more. Found in the paper’s title, which its main theorem states: uniform, convexity, diamond, asymptotic and midpoint. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: least and infinite.

Statements in Asymptotic midpoint uniform convexity and unbounded diamond distortion in a reflexive tree space come first because the release’s name for this Lean statement 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

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: extension, remain, convention, including, exactly, natural and 16 more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: tree.
Theorem 7.12 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: positivity, maximum, convexity, modulus, codimensional, convex and ten more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: uniformly, asymptotically, banach and infinite.
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: sided, vertex, distinct, path, modulus, root and twelve more.
Proposition 5.2 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: beta, path, alpha, branching, separable, root and ten more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: uniformly and asymptotically.
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, distortion and embedding. Found in the paper’s title, which its main theorem states: countably, branching, diamonds and midpoint.
Proposition 8.4 of Midpoint lenses in segment spaces. Words of the Lean statement, its names split into words, that also appear in the theorem: continuous, natural, integer, positive, dimensional, dual and six more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: uniformly, asymptotically, banach and infinite.

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 102 s; peak memory 6.3 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 constructs a Banach space whose midpoint geometry is asymptotically uniformly convex even though no equivalent norm is asymptotically uniformly convex in the usual one-sided sense. The formalization proves that the full dual of the specified segment-norm completion is infinite-dimensional, separable, and reflexive; its averaged midpoint modulus is at least $\sqrt{1+t^2/12}-1$ for every $t>0$, while every equivalent norm fails asymptotic uniform convexity. It also proves that an embedding of a depth-$k$ countably branching diamond has distortion at least $\sqrt{1+k/12}$.”
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

468 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: 160 to its own declarations, 215 to Mathlib’s documentation. 36 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.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves main_theorem exactly as written here. To compare that statement with the paper, also read the definitions and lemmas listed below. Atlas found them by following names in the source text and lists each after the names it uses. For x.foo, Atlas includes every foo defined here for a type, even when the statement may use only one of them.

It reaches 25 of the file’s 43 other declarations; with the theorem itself they span 89 lines. 9 of them are named in the statement itself.

Names in the namespace OAI.ForestSpace are listed without it.

def Cofiniteline 19library
def aucValuesline 23library
def aucInnerline 26library
def aucCenterline 29library
def aucModulusline 34library
def averageValuesline 38library
def averageInnerline 42library
def averageCenterline 45library
def averageModulusline 49in the statementlibrary
structure EquivalentNormline 56in the statementlibrary
def Reflexiveline 65in the statementlibrary
def IsAUCline 69in the statementlibrary
def TestVectorline 226library
abbrev Jline 332library
abbrev FullDualline 335in the statementlibrary
structure Wordline 347library
structure Vertexline 364in the statementlibrary
structure Diamond.Networkline 388in the statementlibrary
def Diamond.Network.graphline 399library
def Diamond.Network.refineline 413library
def Diamond.Network.iterateline 422library
def Diamond.baseline 429library
def Diamond.diamondline 437in the statementlibrary
def Diamond.distanceline 440library
def Diamond.EmbeddingBoundline 444in the statementlibrary
theorem main_theoremline 453selected theoremlibrary

Other Lean rules that may affect the statement 10

Lean can use instances, notation and attributes without naming them in the statement. These are rules for interpreting or finding mathematical objects. Atlas’s scan of names cannot follow all such uses; read the listed source as well.

instance AddCommGroup (TestVector V) line 232
instance Module ℝ (TestVector V) line 234
instance CoeFun (TestVector V) (fun _ => V → ℝ) line 238
instance Norm (TestVector V) line 242
instance NormedAddCommGroup (TestVector V) line 244
instance NormedSpace ℝ (TestVector V) line 326
instance fullDualAddCommGroup line 341
instance PartialOrder Word line 353
instance PartialOrder Vertex line 373
attribute [local instance] Classical.propDecidable line 12

9 declarations the statement does not reach by name

These definitions and lemmas were not found by following names in the statement. Lean may still use them through rules such as instances or macros.

