Lean statement

Tree-potential midpoint convexity and renorming obstruction

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.BoundedTreePotentials.TreeCalculus.main_counterexample

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

See the closest candidates 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 TreeCalculus.main_counterexample (r : Bool) (k : QuadraticKind) :    (CompleteSpace (TestCompletion (treeTestFamily r k)) ∧      TopologicalSpace.SeparableSpace (TestCompletion (treeTestFamily r k)) ∧      ¬Module.Finite ℝ (TestCompletion (treeTestFamily r k))) ∧    (∀ t : ℝ, 0 < t → t < 1 →      Real.sqrt (1 + t ^ 2 / 4) - 1 ≤        averagedModulusReal (TestCompletion (treeTestFamily r k)) t) ∧    (∀ (Y : Type u) [NormedAddCommGroup Y] [NormedSpace ℝ Y],      (TestCompletion (treeTestFamily r k) ≃L[ℝ] Y) → ¬IsAUCReal Y) := by  sorry

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

Definitions needed to read it

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

structure OAI.BoundedTreePotentials.TestFamily line 17

structure TestFamily (I : Type*) where  carrier : Set (I → ℝ)  zero_mem : (0 : I → ℝ) ∈ carrier  neg_mem : ∀ f ∈ carrier, -f ∈ carrier  coeff_bound : ∀ f ∈ carrier, ∀ i, |f i| ≤ 1  singleton_mem : ∀ i, (fun j => if j = i then (1 : ℝ) else 0) ∈ carrier

definition OAI.BoundedTreePotentials.TestVector line 77
```
def TestVector (_ : TestFamily I) := I →₀ ℝ
```

abbreviation OAI.BoundedTreePotentials.TestCompletion line 129

abbrev TestCompletion (K : TestFamily I) := UniformSpace.Completion (TestVector K)

abbreviation OAI.BoundedTreePotentials.TreeCalculus.Node line 135
```
abbrev Node := List ℕ
```

definition OAI.BoundedTreePotentials.TreeCalculus.potential line 137

def potential (f : Node → ℝ) (s : Node) : ℝ := (s.inits.map f).sum

inductive type OAI.BoundedTreePotentials.TreeCalculus.QuadraticKind line 139

inductive QuadraticKind where  | sibling | antichain | global  deriving DecidableEq

definition OAI.BoundedTreePotentials.TreeCalculus.QuadraticGroup line 143

def QuadraticGroup : QuadraticKind → Finset Node → Prop  | .sibling, B => ∃ r, ∀ s ∈ B, ∃ j, s = r ++ [j]  | .antichain, B => (B : Set Node).Pairwise (fun s t => ¬s <+: t ∧ ¬t <+: s)  | .global, _ => True

definition OAI.BoundedTreePotentials.TreeCalculus.QuadraticBudget line 148

def QuadraticBudget (k : QuadraticKind) (f : Node → ℝ) : Prop :=  ∀ B : Finset Node, QuadraticGroup k B → ∑ s ∈ B, f s ^ 2 ≤ 1

definition OAI.BoundedTreePotentials.TreeCalculus.CoefficientSupport line 151

def CoefficientSupport : QuadraticKind → (Node → ℝ) → Prop  | .sibling, _ => True  | .antichain, f => (Function.support f).Finite  | .global, f => Memℓp f 2

structure OAI.BoundedTreePotentials.TreeCalculus.IsTreeTest line 156

structure IsTreeTest (includeRoot : Bool) (k : QuadraticKind) (f : Node → ℝ) : Prop where  root_zero : includeRoot = false → f [] = 0  potential_bound : ∀ s, |potential f s| ≤ 1  quadratic_budget : QuadraticBudget k f  coefficient_support : CoefficientSupport k f

abbreviation OAI.BoundedTreePotentials.TreeCalculus.TreeCoordinate line 162

abbrev TreeCoordinate (includeRoot : Bool) := {s : Node // includeRoot = true ∨ s ≠ []}

definition OAI.BoundedTreePotentials.TreeCalculus.liftCoefficients line 164

def liftCoefficients (r : Bool) (f : TreeCoordinate r → ℝ) (s : Node) : ℝ :=  if h : r = true ∨ s ≠ [] then f ⟨s, h⟩ else 0

definition OAI.BoundedTreePotentials.TreeCalculus.treeTestFamily line 167

def treeTestFamily (r : Bool) (k : QuadraticKind) : TestFamily (TreeCoordinate r) := by

