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Lean statement

Recursive-potential spaces and midpoint convexity

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.ComparatorModel.RecursivePotentials.main

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

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

The Lean statement

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

def MainClaim : Prop :=  ∃ h0 : FiniteRootedTree.Heads.{0}, letI := h0;  ∃ h1 : TreeVector.Construction.{0}, letI := h1;  ∃ h2 : ZeroTreeVector.Construction.{0}, letI := h2;  ∃ h3 : AggregationVector.Construction.{0}, letI := h3;  ∃ h4 : AggregationVector.LpConstruction.{0}, letI := h4;  rec_mainStatement ∧  rec_completionStatement ∧  stop_all_radiiStatement ∧  XJoined_separated_familiesStatement ∧  Variable.spaceStatement ∧  Variable.lp_model_exactStatement ∧  Variable.one_sided_modulus_zeroStatement ∧  Variable.no_equivalent_AUCStatement ∧  Variable.sixth_stabilityStatement ∧  Variable.cubic_stabilityStatement ∧  Variable.weak_tailStatementtheorem main : MainClaim := 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.RecursivePotentials.leastPair line 18

    def leastPair (x a b : ℝ) : ℝ × ℝ := (max a (x + b), max b (-x + a))

  • structure OAI.RecursivePotentials.FiniteRootedTree line 20

    structure FiniteRootedTree (ι : Type uIndex) where  root : ι  parent : ι → ι  depth : ι → ℕ  parent_root : parent root = root  depth_root : depth root = 0  parent_depth : ∀ i, i ≠ root → depth (parent i) < depth i

  • definition OAI.RecursivePotentials.FiniteRootedTree.children line 35

    def children (s : ι) : Finset ι :=  Finset.univ.filter fun c => c ≠ T.root ∧ T.parent c = s

  • definition OAI.RecursivePotentials.FiniteRootedTree.maxDepth line 38

    def maxDepth : ℕ := Finset.univ.sup T.depth

  • definition OAI.RecursivePotentials.FiniteRootedTree.fields line 40

    noncomputable def fields (v : ι → ℝ) (s : ι) : ℝ × ℝ :=
    Show the remaining 11 lines
      leastPair (v s)    (Real.sqrt (∑ c : {c // c ∈ T.children s}, (fields v c.1).1 ^ 2))    (Real.sqrt (∑ c : {c // c ∈ T.children s}, (fields v c.1).2 ^ 2))termination_by T.maxDepth - T.depth sdecreasing_by  all_goals    have hc := (Finset.mem_filter.mp c.2).2    have hd := T.parent_depth c.1 hc.1    rw [hc.2] at hd    have hm : T.depth c.1 ≤ T.maxDepth := Finset.le_sup (Finset.mem_univ c.1)    omega

  • definition OAI.RecursivePotentials.FiniteRootedTree.Initial line 53

    def Initial (H : Finset ι) : Prop := T.root ∈ H ∧ ∀ s ∈ H, T.parent s ∈ H

  • definition OAI.RecursivePotentials.FiniteRootedTree.ancestors line 59

    def ancestors (s : ι) : Finset ι :=  if _hs : s = T.root then {s} else insert s (ancestors (T.parent s))termination_by T.depth sdecreasing_by exact T.parent_depth s _hs

  • definition OAI.RecursivePotentials.FiniteRootedTree.ancestorClosure line 64

    def ancestorClosure (S : Finset ι) : Finset ι :=  insert T.root (S.biUnion T.ancestors)

  • definition OAI.RecursivePotentials.FiniteRootedTree.restrict line 67

    def restrict (H : Finset ι) (hH : T.Initial H) : FiniteRootedTree H where  root := ⟨T.root, hH.1⟩  parent s := ⟨T.parent s, hH.2 s.1 s.2⟩  depth s := T.depth s  parent_root := Subtype.ext T.parent_root  depth_root := T.depth_root  parent_depth s hs := T.parent_depth s (fun h => hs (Subtype.ext h))