def IsSegment line 73
def segSum line 78
def TestBound line 81
def testValues line 93
def squareSup line 98
def rho line 102
def l1size line 104
def segmentSeminorm line 109
def toFinsupp line 236

import Mathlibnamespace OAInoncomputable sectionopen scoped BigOperatorsopen Finsetnamespace ForestSpaceattribute [local instance] Classical.propDecidablesection ModulusDefinitionsvariable (E : Type) [NormedAddCommGroup E] [NormedSpace ℝ E]/-- The closed finite-codimensional subspaces. -/def Cofinite : Type :=  {F : Submodule ℝ E // IsClosed (F : Set E) ∧ FiniteDimensional ℝ (E ⧸ F)}/-- The inner one-sided excess values and their infimum. -/def aucValues (N : E → ℝ) (t : ℝ) (x : E) (F : Cofinite E) : Set ℝ :=  {a | ∃ y : E, y ∈ F.val ∧ N y = 1 ∧ a = N (x + t • y) - 1}noncomputable def aucInner (N : E → ℝ) (t : ℝ) (x : E) (F : Cofinite E) : ℝ :=  sInf (aucValues E N t x F)noncomputable def aucCenter (N : E → ℝ) (t : ℝ) (x : E) : ℝ :=  sSup (Set.range (aucInner E N t x))/-- The one-sided asymptotic uniform convexity modulus, with the orderof infimum, supremum and infimum. -/noncomputable def aucModulus (N : E → ℝ) (t : ℝ) : ℝ :=  sInf {a | ∃ x : E, N x = 1 ∧ a = aucCenter E N t x}/-- The inner averaged signed excess; it is not replaced by a maximum. -/def averageValues (N : E → ℝ) (t : ℝ) (x : E) (F : Cofinite E) : Set ℝ :=  {a | ∃ y : E, y ∈ F.val ∧ N y = 1 ∧    a = (N (x+t•y) + N (x-t•y))/((2 : ℕ) : ℝ) - 1}noncomputable def averageInner (N : E → ℝ) (t : ℝ) (x : E) (F : Cofinite E) : ℝ :=  sInf (averageValues E N t x F)noncomputable def averageCenter (N : E → ℝ) (t : ℝ) (x : E) : ℝ :=  sSup (Set.range (averageInner E N t x))/-- The averaged asymptotic midpoint modulus. -/noncomputable def averageModulus (N : E → ℝ) (t : ℝ) : ℝ :=  sInf {a | ∃ x : E, N x = 1 ∧ a = averageCenter E N t x}end ModulusDefinitions/-- An equivalent norm is kept as its exact real seminorm, with explicit positivecomparison constants. The lower inequality makes it a norm. -/structure EquivalentNorm (E : Type) [NormedAddCommGroup E] [NormedSpace ℝ E]    (α β : ℝ) where  N : Seminorm ℝ E  alpha_pos : 0 < α  alpha_le_beta : α ≤ β  lower : ∀ x : E, α*‖x‖ ≤ N x  upper : ∀ x : E, N x ≤ β*‖x‖/-- Canonical (not merely abstract) reflexivity. -/def Reflexive (E : Type) [NormedAddCommGroup E] [NormedSpace ℝ E] : Prop :=  Function.Surjective (NormedSpace.inclusionInDoubleDual ℝ E)/-- AUC requires positivity at every strictly positive parameter. -/def IsAUC {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] (N : E → ℝ) : Prop :=  ∀ t : ℝ, 0 < t → 0 < aucModulus E N t/-- A finite, order-convex chain. The empty chain is a harmless zero test. -/def IsSegment {V : Type} [PartialOrder V] (S : Finset V) : Prop :=  (∀ a ∈ S, ∀ b ∈ S, a ≤ b ∨ b ≤ a) ∧  (∀ a ∈ S, ∀ b ∈ S, ∀ c, a ≤ c → c ≤ b → c ∈ S)/-- The sum of coordinates over a segment. -/def segSum {V : Type} (u : V → ℝ) (S : Finset V) : ℝ := ∑ v ∈ S, u v/-- Squared bound on every disjoint family of segment tests. -/def TestBound {V : Type} [PartialOrder V] (u : V → ℝ) (C : ℝ) : Prop :=  ∀ (ι : Type) (_ : Fintype ι) (S : ι → Finset V),    (∀ i, IsSegment (S i)) → Pairwise (fun i j => Disjoint (S i) (S j)) →    ∑ i, segSum u (S i) ^ 2 ≤ Csection SegmentNormvariable {V : Type} [PartialOrder V]/-- All squared disjoint-segment tests. Quantifying over finite index typesis equivalent to families S₁,...,Sₘ; a zero-sized familyadds only zero to the set of test values. -/def testValues (u : V → ℝ) : Set ℝ :=  {r | ∃ (ι : Type) (_ : Fintype ι) (S : ι → Finset V),    (∀ i, IsSegment (S i)) ∧ Pairwise (fun i j => Disjoint (S i) (S j)) ∧    r = ∑ i, segSum u (S i) ^ 2}noncomputable def squareSup (u : V → ℝ) : ℝ := sSup (testValues u)/-- Exactly the disjoint-segment norm, written as the square root of itssquared testing supremum. -/noncomputable def rho (u : V → ℝ) : ℝ := Real.sqrt (squareSup u)noncomputable def l1size (u : V →₀ ℝ) : ℝ := ∑ v ∈ u.support, |u v|end SegmentNorm/-- The disjoint-segment seminorm on finite coordinate vectors. -/def segmentSeminorm (V : Type) [PartialOrder V] : Seminorm ℝ (V →₀ ℝ) := by  have segSum_zero [PartialOrder V] (S : Finset V) : segSum (0 : V → ℝ) S = 0 := by    simp [segSum]  have segSum_add [PartialOrder V] (u v : V → ℝ) (S : Finset V) :      segSum (u + v) S = segSum u S + segSum v S := by    simp [segSum, Finset.sum_add_distrib]  have segSum_smul [PartialOrder V] (a : ℝ) (u : V → ℝ) (S : Finset V) :      segSum (a • u) S = a * segSum u S := by    simp [segSum, Finset.mul_sum]  have TestBound.nonneg {u : V → ℝ} {C : ℝ} (h : TestBound u C) : 0 ≤ C := by    simpa using h PEmpty inferInstance PEmpty.elim (by simp) (by simp [Pairwise])  have TestBound.zero : TestBound (0 : V → ℝ) 0 := by    intro ι inst S hS hd    simp [segSum]  have TestBound.smul {u : V → ℝ} {C : ℝ} (h : TestBound u C) (a : ℝ) :      TestBound (a • u) (a ^ 2 * C) := by    intro ι inst S hS hd    have hh := mul_le_mul_of_nonneg_left (h ι inst S hS hd) (sq_nonneg a)    simpa only [segSum_smul, mul_pow, Finset.mul_sum] using hh  have TestBound.add {u v : V → ℝ} {A B : ℝ}      (hu : TestBound u (A ^ 2)) (hv : TestBound v (B ^ 2))      (hA : 0 ≤ A) (hB : 0 ≤ B) : TestBound (u + v) ((A + B) ^ 2) := by    intro ι inst S hS hd    have h₁ := hu ι inst S hS hd    have h₂ := hv ι inst S hS hd    have hc := Real.sum_mul_le_sqrt_mul_sqrt (univ : Finset ι)      (fun i => segSum u (S i)) (fun i => segSum v (S i))    have hs₁ : Real.sqrt (∑ i, segSum u (S i) ^ 2) ≤ A :=      (Real.sqrt_le_iff).2 ⟨hA, h₁⟩    have hs₂ : Real.sqrt (∑ i, segSum v (S i) ^ 2) ≤ B :=      (Real.sqrt_le_iff).2 ⟨hB, h₂⟩    have hc' : ∑ i, segSum u (S i) * segSum v (S i) ≤ A * B :=      hc.trans (mul_le_mul hs₁ hs₂ (Real.sqrt_nonneg _) hA)    have he : (∑ i, segSum (u + v) (S i) ^ 2) =        (∑ i, segSum u (S i) ^ 2) + 2 * (∑ i, segSum u (S i) * segSum v (S i)) +        (∑ i, segSum v (S i) ^ 2) := by      simp only [segSum_add, add_sq, Finset.sum_add_distrib, Finset.mul_sum]      congr 1      congr 1      apply Finset.sum_congr rfl      intro i hi      ring    rw [he]    nlinarith  have zero_mem_testValues (u : V → ℝ) : 0 ∈ testValues u := by    refine ⟨PEmpty, inferInstance, PEmpty.elim, by simp, by simp [Pairwise], ?_⟩    simp [segSum]  have TestBound.bddAbove {u : V → ℝ} {C : ℝ} (h : TestBound u C) :      BddAbove (testValues u) := by    refine ⟨C, ?