Show the remaining 145 lines

  have potential_root (f : Node → ℝ) : potential f [] = f [] := by simp [potential]  have potential_child (f : Node → ℝ) (s : Node) (j : ℕ) :      potential f (s ++ [j]) = potential f s + f (s ++ [j]) := by    simp [potential, List.inits_append]  have potential_smul (a : ℝ) (f : Node → ℝ) (s : Node) :      potential (a • f) s = a * potential f s := by    induction s using List.reverseRecOn with    | nil => simp [potential_root]    | append_singleton s j ih => simp only [potential_child, ih, Pi.smul_apply, smul_eq_mul]; ring  have quadratic_singleton (k : QuadraticKind) {s : Node} (hs : s ≠ []) :      QuadraticGroup k {s} := by    cases k with    | sibling =>      cases s using List.reverseRecOn with      | nil => exact (hs rfl).elim      | append_singleton s j =>        refine ⟨s, ?_⟩        intro t ht        have ht' : t = s ++ [j] := by simpa using ht        exact ⟨j, ht'⟩    | antichain => simp [QuadraticGroup]    | global => trivial  have test_coefficient_bound {r : Bool} {k : QuadraticKind} {f : Node → ℝ}      (hf : IsTreeTest r k f) (s : Node) : |f s| ≤ 1 := by    by_cases hs : s = []    · subst s; simpa only [potential_root] using hf.potential_bound []    · have h := hf.quadratic_budget {s} (quadratic_singleton k hs)      rw [Finset.sum_singleton] at h      exact (sq_le_one_iff_abs_le_one (f s)).mp h  have quadraticBudget_dominate {k : QuadraticKind} {f g : Node → ℝ}      (hf : QuadraticBudget k f) (h : ∀ s, |g s| ≤ |f s|) : QuadraticBudget k g := by    intro B hB    apply le_trans _ (hf B hB)    apply Finset.sum_le_sum    intro s hs    exact (sq_le_sq).mpr (h s)  have coefficientSupport_dominate {k : QuadraticKind} {f g : Node → ℝ}      (hf : CoefficientSupport k f) (h : ∀ s, |g s| ≤ |f s|) : CoefficientSupport k g := by    cases k with    | sibling => trivial    | antichain =>      apply hf.subset      intro s hs      change g s ≠ 0 at hs      change f s ≠ 0      intro hzero      have hh := h s      rw [hzero, abs_zero] at hh      exact hs (abs_nonpos_iff.mp hh)    | global =>      exact (hf.norm).mono (by intro s; simpa only [Real.norm_eq_abs] using h s)  have zero_isTreeTest (r : Bool) (k : QuadraticKind) : IsTreeTest r k 0 := by    constructor    · simp    · intro s      change |(s.inits.map fun _ => (0 : ℝ)).sum| ≤ 1      simp    · intro B hB; simp    · cases k with      | sibling => trivial      | antichain => simp [CoefficientSupport]      | global => exact zero_memℓp  have neg_isTreeTest {r : Bool} {k : QuadraticKind} {f : Node → ℝ}      (hf : IsTreeTest r k f) : IsTreeTest r k (-f) := by    constructor    · intro hr; simpa using congrArg Neg.neg (hf.root_zero hr)    · intro s      have he : -f = (-1 : ℝ) • f := by ext s; simp      rw [he, potential_smul, neg_one_mul, abs_neg]      exact hf.potential_bound s    · exact quadraticBudget_dominate hf.quadratic_budget (by intro s; simp)    · exact coefficientSupport_dominate hf.coefficient_support (by intro s; simp)  have inits_nodup (s : Node) : s.inits.Nodup := by    induction s using List.reverseRecOn with    | nil => simp    | append_singleton s j ih =>      have he : (s ++ [j]).inits = s.inits ++ [s ++ [j]] := by simp [List.inits_append]      rw [he, List.nodup_append]      refine ⟨ih, by simp, ?_⟩      intro a ha b hb hab      have hb' : b = s ++ [j] := by simpa using hb      have hp := ((List.mem_inits a s).mp ha).length_le      rw [hab, hb', List.length_append, List.length_singleton] at hp      exact Nat.not_succ_le_self _ hp  have potential_single (i s : Node) :      potential (Pi.single i (1 : ℝ)) s = if i <+: s then 1 else 0 := by    rw [potential, ← List.sum_toFinset _ (inits_nodup s)]    simp only [Pi.single_apply]    by_cases h : i <+: s    · have hm : i ∈ s.inits.toFinset := List.mem_toFinset.mpr ((List.mem_inits i s).mpr h)      simp [hm, h]    · have hm : i ∉ s.inits.toFinset := fun hm => h ((List.mem_inits i s).mp (List.mem_toFinset.mp hm))      simp [hm, h]  have single_isTreeTest (r : Bool) (k : QuadraticKind) {i : Node}      (hi : r = true ∨ i ≠ []) : IsTreeTest r k (Pi.single i (1 : ℝ)) := by    constructor    · intro hr      have hn : [] ≠ i := by        rcases hi with hi | hi        · simp_all        · exact Ne.symm hi      exact Pi.single_eq_of_ne hn _    · intro s      rw [potential_single]      split_ifs <;> norm_num    · intro B hB      simp only [Pi.single_apply, ite_pow, one_pow, zero_pow (by decide : (2 : ℕ) ≠ 0)]      by_cases h : i ∈ B <;> simp [h]    · cases k with      | sibling => trivial      | antichain =>        change (Function.support (Pi.single i (1 : ℝ))).Finite        exact (Set.finite_singleton i).subset (by intro s hs; simpa [Function.mem_support, Pi.single_apply] using hs)      | global => exact (lp.single (E := fun _ : Node => ℝ) 2 i (1 : ℝ)).property  have liftCoefficients_coord {r : Bool} (f : TreeCoordinate r → ℝ) (s : TreeCoordinate r) :      liftCoefficients r f s.val = f s := by simp [liftCoefficients, s.property]  have liftCoefficients_zero (r : Bool) : liftCoefficients r 0 = 0 := by    ext s; simp [liftCoefficients]  have liftCoefficients_neg (r : Bool) (f : TreeCoordinate r → ℝ) :      liftCoefficients r (-f) = -liftCoefficients r f := by    ext s    by_cases h : r = true ∨ s ≠ [] <;> simp [liftCoefficients, h]  exact {    carrier := {f | IsTreeTest r k (liftCoefficients r f)}    zero_mem := by      change IsTreeTest r k (liftCoefficients r 0)      rw [liftCoefficients_zero]      exact zero_isTreeTest r k    neg_mem := by      intro f hf      change IsTreeTest r k (liftCoefficients r (-f))      rw [liftCoefficients_neg]      exact neg_isTreeTest hf    coeff_bound := by      intro f hf i      simpa only [liftCoefficients_coord] using test_coefficient_bound hf i.val    singleton_mem := by      rintro ⟨i, hi⟩      change IsTreeTest r k (liftCoefficients r _)      convert single_isTreeTest r k hi using 1      ext s      by_cases h : r = true ∨ s ≠ []      · simp [liftCoefficients, h, Pi.single_apply, Subtype.mk.injEq]      · have hn : s ≠ i := by rintro rfl; exact h hi        simp [liftCoefficients, h, hn] }