  • definition OAI.RecursivePotentials.FiniteRootedTree.extend line 75

    noncomputable def extend (H : Finset ι) (f : H → ℝ) : ι →₀ ℝ :=  Finsupp.onFinset H (fun s => if hs : s ∈ H then f ⟨s, hs⟩ else 0)    (fun s hs => by by_contra hh; simp [hh] at hs)

  • definition OAI.RecursivePotentials.FiniteRootedTree.head line 79

    def head (v : ι →₀ ℝ) : Finset ι := T.ancestorClosure v.support

  • definition OAI.RecursivePotentials.FiniteRootedTree.projectHead line 81

    def projectHead (_T : FiniteRootedTree ι) (H : Finset ι) (v : ι →₀ ℝ) : ι →₀ ℝ := v.filter (· ∈ H)

  • structure OAI.RecursivePotentials.CofiniteSubspace line 85

    structure CofiniteSubspace (E : Type uSpace) [NormedAddCommGroup E] [NormedSpace ℝ E] where  space : Submodule ℝ E  closed : IsClosed (space : Set E)  finiteQuotient : Module.Finite ℝ (E ⧸ space)

  • definition OAI.RecursivePotentials.averagedGain line 94

    noncomputable def averagedGain (x : E) (t : ℝ) (y : E) : ℝ := (‖x + t • y‖ + ‖x - t • y‖) / 2 - 1

  • definition OAI.RecursivePotentials.averagedInner line 96

    noncomputable def averagedInner (x : E) (t : ℝ) (F : CofiniteSubspace E) : ℝ :=  sInf {r : ℝ | ∃ y ∈ F.space, 1 ≤ ‖y‖ ∧ r = averagedGain x t y}

  • definition OAI.RecursivePotentials.averagedModulus line 99

    noncomputable def averagedModulus (E : Type uSpace) [NormedAddCommGroup E] [NormedSpace ℝ E]    (t : ℝ) : ℝ :=  sInf {r : ℝ | ∃ x : E, ‖x‖ = 1 ∧ r = sSup (Set.range (averagedInner x t))}

  • definition OAI.RecursivePotentials.sequenceTree line 105

    def sequenceTree : FiniteRootedTree (List ℕ) where  root := []  parent := List.dropLast  depth := List.length  parent_root := rfl  depth_root := rfl  parent_depth s hs := by    rw [List.length_dropLast]    have h := List.length_pos_iff_ne_nil.mpr hs    omega

  • abbreviation OAI.RecursivePotentials.ComponentNode line 116

    abbrev ComponentNode (h : ℕ+) := {s : List ℕ // s.length ≤ h.val}

  • abbreviation OAI.RecursivePotentials.JoinedNode line 118

    abbrev JoinedNode := Option (Σ h : ℕ+, ComponentNode h)

  • definition OAI.RecursivePotentials.joinedTree line 120

    def joinedTree : FiniteRootedTree JoinedNode where
    Show the remaining 23 lines
      root := none  parent    | none => none    | some ⟨h, s⟩ => if s.1 = [] then none else        some ⟨h, ⟨s.1.dropLast, by rw [List.length_dropLast]; have := s.2; omega⟩⟩  depth    | none => 0    | some ⟨_, s⟩ => s.1.length + 1  parent_root := rfl  depth_root := rfl  parent_depth s hs := by    cases s with    | none => exact (hs rfl).elim    | some s =>      obtain ⟨h, s⟩ := s      dsimp      by_cases he : s.1 = []      · simp [he]      · rw [ite_eq_right he]        dsimp        rw [List.length_dropLast]        have hpos := List.length_pos_iff_ne_nil.mpr he        omega

  • definition OAI.RecursivePotentials.WeaklyNull line 147

    def WeaklyNull {E : Type uSpace} [NormedAddCommGroup E] [NormedSpace ℝ E] (d : ℕ → E) : Prop :=  ∀ f : E →L[ℝ] ℝ, Tendsto (fun k => f (d k)) atTop (𝓝 0)