_⟩    rintro r ⟨ι, inst, S, hs, hd, rfl⟩    exact h ι inst S hs hd  have TestBound.squareSup_le {u : V → ℝ} {C : ℝ} (h : TestBound u C) :      squareSup u ≤ C := by    apply csSup_le ⟨0, zero_mem_testValues u⟩    rintro r ⟨ι, inst, S, hs, hd, rfl⟩    exact h ι inst S hs hd  have testBound_squareSup {u : V → ℝ} (hu : ∃ C, TestBound u C) :      TestBound u (squareSup u) := by    intro ι inst S hs hd    exact le_csSup (TestBound.bddAbove hu.choose_spec) ⟨ι, inst, S, hs, hd, rfl⟩  have squareSup_nonneg {u : V → ℝ} (hu : ∃ C, TestBound u C) : 0 ≤ squareSup u :=    TestBound.nonneg (testBound_squareSup hu)  have rho_nonneg (u : V → ℝ) : 0 ≤ rho u := Real.sqrt_nonneg _  have rho_sq {u : V → ℝ} (hu : ∃ C, TestBound u C) : rho u ^ 2 = squareSup u :=    Real.sq_sqrt (squareSup_nonneg hu)  have testBound_rho_sq {u : V → ℝ} (hu : ∃ C, TestBound u C) : TestBound u (rho u ^ 2) := by    rw [rho_sq hu]    exact testBound_squareSup hu  have rho_le_of_testBound {u : V → ℝ} {C : ℝ} (hC : 0 ≤ C)      (hu : TestBound u (C ^ 2)) : rho u ≤ C :=    Real.sqrt_le_iff.mpr ⟨hC, TestBound.squareSup_le hu⟩  have rho_add_le {u v : V → ℝ} (hu : ∃ C, TestBound u C) (hv : ∃ C, TestBound v C) :      rho (u + v) ≤ rho u + rho v :=    rho_le_of_testBound (add_nonneg (rho_nonneg u) (rho_nonneg v))      (TestBound.add (testBound_rho_sq hu) (testBound_rho_sq hv) (rho_nonneg u) (rho_nonneg v))  have rho_smul_le {u : V → ℝ} (hu : ∃ C, TestBound u C) (a : ℝ) :      rho (a • u) ≤ |a| * rho u := by    apply rho_le_of_testBound (mul_nonneg (abs_nonneg a) (rho_nonneg u))    have hh := TestBound.smul (testBound_rho_sq hu) a    simpa only [mul_pow, sq_abs] using hh  have rho_zero : rho (0 : V → ℝ) = 0 := by    apply le_antisymm _ (rho_nonneg _)    apply rho_le_of_testBound (C := 0) (by norm_num)    simpa using (TestBound.zero : TestBound (0 : V → ℝ) 0)  have l1size_nonneg [PartialOrder V] (u : V →₀ ℝ) : 0 ≤ l1size u :=    sum_nonneg (fun vertex _ => abs_nonneg (u vertex))  have finset_abs_sum_le_l1size [PartialOrder V] (u : V →₀ ℝ) (S : Finset V) :      ∑ v ∈ S, |u v| ≤ l1size u := by    have he : (∑ v ∈ S ∪ u.support, |u v|) = l1size u := by      symm      apply sum_subset subset_union_right      intro v hv hn      rw [Finsupp.notMem_support_iff.mp hn, abs_zero]    rw [← he]    exact sum_le_sum_of_subset_of_nonneg subset_union_left (fun v hv hn => abs_nonneg _)  have testBound_l1size (u : V →₀ ℝ) : TestBound u (l1size u ^ 2) := by    intro ι inst S hS hd    have habs : (∑ i, |segSum u (S i)|) ≤ l1size u := by      calc        (∑ i, |segSum u (S i)|) ≤ ∑ i, ∑ v ∈ S i, |u v| :=          sum_le_sum fun i hi => abs_sum_le_sum_abs _ _        _ = ∑ v ∈ univ.biUnion S, |u v| :=          (sum_biUnion (by intro i hi j hj hij; exact hd hij)).symm        _ ≤ l1size u := finset_abs_sum_le_l1size u _    have hsq := sum_sq_le_sq_sum_of_nonneg      (s := (univ : Finset ι)) (f := fun i => |segSum u (S i)|) (fun i hi => abs_nonneg _)    simp only [sq_abs] at hsq    have hnonneg : 0 ≤ ∑ i, |segSum u (S i)| := sum_nonneg (fun i hi => abs_nonneg _)    nlinarith [l1size_nonneg u]  have finsupp_has_testBound (u : V →₀ ℝ) : ∃ C, TestBound