structure OAI.BoundedTreePotentials.ClosedFiniteCodim line 320

structure ClosedFiniteCodim where  space : Submodule ℝ E  closed : IsClosed (space : Set E)  finiteCodim : Module.Finite ℝ (E ⧸ space)

definition OAI.BoundedTreePotentials.averagedModulusReal line 325

def averagedModulusReal (t : ℝ) : ℝ :=  ⨅ x : {x : E // ‖x‖=1},⨆ F : ClosedFiniteCodim E,    ⨅ y : {y : E // y ∈ F.space ∧ 1≤‖y‖},      (‖x.val+t • y.val‖+‖x.val-t • y.val‖)/((2 : ℕ) : ℝ)-1

definition OAI.BoundedTreePotentials.oneSidedModulusReal line 330

def oneSidedModulusReal (t : ℝ) : ℝ :=  ⨅ x : {x : E // ‖x‖=1},⨆ F : ClosedFiniteCodim E,    ⨅ y : {y : E // y ∈ F.space ∧ ‖y‖=1},‖x.val+t • y.val‖-1

definition OAI.BoundedTreePotentials.IsAUCReal line 334

def IsAUCReal : Prop := ∀ t : ℝ, 0 < t → 0 < oneSidedModulusReal E t

No confident match with a paper statement

No paper theorem is paired with this Lean statement.

Why none: no paper statement matches the Lean text clearly enough to pair, so the closest candidates are listed below.

Statements in Midpoint convexity from bounded tree potentials and path costs come first because the release’s name for this Lean statement shares words with that paper’s title.