  • structure OAI.RecursivePotentials.EquivalentNorm line 150

    structure EquivalentNorm (E : Type uSpace) [NormedAddCommGroup E] [NormedSpace ℝ E] where  N : Seminorm ℝ E  lower : ℝ  upper : ℝ  lower_pos : 0 < lower  lower_le_upper : lower ≤ upper  lower_bound : ∀ x, lower * ‖x‖ ≤ N x  upper_bound : ∀ x, N x ≤ upper * ‖x‖

  • definition OAI.RecursivePotentials.EquivalentNorm.inner line 163

    noncomputable def inner (x : E) (t : ℝ) (F : CofiniteSubspace E) : ℝ :=  sInf {r : ℝ | ∃ y ∈ F.space, A.N y = 1 ∧ r = A.N (x + t • y) - 1}

  • definition OAI.RecursivePotentials.EquivalentNorm.modulus line 166

    noncomputable def modulus (t : ℝ) : ℝ :=  sInf {r : ℝ | ∃ x : E, A.N x = 1 ∧ r = sSup (Set.range (A.inner x t))}

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

Possible paper matches for the parts of the Lean statement

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

Statements in Midpoint convexity from two recursive potentials come first because the release’s name for this Lean statement shares words with that paper’s title.

The Lean statement joins several assertions with “and”. Atlas selected possible matches to two paper results from their shared wording, listed below. This does not establish that their mathematical meanings agree.

Theorem 2.1 of Midpoint convexity from two recursive potentials

Words of the Lean statement, its names split into words, that also appear in the theorem: 3/128, sigma, initial, auc, equivalent, vector, supported, one and two more. Found in the paper’s title, which its main theorem states: recursive and 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 6.3 of Midpoint convexity from two recursive potentials

Words of the Lean statement, its names split into words, that also appear in the theorem: min, gamma, 1/2, separated, families and one. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: midpoint.

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
$\eta(\varepsilon)=\min\{1/2,\gamma(2,\varepsilon/16)\}$.

Other possible paper matches, closest wording first

  • Proposition 7.1 of Midpoint convexity from two recursive potentials. Words of the Lean statement, its names split into words, that also appear in the theorem: total, separable, dense, complete, projection, reflexive and eight more.
  • Proposition 2.2 of Midpoint convexity from two recursive potentials. Words of the Lean statement, its names split into words, that also appear in the theorem: fields, extend, completion, formulas, dense, root and eight more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: banach.
  • Proposition 9.1 of Midpoint lenses in segment spaces. Words of the Lean statement, its names split into words, that also appear in the theorem: using, sided, fact, children, child, modulus and 13 more.
  • Theorem 7.4 of Midpoint convexity from two recursive potentials. Words of the Lean statement, its names split into words, that also appear in the theorem: zero, total, component, support, one and finite. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: ancestral.
  • Corollary 5.2 of Midpoint convexity from two recursive potentials. Words of the Lean statement, its names split into words, that also appear in the theorem: averaged, sigma and modulus. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: positive.
  • 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: children, separable, root, weakly, null, pairing and seven more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: uniformly, asymptotically and convex.

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 176 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 three Banach spaces from recursive tree potentials: a root-sum space, a zero-root space, and a joined space. The formalization proves that all three are infinite-dimensional and have no equivalent asymptotically uniformly convex norm. The root-sum and joined spaces have averaged midpoint modulus at least $t^3/128$ for $0<t<1$, and the root-sum modulus is positive for every $t>0$. The zero-root estimate retains its finite ancestral support and tail hypotheses, and the zero-root and joined spaces are reflexive.”

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

964 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: 550 to its own declarations, 290 to Mathlib’s documentation. 142 declared names also link to the same declaration in OpenAI’s library. 9 identifiers stay plain because they resolve to no single declaration: variables Lean binds automatically, notation, or names the scan of the pinned sources did not find. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