u C := ⟨_, testBound_l1size u⟩  exact Seminorm.ofSMulLE (fun u : V →₀ ℝ => rho u) rho_zero    (fun u v => rho_add_le (finsupp_has_testBound u) (finsupp_has_testBound v))    (fun a u => by simpa only [Real.norm_eq_abs, Finsupp.coe_smul] using rho_smul_le (finsupp_has_testBound u) a)/-- Finite coordinate vectors with the segment norm (a type synonym preventsclashes with any other norm on finitely supported functions). -/def TestVector (V : Type) := V →₀ ℝnamespace TestVectorvariable {V : Type}instance : AddCommGroup (TestVector V) := inferInstanceAs (AddCommGroup (V →₀ ℝ))instance : Module ℝ (TestVector V) := inferInstanceAs (Module ℝ (V →₀ ℝ))def toFinsupp (u : TestVector V) : V →₀ ℝ := uinstance : CoeFun (TestVector V) (fun _ => V → ℝ) := ⟨fun u => u.toFinsupp⟩variable [PartialOrder V]instance : Norm (TestVector V) := ⟨fun u => rho u⟩instance : NormedAddCommGroup (TestVector V) := by  have isSegment_singleton (v : V) : IsSegment ({v} : Finset V) := by    constructor    · intro a ha b hb      simp only [mem_singleton] at ha hb      subst a      subst b      exact Or.inl le_rfl    · intro a ha b hb c hac hcb      simp only [mem_singleton] at ha hb ⊢      subst a      subst b      exact le_antisymm hcb hac  have TestBound.nonneg {u : V → ℝ} {C : ℝ} (h : TestBound u C) : 0 ≤ C := by    simpa using h PEmpty inferInstance PEmpty.elim (by simp) (by simp [Pairwise])  have TestBound.single {u : V → ℝ} {C : ℝ} (h : TestBound u C)      {S : Finset V} (hS : IsSegment S) : segSum u S ^ 2 ≤ C := by    simpa using h Unit inferInstance (fun _ => S) (fun _ => hS)      (by simp [Pairwise])  have TestBound.bddAbove {u : V → ℝ} {C : ℝ} (h : TestBound u C) :      BddAbove (testValues u) := by    refine ⟨C, ?_⟩    rintro r ⟨ι, inst, S, hs, hd, rfl⟩    exact h ι inst S hs hd  have testBound_squareSup {u : V → ℝ} (hu : ∃ C, TestBound u C) :      TestBound u (squareSup u) := by    intro ι inst S hs hd    exact le_csSup (TestBound.bddAbove hu.choose_spec) ⟨ι, inst, S, hs, hd, rfl⟩  have squareSup_nonneg {u : V → ℝ} (hu : ∃ C, TestBound u C) : 0 ≤ squareSup u :=    TestBound.nonneg (testBound_squareSup hu)  have rho_nonneg (u : V → ℝ) : 0 ≤ rho u := Real.sqrt_nonneg _  have rho_sq {u : V → ℝ} (hu : ∃ C, TestBound u C) : rho u ^ 2 = squareSup u :=    Real.sq_sqrt (squareSup_nonneg hu)  have testBound_rho_sq {u : V → ℝ} (hu : ∃ C, TestBound u C) : TestBound u (rho u ^ 2) := by    rw [rho_sq hu]    exact testBound_squareSup hu  have coordinate_le_rho {u : V → ℝ} (hu : ∃ C, TestBound u C) (v : V) :      |u v| ≤ rho u := by    have hh := TestBound.single (testBound_rho_sq hu) (isSegment_singleton v)    simp only [segSum, sum_singleton] at hh    exact (sq_le_sq₀ (abs_nonneg _) (rho_nonneg _)).mp (by simpa only [sq_abs] using hh)  have l1size_nonneg [PartialOrder V] (u : V →₀ ℝ) : 0 ≤ l1size u :=    sum_nonneg (fun vertex _ => abs_nonneg (u vertex))  have finset_abs_sum_le_l1size [PartialOrder V] (u : V →₀ ℝ) (S : Finset V) :      ∑ v ∈ S, |u v| ≤ l1size u := by    have he : (∑ v ∈ S ∪ u.support, |u v|) = l1size u := by      symm      apply sum_subset subset_union_right      intro v hv hn      rw [Finsupp.notMem_support_iff.mp