Other possible paper matches, closest wording first

Theorem 2.1 of Midpoint convexity from bounded tree potentials and path costs. Words of the Lean statement, its names split into words, that also appear in the theorem: 2/4, separable, auc, completed, real, spaces and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: banach and infinite.
Proposition 5.1 of Midpoint convexity from bounded tree potentials and path costs. Words of the Lean statement, its names split into words, that also appear in the theorem: global, linear, pairing, test, complete, coordinate and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: cost.
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: separable, root, pairing, coordinate, vector, one and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: path, uniformly, asymptotically and convex.
Theorem 4.1 of Midpoint convexity from bounded tree potentials and path costs. Words of the Lean statement, its names split into words, that also appear in the theorem: auc, tests, real, space, norm and sqrt. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: banach.
Proposition 7.1 of Midpoint convexity from bounded tree potentials and path costs. Words of the Lean statement, its names split into words, that also appear in the theorem: node, support, spaces, norm, finite and sqrt. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: banach.

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 253 s; peak memory 6.3 GiB).
Reported 11 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

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

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

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

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

Do family 331’s Lean statements express the advertised result?

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

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

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

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The paper constructs Banach spaces from bounded tree potentials and path costs to separate averaged midpoint convexity from asymptotic uniform convexity. For the specified tree-potential completions, the formalization proves completeness, separability, infinite dimension, and an averaged midpoint modulus of at least $\sqrt{1+t^2/4}-1$ for $0<t<1$. No space linearly isomorphic to one of these completions is asymptotically uniformly convex.”
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

354 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: 148 to its own declarations, 139 to Mathlib’s documentation. 21 declared names also link to the same declaration in OpenAI’s library. 2 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_counterexample 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 17 of the file’s 27 other declarations; with the theorem itself they span 199 lines. 5 of them are named in the statement itself.

Names in the namespace OAI.BoundedTreePotentials.TreeCalculus are listed without it.

structure OAI.BoundedTreePotentials.TestFamilyline 17library
def OAI.BoundedTreePotentials.TestVectorline 77library
abbrev OAI.BoundedTreePotentials.TestCompletionline 129in the statementlibrary
abbrev Nodeline 135library
def potentialline 137library
inductive QuadraticKindline 139in the statementlibrary
def QuadraticGroupline 143library
def QuadraticBudgetline 148library
def CoefficientSupportline 151library
structure IsTreeTestline 156library
abbrev TreeCoordinateline 162library
def liftCoefficientsline 164library
def treeTestFamilyline 167in the statementlibrary
structure OAI.BoundedTreePotentials.ClosedFiniteCodimline 320library
def OAI.BoundedTreePotentials.averagedModulusRealline 325in the statementlibrary
def OAI.BoundedTreePotentials.oneSidedModulusRealline 330library
def OAI.BoundedTreePotentials.IsAUCRealline 334in the statementlibrary
theorem main_counterexampleline 339selected theoremlibrary

Other Lean rules that may affect the statement 7

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 CoeSort (TestFamily I) (Type _) line 24
instance (K : TestFamily I) : Nonempty K.carrier line 25
instance (K : TestFamily I) : AddCommGroup (TestVector K) line 79
instance (K : TestFamily I) : Module ℝ (TestVector K) line 80
instance (K : TestFamily I) : Norm (TestVector K) line 81
instance (K : TestFamily I) : NormedAddCommGroup (TestVector K) line 83
instance (K : TestFamily I) : NormedSpace ℝ (TestVector K) line 126

3 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 finitePairing line 27
def testNorm line 30
def testSeminorm line 33