hn, abs_zero]    rw [← he]    exact sum_le_sum_of_subset_of_nonneg subset_union_left (fun v hv hn => abs_nonneg _)  have testBound_l1size (u : V →₀ ℝ) : TestBound u (l1size u ^ 2) := by    intro ι inst S hS hd    have habs : (∑ i, |segSum u (S i)|) ≤ l1size u := by      calc        (∑ i, |segSum u (S i)|) ≤ ∑ i, ∑ v ∈ S i, |u v| :=          sum_le_sum fun i hi => abs_sum_le_sum_abs _ _        _ = ∑ v ∈ univ.biUnion S, |u v| :=          (sum_biUnion (by intro i hi j hj hij; exact hd hij)).symm        _ ≤ l1size u := finset_abs_sum_le_l1size u _    have hsq := sum_sq_le_sq_sum_of_nonneg      (s := (univ : Finset ι)) (f := fun i => |segSum u (S i)|) (fun i hi => abs_nonneg _)    simp only [sq_abs] at hsq    have hnonneg : 0 ≤ ∑ i, |segSum u (S i)| := sum_nonneg (fun i hi => abs_nonneg _)    nlinarith [l1size_nonneg u]  have finsupp_has_testBound (u : V →₀ ℝ) : ∃ C, TestBound u C := ⟨_, testBound_l1size u⟩  exact NormedAddCommGroup.ofCore (𝕜 := ℝ) {    norm_nonneg u := apply_nonneg (segmentSeminorm V) u.toFinsupp    norm_triangle u v := map_add_le_add (segmentSeminorm V) u.toFinsupp v.toFinsupp    norm_smul a u := map_smul_eq_mul (segmentSeminorm V) a u.toFinsupp    norm_eq_zero_iff u := ⟨by      intro h      apply Finsupp.ext      intro v      have hh := coordinate_le_rho (finsupp_has_testBound u.toFinsupp) v      change rho u.toFinsupp = 0 at h      rw [h] at hh      exact abs_nonpos_iff.mp hh, by      rintro rfl      exact map_zero (segmentSeminorm V)⟩ }instance : NormedSpace ℝ (TestVector V) where  norm_smul_le a u := le_of_eq (map_smul_eq_mul (segmentSeminorm V) a u.toFinsupp)end TestVector/-- J(T), completion of the finite-support disjoint-segment norm. -/abbrev J (V : Type) [PartialOrder V] := UniformSpace.Completion (TestVector V)/-- The full continuous dual, used for the finite-height forest example. -/abbrev FullDual (V : Type) [PartialOrder V] := StrongDual ℝ (J V)section CoordinateCompletionvariable {W : no_index Type} [PartialOrder W]instance fullDualAddCommGroup : AddCommGroup (FullDual W) :=  inferInstanceAs (AddCommGroup (StrongDual ℝ (J W)))end CoordinateCompletion/-- Finite words, ordered by extension, including the empty root. -/structure Word where  letters : List ℕ  deriving DecidableEqnamespace Wordinstance : PartialOrder Word where  le a b := a.letters <+: b.letters  le_refl _ := List.prefix_rfl  le_trans _ _ _ := List.IsPrefix.trans  le_antisymm a b hab hba := by    have h := hab.eq_of_length (Nat.le_antisymm hab.length_le hba.length_le)    cases a; cases b; cases h; rflend Word/-- A disjoint forest: one copy of `Nat^{≤h}` for every `h≥1`. -/structure Vertex where  height : ℕ  positive : 1 ≤ height  word : Word  bounded : word.letters.length ≤ height  deriving DecidableEqnamespace Vertexinstance : PartialOrder Vertex where  le a b := a.height = b.height ∧ a.word ≤ b.word  le_refl _ := ⟨rfl,le_rfl⟩  le_trans a b c hab hbc := ⟨hab.1.trans hbc.1,hab.2.trans hbc.2⟩  le_antisymm a b hab hba := by    obtain ⟨hh, hw⟩ := hab    have he := le_antisymm hw hba.2    cases a; cases b; cases hh; cases he; rflend Vertexnamespace Diamond/-- An oriented edge enumeration; only the