import Mathlibnamespace OAInoncomputable sectionuniverse uopen Setopen scoped ENNReal Classicalnamespace BoundedTreePotentialssection TestNormsvariable {I : Type*}structure TestFamily (I : Type*) where  carrier : Set (I → ℝ)  zero_mem : (0 : I → ℝ) ∈ carrier  neg_mem : ∀ f ∈ carrier, -f ∈ carrier  coeff_bound : ∀ f ∈ carrier, ∀ i, |f i| ≤ 1  singleton_mem : ∀ i, (fun j => if j = i then (1 : ℝ) else 0) ∈ carrierinstance : CoeSort (TestFamily I) (Type _) := ⟨fun K => K.carrier⟩instance (K : TestFamily I) : Nonempty K.carrier := ⟨⟨0, K.zero_mem⟩⟩def finitePairing (f : I → ℝ) : (I →₀ ℝ) →ₗ[ℝ] ℝ :=  Finsupp.linearCombination ℝ fdef testNorm (K : TestFamily I) (x : I →₀ ℝ) : ℝ :=  ⨆ f : K, |finitePairing f.val x|def testSeminorm (K : TestFamily I) : Seminorm ℝ (I →₀ ℝ) := by  have finitePairing_apply (f : I → ℝ) (x : I →₀ ℝ) :      finitePairing f x = ∑ i ∈ x.support, x i * f i := rfl  have finitePairing_abs_bound (K : TestFamily I) (f : K) (x : I →₀ ℝ) :      |finitePairing f.val x| ≤ ∑ i ∈ x.support, |x i| := by    rw [finitePairing_apply]    calc      |∑ i ∈ x.support, x i * f.val i| ≤ ∑ i ∈ x.support, |x i * f.val i| :=        Finset.abs_sum_le_sum_abs _ _      _ ≤ ∑ i ∈ x.support, |x i| := by        apply Finset.sum_le_sum        intro i hi        rw [abs_mul]        exact mul_le_of_le_one_right (abs_nonneg _) (K.coeff_bound _ f.property i)  have testNorm_bddAbove (K : TestFamily I) (x : I →₀ ℝ) :      BddAbove (range fun f : K => |finitePairing f.val x|) :=    ⟨∑ i ∈ x.support, |x i|, by rintro _ ⟨f, rfl⟩; exact finitePairing_abs_bound K f x⟩  have finitePairing_le_testNorm (K : TestFamily I) (f : K) (x : I →₀ ℝ) :      |finitePairing f.val x| ≤ testNorm K x := le_ciSup (testNorm_bddAbove K x) f  have testNorm_nonneg (K : TestFamily I) (x : I →₀ ℝ) : 0 ≤ testNorm K x :=    (abs_nonneg _).trans (finitePairing_le_testNorm K ⟨0, K.zero_mem⟩ x)  have testNorm_le_l1 (K : TestFamily I) (x : I →₀ ℝ) :      testNorm K x ≤ ∑ i ∈ x.support, |x i| := by    apply ciSup_le    exact fun f => finitePairing_abs_bound K f x  have testNorm_zero (K : TestFamily I) : testNorm K 0 = 0 := by    apply le_antisymm    · simpa using testNorm_le_l1 K 0    · exact testNorm_nonneg K 0  have testNorm_add_le (K : TestFamily I) (x y : I →₀ ℝ) :      testNorm K (x + y) ≤ testNorm K x + testNorm K y := by    apply ciSup_le    intro f    rw [map_add]    exact (abs_add_le _ _).trans (add_le_add      (finitePairing_le_testNorm K f x) (finitePairing_le_testNorm K f y))  have testNorm_smul_le (K : TestFamily I) (a : ℝ) (x : I →₀ ℝ) :      testNorm K (a • x) ≤ ‖a‖ * testNorm K x := by    apply ciSup_le    intro f    rw [map_smul, smul_eq_mul, abs_mul, Real.norm_eq_abs]    exact mul_le_mul_of_nonneg_left (finitePairing_le_testNorm K f x) (abs_nonneg a)  exact Seminorm.ofSMulLE (testNorm K) (testNorm_zero K) (testNorm_add_le K) (testNorm_smul_le K)def TestVector (_ : TestFamily I) := I →₀ ℝinstance (K : TestFamily I) : AddCommGroup (TestVector K) := inferInstanceAs (AddCommGroup (I →₀ ℝ))instance (K : TestFamily I) : Module ℝ (TestVector K) := inferInstanceAs (Module ℝ (I →₀ ℝ))instance (K : TestFamily I) : Norm (TestVector K) := ⟨testNorm K⟩instance (K : TestFamily I) : NormedAddCommGroup (TestVector K) := by  have finitePairing_apply (f : I → ℝ) (x : I →₀ ℝ) :      finitePairing f x = ∑ i ∈ x.support, x i * f i := rfl  have finitePairing_abs_bound (K : TestFamily I) (f : K) (x : I →₀ ℝ) :      |finitePairing f.val x| ≤ ∑ i ∈ x.support, |x i| := by    rw [finitePairing_apply]    calc      |∑ i ∈ x.support, x i * f.val i| ≤ ∑ i ∈ x.support, |x i * f.val i| :=        Finset.abs_sum_le_sum_abs _ _      _ ≤ ∑ i ∈ x.support, |x i| := by        apply Finset.sum_le_sum        intro i hi        rw [abs_mul]        exact mul_le_of_le_one_right (abs_nonneg _) (K.coeff_bound _ f.property i)  have testNorm_bddAbove (K : TestFamily I) (x : I →₀ ℝ) :      BddAbove (range fun f : K => |finitePairing f.val x|) :=    ⟨∑ i ∈ x.support, |x i|, by rintro _ ⟨f, rfl⟩; exact finitePairing_abs_bound K f x⟩  have finitePairing_le_testNorm (K : TestFamily I) (f : K) (x : I →₀ ℝ) :      |finitePairing f.val x| ≤ testNorm K x := le_ciSup (testNorm_bddAbove K x) f  have finitePairing_single_test (x : I →₀ ℝ) (i : I) :      finitePairing (fun j => if j = i then (1 : ℝ) else 0) x = x i := by    classical    rw [finitePairing_apply]    simp only [mul_ite, mul_one, mul_zero]    by_cases hi : i ∈ x.support    · simp [hi]    · simp [hi, Finsupp.notMem_support_iff.mp hi]  have coordinate_le_testNorm (K : TestFamily I) (x : I →₀ ℝ) (i : I) :      |x i| ≤ testNorm K x := by    convert finitePairing_le_testNorm K ⟨_, K.singleton_mem i⟩ x using 1    rw [finitePairing_single_test]  exact NormedAddCommGroup.ofCore (𝕜 := ℝ) {    norm_nonneg := apply_nonneg (testSeminorm K)    norm_smul := map_smul_eq_mul (testSeminorm K)    norm_triangle := map_add_le_add (testSeminorm K)    norm_eq_zero_iff := fun x => ⟨by      intro h      apply Finsupp.ext      intro i      exact abs_nonpos_iff.mp (h ▸ coordinate_le_testNorm K x i), by      rintro rfl      exact map_zero (testSeminorm K)⟩ }instance (K : TestFamily I) : NormedSpace ℝ (TestVector K) where  norm_smul_le a x := le_of_eq (map_smul_eq_mul (testSeminorm K) a x)abbrev TestCompletion (K : TestFamily I) := UniformSpace.Completion (TestVector K)end TestNormsnamespace TreeCalculusabbrev Node := List ℕdef potential (f : Node → ℝ) (s : Node) : ℝ := (s.inits.map f).suminductive QuadraticKind where  | sibling | antichain | global  deriving DecidableEqdef QuadraticGroup : QuadraticKind → Finset Node → Prop  | .sibling, B => ∃ r, ∀ s ∈ B, ∃ j, s = r ++ [j]  | .antichain, B => (B : Set Node).Pairwise (fun s t => ¬s <+: t ∧ ¬t <+: s)  | .global, _ => Truedef QuadraticBudget (k : QuadraticKind) (f : Node → ℝ) : Prop :=  ∀ B : Finset Node, QuadraticGroup k B → ∑ s ∈ B, f s ^ 2 ≤ 1def CoefficientSupport : QuadraticKind → (Node → ℝ) → Prop  | .sibling, _ => True  | .antichain, f => (Function.support f).Finite  | .global, f => Memℓp f 2structure IsTreeTest (includeRoot : Bool) (k : QuadraticKind) (f : Node → ℝ) : Prop where  root_zero : includeRoot = false → f [] = 0  potential_bound : ∀ s, |potential f s| ≤ 1  quadratic_budget : QuadraticBudget k f  coefficient_support : CoefficientSupport k fabbrev TreeCoordinate (includeRoot : Bool) := {s : Node // includeRoot = true ∨ s ≠ []}def liftCoefficients (r : Bool) (f : TreeCoordinate r → ℝ) (s : Node) : ℝ :=  if h : r = true ∨ s ≠ [] then f ⟨s, h⟩ else 0def treeTestFamily (r : Bool) (k : QuadraticKind) : TestFamily (TreeCoordinate r) := by  have potential_root (f : Node → ℝ) : potential f [] = f [] := by simp [potential]  have potential_child (f : Node → ℝ) (s : Node) (j : ℕ) :      potential f (s ++ [j]) = potential f s + f (s ++ [j]) := by    simp [potential, List.inits_append]  have potential_smul (a : ℝ) (f : Node → ℝ) (s : Node) :      potential (a • f) s = a * potential f s := by    induction s using List.reverseRecOn with    | nil => simp [potential_root]    | append_singleton s j ih => simp only [potential_child, ih, Pi.smul_apply, smul_eq_mul]; ring  have quadratic_singleton (k : QuadraticKind) {s : Node} (hs : s ≠ []) :      QuadraticGroup k {s} := by    cases k with    | sibling =>      cases s using List.reverseRecOn with      | nil => exact (hs rfl).elim      | append_singleton s j =>        refine ⟨s, ?_⟩        intro t ht        have ht' : t = s ++ [j] := by simpa using ht        exact ⟨j, ht'⟩    | antichain => simp [QuadraticGroup]    | global => trivial  have test_coefficient_bound {r : Bool} {k : QuadraticKind} {f : Node → ℝ}      (hf : IsTreeTest r k f) (s : Node) : |f s| ≤ 1 := by    by_cases hs : s = []    · subst s; simpa only [potential_root] using hf.potential_bound []    · have h := hf.quadratic_budget {s} (quadratic_singleton k hs)      rw [Finset.sum_singleton] at h      exact (sq_le_one_iff_abs_le_one (f s)).mp h  have quadraticBudget_dominate {k : QuadraticKind} {f g : Node → ℝ}      (hf : QuadraticBudget k f) (h : ∀ s, |g s| ≤ |f s|) : QuadraticBudget k g := by    intro B hB    apply le_trans _ (hf B hB)    apply Finset.sum_le_sum    intro s hs    exact (sq_le_sq).mpr (h s)  have coefficientSupport_dominate {k : QuadraticKind} {f g : Node → ℝ}      (hf : CoefficientSupport k f) (h : ∀ s, |g s| ≤ |f s|) : CoefficientSupport k g := by    cases k with    | sibling => trivial    | antichain =>      apply hf.subset      intro s hs      change g s ≠ 0 at hs      change f s ≠ 0      intro hzero      have hh := h s      rw [hzero, abs_zero] at hh      exact hs (abs_nonpos_iff.mp hh)    | global =>      exact (hf.norm).mono (by intro s; simpa only [Real.norm_eq_abs] using h s)  have zero_isTreeTest (r : Bool) (k : QuadraticKind) : IsTreeTest r k 0 := by    constructor    · simp    · intro s      change |(s.inits.map fun _ => (0 : ℝ)).sum| ≤ 1      simp    · intro B hB; simp    · cases k with      | sibling => trivial      | antichain => simp [CoefficientSupport]      | global => exact zero_memℓp  have neg_isTreeTest {r : Bool} {k : QuadraticKind} {f : Node → ℝ}      (hf : IsTreeTest r k f) : IsTreeTest r k (-f) := by    constructor    · intro hr; simpa using congrArg Neg.neg (hf.root_zero hr)    · intro s      have he : -f = (-1 : ℝ) • f := by ext s; simp      rw [he, potential_smul, neg_one_mul, abs_neg]      exact hf.potential_bound s    · exact quadraticBudget_dominate hf.quadratic_budget (by intro s; simp)    · exact coefficientSupport_dominate hf.coefficient_support (by intro s; simp)  have inits_nodup (s : Node) : s.inits.Nodup := by    induction s using List.reverseRecOn with    | nil => simp    | append_singleton s j ih =>      have he : (s ++ [j]).inits = s.inits ++ [s ++ [j]] := by simp [List.inits_append]      rw [he, List.nodup_append]      refine ⟨ih, by simp, ?_⟩      intro a ha b hb hab      have hb' : b = s ++ [j] := by simpa using hb      have hp := ((List.mem_inits a s).mp ha).length_le      rw [hab, hb', List.length_append, List.length_singleton] at hp      exact Nat.not_succ_le_self _ hp  have potential_single (i s : Node) :      potential (Pi.single i (1 : ℝ)) s = if i <+: s then 1 else 0 := by    rw [potential, ← List.sum_toFinset _ (inits_nodup s)]    simp only [Pi.single_apply]    by_cases h : i <+: s    · have hm : i ∈ s.inits.toFinset := List.mem_toFinset.mpr ((List.mem_inits i s).mpr h)      simp [hm, h]    · have hm : i ∉ s.inits.toFinset := fun hm => h ((List.mem_inits i s).mp (List.mem_toFinset.mp hm))      simp [hm, h]  have single_isTreeTest (r : Bool) (k : QuadraticKind) {i : Node}      (hi : r = true ∨ i ≠ []) : IsTreeTest r k (Pi.single i (1 : ℝ)) := by    constructor    · intro hr      have hn : [] ≠ i := by        rcases hi with hi | hi        · simp_all        · exact Ne.symm hi      exact Pi.single_eq_of_ne hn _    · intro s      rw [potential_single]      split_ifs <;> norm_num    · intro B hB      simp only [Pi.single_apply, ite_pow, one_pow, zero_pow (by decide : (2 : ℕ) ≠ 0)]      by_cases h : i ∈ B <;> simp [h]    · cases k with      | sibling => trivial      | antichain =>        change (Function.support (Pi.single i (1 : ℝ))).Finite        exact (Set.finite_singleton i).subset (by intro s hs; simpa [Function.mem_support, Pi.single_apply] using hs)      | global => exact (lp.single (E := fun _ : Node => ℝ) 2 i (1 : ℝ)).property  have liftCoefficients_coord {r : Bool} (f : TreeCoordinate r → ℝ) (s : TreeCoordinate r) :      liftCoefficients r f s.val = f s := by simp [liftCoefficients, s.property]  have liftCoefficients_zero (r : Bool) : liftCoefficients r 0 = 0 := by    ext s; simp [liftCoefficients]  have liftCoefficients_neg (r : Bool) (f : TreeCoordinate r → ℝ) :      liftCoefficients r (-f) = -liftCoefficients r f := by    ext s    by_cases h : r = true ∨ s ≠ [] <;> simp [liftCoefficients, h]  exact {    carrier := {f | IsTreeTest r k (liftCoefficients r f)}    zero_mem := by      change IsTreeTest r k (liftCoefficients r 0)      rw [liftCoefficients_zero]      exact zero_isTreeTest r k    neg_mem := by      intro f hf      change IsTreeTest r k (liftCoefficients r (-f))      rw [liftCoefficients_neg]      exact neg_isTreeTest hf    coeff_bound := by      intro f hf i      simpa only [liftCoefficients_coord] using test_coefficient_bound hf i.val    singleton_mem := by      rintro ⟨i, hi⟩      change IsTreeTest r k (liftCoefficients r _)      convert single_isTreeTest r k hi using 1      ext s      by_cases h : r = true ∨ s ≠ []      · simp [liftCoefficients, h, Pi.single_apply, Subtype.mk.injEq]      · have hn : s ≠ i := by rintro rfl; exact h hi        simp [liftCoefficients, h, hn] }end TreeCalculussection AsymptoticModulivariable (E : Type*) [NormedAddCommGroup E] [NormedSpace ℝ E]structure ClosedFiniteCodim where  space : Submodule ℝ E  closed : IsClosed (space : Set E)  finiteCodim : Module.Finite ℝ (E ⧸ space)def averagedModulusReal (t : ℝ) : ℝ :=  ⨅ x : {x : E // ‖x‖=1},⨆ F : ClosedFiniteCodim E,    ⨅ y : {y : E // y ∈ F.space ∧ 1≤‖y‖},      (‖x.val+t • y.val‖+‖x.val-t • y.val‖)/((2 : ℕ) : ℝ)-1def oneSidedModulusReal (t : ℝ) : ℝ :=  ⨅ x : {x : E // ‖x‖=1},⨆ F : ClosedFiniteCodim