associated undirected unit-edgesimple graph is used for distances. -/structure Network where  V : Type  E : Type  left : E → V  right : E → V  distinct : ∀ e, left e ≠ right enamespace Networkvariable (A : Network)def graph : SimpleGraph A.V where  Adj u v := ∃ e : A.E, (A.left e = u ∧ A.right e = v) ∨    (A.right e = u ∧ A.left e = v)  symm := ⟨by    rintro u v ⟨e,h | h⟩    · exact ⟨e,Or.inr ⟨h.2,h.1⟩⟩    · exact ⟨e,Or.inl ⟨h.2,h.1⟩⟩⟩  loopless := ⟨by    rintro u ⟨e,h | h⟩    · exact A.distinct e (h.1.trans h.2.symm)    · exact A.distinct e (h.2.trans h.1.symm)⟩/-- Every edge is replaced by countably many internally disjoint two-edgepaths. All old vertices remain, and each pair (edge,index) is a new midpoint. -/@[reducible] def refine : Network where  V := Sum A.V (A.E × ℕ)  E := A.E × ℕ × Bool  left e := if e.2.2 then Sum.inr (e.1,e.2.1) else Sum.inl (A.left e.1)  right e := if e.2.2 then Sum.inl (A.right e.1) else Sum.inr (e.1,e.2.1)  distinct := by rintro ⟨e,j,b⟩; cases b <;> simp/-- Iterating the very same edge replacement, starting from an arbitrarynetwork. This recursion groups the first replacement before the remaining ones. -/@[reducible] def iterate : Network → ℕ → Network  | A,0 => A  | A,n+1 => iterate A.refine nend Network/-- D₀ is precisely one edge. -/@[reducible] def base : Network where  V := Bool  E := Unit  left _ := false  right _ := true  distinct _ := by decide/-- Countably branching diamonds, all current edges of length one. -/@[reducible] def diamond (k : ℕ) := base.iterate k/-- The shortest-path distance, via Mathlib's actual graph distance. -/def distance (k : ℕ) (u v : (diamond k).V) : ℝ := ((diamond k).graph.dist u v : ℝ)variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E]/-- The distortion convention, including its arbitrary positive scale. -/def EmbeddingBound (k : ℕ) (f : (diamond k).V → E) (C : ℝ) : Prop :=  ∃ s : ℝ, 0 < s ∧ ∀ u v,    s*distance k u v ≤ ‖f u-f v‖ ∧ ‖f u-f v‖ ≤ C*s*distance k u vend Diamond/-- The spaceis the full continuous dual of the specified segment-norm completion, not thecoordinate completion in place of that dual. -/theorem main_theorem :    (¬FiniteDimensional ℝ (FullDual Vertex)) ∧    TopologicalSpace.SeparableSpace (FullDual Vertex) ∧    Reflexive (FullDual Vertex) ∧    (∀ (α β : ℝ) (R : EquivalentNorm (FullDual Vertex) α β), ¬IsAUC R.N) ∧    (∀ t : ℝ, 0 < t →      Real.sqrt (1+t^2/12)-1 ≤ averageModulus (FullDual Vertex) norm t) ∧    (∀ (k : ℕ) (f : (Diamond.diamond k).V → FullDual Vertex) (C : ℝ),      Diamond.EmbeddingBound k f C → Real.sqrt (1+(k:ℝ)/12) ≤ C) := by  sorryend ForestSpaceendend 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

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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

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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

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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

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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 $S=0$.

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