E,    ⨅ y : {y : E // y ∈ F.space ∧ ‖y‖=1},‖x.val+t • y.val‖-1def IsAUCReal : Prop := ∀ t : ℝ, 0 < t → 0 < oneSidedModulusReal E tend AsymptoticModuliopen TreeCalculus intheorem TreeCalculus.main_counterexample (r : Bool) (k : QuadraticKind) :    (CompleteSpace (TestCompletion (treeTestFamily r k)) ∧      TopologicalSpace.SeparableSpace (TestCompletion (treeTestFamily r k)) ∧      ¬Module.Finite ℝ (TestCompletion (treeTestFamily r k))) ∧    (∀ t : ℝ, 0 < t → t < 1 →      Real.sqrt (1 + t ^ 2 / 4) - 1 ≤        averagedModulusReal (TestCompletion (treeTestFamily r k)) t) ∧    (∀ (Y : Type u) [NormedAddCommGroup Y] [NormedSpace ℝ Y],      (TestCompletion (treeTestFamily r k) ≃L[ℝ] Y) → ¬IsAUCReal Y) := by  sorryend BoundedTreePotentialsendend 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

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

Distortion of countably branching diamonds from midpoint and tree energies

Theorem

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

D^2\ge1+k/4.

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

Show the TeX

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

Exact asymptotic moduli in a Daugavet subspace of L1

Theorem

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

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

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

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

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

Midpoint convexity from bounded tree potentials and path costs

Theorem

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

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

None of these spaces admits an equivalent AUC norm.

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

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

Theorem

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

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

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

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

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

Theorem (Clipping estimate on a head and its complement)

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

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

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

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

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

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

Theorem

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

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

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

Show the TeX

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

Midpoint convexity from two recursive potentials

Theorem

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

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

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

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

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

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

Show the TeX

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

Theorem (Separated midpoint families)

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

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

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

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

Show the TeX

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

Theorem

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

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

The right side is zero when S=0.

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

Show the TeX

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

Configuration and file facts

File: lean/ComparatorChallenges/BoundedTreePotentials.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Analysis/TreePotential/Main.lean
Allowed axioms: propext, Quot.sound, Classical.choiceThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports: Mathlib
Imports from outside Mathlib: None
Local vocabulary: 27 local vocabulary declarations
Lines of Lean, without comments or blank lines: 295
Second proof-verification program (nanoda): Off in OpenAI's configuration.

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