import Mathlibnamespace OAInoncomputable section/-!The final claim constructs the spaces and states all eleven original results.`Analysis.RecursivePotentials.Challenge.main_iff_original` verifies statement preservation.-/universe uIndex uSpace uOther uDomain uTarget uFiberopen scoped BigOperatorsnamespace RecursivePotentialsdef leastPair (x a b : ℝ) : ℝ × ℝ := (max a (x + b), max b (-x + a))structure FiniteRootedTree (ι : Type uIndex) where  root : ι  parent : ι → ι  depth : ι → ℕ  parent_root : parent root = root  depth_root : depth root = 0  parent_depth : ∀ i, i ≠ root → depth (parent i) < depth inamespace FiniteRootedTreesectionvariable {ι : Type uIndex} [indexFinite : Fintype ι] [indexDecidable : DecidableEq ι]    (T : FiniteRootedTree ι)def children (s : ι) : Finset ι :=  Finset.univ.filter fun c => c ≠ T.root ∧ T.parent c = sdef maxDepth : ℕ := Finset.univ.sup T.depthnoncomputable def fields (v : ι → ℝ) (s : ι) : ℝ × ℝ :=  leastPair (v s)    (Real.sqrt (∑ c : {c // c ∈ T.children s}, (fields v c.1).1 ^ 2))    (Real.sqrt (∑ c : {c // c ∈ T.children s}, (fields v c.1).2 ^ 2))termination_by T.maxDepth - T.depth sdecreasing_by  all_goals    have hc := (Finset.mem_filter.mp c.2).2    have hd := T.parent_depth c.1 hc.1    rw [hc.2] at hd    have hm : T.depth c.1 ≤ T.maxDepth := Finset.le_sup (Finset.mem_univ c.1)    omegadef Initial (H : Finset ι) : Prop := T.root ∈ H ∧ ∀ s ∈ H, T.parent s ∈ Hendvariable {ι : Type uIndex} [indexDecidable : DecidableEq ι] (T : FiniteRootedTree ι)def ancestors (s : ι) : Finset ι :=  if _hs : s = T.root then {s} else insert s (ancestors (T.parent s))termination_by T.depth sdecreasing_by exact T.parent_depth s _hsdef ancestorClosure (S : Finset ι) : Finset ι :=  insert T.root (S.biUnion T.ancestors)def restrict (H : Finset ι) (hH : T.Initial H) : FiniteRootedTree H where  root := ⟨T.root, hH.1⟩  parent s := ⟨T.parent s, hH.2 s.1 s.2⟩  depth s := T.depth s  parent_root := Subtype.ext T.parent_root  depth_root := T.depth_root  parent_depth s hs := T.parent_depth s (fun h => hs (Subtype.ext h))noncomputable def extend (H : Finset ι) (f : H → ℝ) : ι →₀ ℝ :=  Finsupp.onFinset H (fun s => if hs : s ∈ H then f ⟨s, hs⟩ else 0)    (fun s hs => by by_contra hh; simp [hh] at hs)def head (v : ι →₀ ℝ) : Finset ι := T.ancestorClosure v.supportdef projectHead (_T : FiniteRootedTree ι) (H : Finset ι) (v : ι →₀ ℝ) : ι →₀ ℝ := v.filter (· ∈ H)end FiniteRootedTreestructure CofiniteSubspace (E : Type uSpace) [NormedAddCommGroup E] [NormedSpace ℝ E] where  space : Submodule ℝ E  closed : IsClosed (space : Set E)  finiteQuotient : Module.Finite ℝ (E ⧸ space)section Modulivariable {E : Type uSpace} [NormedAddCommGroup E] [NormedSpace ℝ E]noncomputable def averagedGain (x : E) (t : ℝ) (y : E) : ℝ := (‖x + t • y‖ + ‖x - t • y‖) / 2 - 1noncomputable def averagedInner (x : E) (t : ℝ) (F : CofiniteSubspace E) : ℝ :=  sInf {r : ℝ | ∃ y ∈ F.space, 1 ≤ ‖y‖ ∧ r = averagedGain x t y}noncomputable def averagedModulus (E : Type uSpace) [NormedAddCommGroup E] [NormedSpace ℝ E]    (t : ℝ) : ℝ :=  sInf {r : ℝ | ∃ x : E, ‖x‖ = 1 ∧ r = sSup (Set.range (averagedInner x t))}end Modulidef sequenceTree : FiniteRootedTree (List ℕ) where  root := []  parent := List.dropLast  depth := List.length  parent_root := rfl  depth_root := rfl  parent_depth s hs := by    rw [List.length_dropLast]    have h := List.length_pos_iff_ne_nil.mpr hs    omegaabbrev ComponentNode (h : ℕ+) := {s : List ℕ // s.length ≤ h.val}abbrev JoinedNode := Option (Σ h : ℕ+, ComponentNode h)def joinedTree : FiniteRootedTree JoinedNode where  root := none  parent    | none => none    | some ⟨h, s⟩ => if s.1 = [] then none else        some ⟨h, ⟨s.1.dropLast, by rw [List.length_dropLast]; have := s.2; omega⟩⟩  depth    | none => 0    | some ⟨_, s⟩ => s.1.length + 1  parent_root := rfl  depth_root := rfl  parent_depth s hs := by    cases s with    | none => exact (hs rfl).elim    | some s =>      obtain ⟨h, s⟩ := s      dsimp      by_cases he : s.1 = []      · simp [he]      · rw [ite_eq_right he]        dsimp        rw [List.length_dropLast]        have hpos := List.length_pos_iff_ne_nil.mpr he        omegaopen Filter Topologydef WeaklyNull {E : Type uSpace} [NormedAddCommGroup E] [NormedSpace ℝ E] (d : ℕ → E) : Prop :=  ∀ f : E →L[ℝ] ℝ, Tendsto (fun k => f (d k)) atTop (𝓝 0)structure EquivalentNorm (E : Type uSpace) [NormedAddCommGroup E] [NormedSpace ℝ E] where  N : Seminorm ℝ E  lower : ℝ  upper : ℝ  lower_pos : 0 < lower  lower_le_upper : lower ≤ upper  lower_bound : ∀ x, lower * ‖x‖ ≤ N x  upper_bound : ∀ x, N x ≤ upper * ‖x‖namespace EquivalentNormvariable {E : Type uSpace} [NormedAddCommGroup E] [NormedSpace ℝ E] (A : EquivalentNorm E)noncomputable def inner (x : E) (t : ℝ) (F : CofiniteSubspace E) : ℝ :=  sInf {r : ℝ | ∃ y ∈ F.space, A.N y = 1 ∧ r = A.N (x + t • y) - 1}noncomputable def modulus (t : ℝ) : ℝ :=  sInf {r : ℝ | ∃ x : E, A.N x = 1 ∧ r = sSup (Set.range (A.inner x t))}def AUC : Prop := ∀ t : ℝ, 0 < t → 0 < A.modulus tend EquivalentNormopen scoped ENNRealdef IsReflexive (E : Type uSpace) [NormedAddCommGroup E] [NormedSpace ℝ E] : Prop :=  Function.Surjective (NormedSpace.inclusionInDoubleDual ℝ E)namespace SquareSumvariable {ι : Type uIndex} [indexDecidable : DecidableEq ι] (E : ι → Type uFiber)    [∀ i, NormedAddCommGroup (E i)] [fiberNormedSpace : ∀ i, NormedSpace ℝ (E i)]abbrev Space : Type _ := ↥(lp E 2)end SquareSumopen UniformSpacedef completionEquivOfDense {E : Type uSpace} {F : Type uOther} [NormedAddCommGroup E]    [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] (f : E →L[ℝ] F)    (K : NNReal) (hK : ∀ x, ‖x‖ ≤ K * ‖f x‖) (hdense : DenseRange f) : Completion E ≃L[ℝ] F := by  let g := f.fromCompletion  have hbound : ∀ x : Completion E, ‖x‖ ≤ K * ‖g x‖ := by    intro x    induction x using Completion.induction_on with    | hp => exact isClosed_le continuous_norm (continuous_const.mul (g.continuous.norm))    | ih x => simpa only [g, ContinuousLinearMap.fromCompletion_apply_coe, Completion.norm_coe] using hK x  have hanti := g.antilipschitz_of_bound hbound  have hclosed := hanti.isClosed_range g.uniformContinuous  have hsurj : Function.Surjective g := by    intro y    have hh : y ∈ closure (Set.range f) := hdense y    have hsub : Set.range f ⊆ Set.range g := by      rintro _ ⟨x, rfl⟩      exact ⟨(x : Completion E), ContinuousLinearMap.fromCompletion_apply_coe f x⟩    have hm := closure_mono hsub hh    rw [hclosed.closure_eq] at hm    exact hm  exact ContinuousLinearEquiv.ofBijective g (LinearMap.ker_eq_bot.mpr hanti.injective)    (LinearMap.range_eq_top.mpr hsurj)def componentTree (h : ℕ+) : FiniteRootedTree (ComponentNode h) where  root := ⟨[], by simp⟩  parent s := ⟨s.1.dropLast, by rw [List.length_dropLast]; have := s.2; omega⟩  depth s := s.1.length  parent_root := Subtype.ext rfl  depth_root := rfl  parent_depth s hs := by    rw [List.length_dropLast]    have hne : s.1 ≠ [] := fun h => hs (Subtype.ext h)    have hp := List.length_pos_iff_ne_nil.mpr hne    omegadef logarithmicGamma (M β : ℝ) : ℝ :=  let h := β ^ 2 / (2 * M ^ 2)  let c := β * Real.sqrt h / 8  let H := 1 + Real.log (M / c)  let K := 4 * (2 + Real.sqrt 6) * H / h  β * h / 8 * Real.exp (-K)open Filteropen scoped Topologystructure ChildExponent where  val : ℝ  one_le : 1 ≤ valnamespace ChildExponentvariable (p : ChildExponent)def agg {ι : Type uIndex} (s : Finset ι) (f : ι → ℝ) : ℝ :=  (∑ i ∈ s, |f i| ^ p.val) ^ (1 / p.val)end ChildExponentsectionvariable {ι : Type uIndex} [indexFinite : Fintype ι] [indexDecidable : DecidableEq ι]    (T : FiniteRootedTree ι) (p : ChildExponent)def FiniteRootedTree.pfields (v : ι → ℝ) (s : ι) : ℝ × ℝ :=  leastPair (v s)    (p.agg Finset.univ (fun c : {c // c ∈ T.children s} => (pfields v c.1).1))    (p.agg Finset.univ (fun c : {c // c ∈ T.children s} => (pfields v c.1).2))termination_by T.maxDepth - T.depth sdecreasing_by  all_goals    have hc := (Finset.mem_filter.mp c.2).2    have hd := T.parent_depth c.1 hc.1    rw [hc.2] at hd    have hm : T.depth c.1 ≤ T.maxDepth := Finset.le_sup (Finset.mem_univ c.1)    omegaendnamespace VariableExponentdef heightExponent (h : ℕ+) : ℝ := 1 + 1 / (h.val : ℝ)def cubicConstant : ℝ := (6 * Real.exp 1)⁻¹ ^ 3def sixthConstant : ℝ := 2 * Real.exp (-(Real.exp 1 - 1)) * (4 : ℝ)⁻¹ ^ 6end VariableExponentopen Filter Topology TopologicalSpaceend RecursivePotentialsnamespace ComparatorModelopen scoped BigOperatorsnamespace RecursivePotentialssectionnoncomputable abbrev leastPair := @_root_.OAI.RecursivePotentials.leastPairnoncomputable abbrev FiniteRootedTree := @_root_.OAI.RecursivePotentials.FiniteRootedTreenamespace FiniteRootedTreesectionvariable {ι : Type uIndex} [indexFinite : Fintype ι] [indexDecidable : DecidableEq ι]    (T : FiniteRootedTree ι)noncomputable abbrev Initial (T : FiniteRootedTree ι) := _root_.OAI.RecursivePotentials.FiniteRootedTree.Initial Tendvariable {ι : Type uIndex} [indexDecidable : DecidableEq ι] (T : FiniteRootedTree ι)noncomputable abbrev restrict (T : FiniteRootedTree ι) := _root_.OAI.RecursivePotentials.FiniteRootedTree.restrict Tnoncomputable abbrev extend (H : Finset ι) := _root_.OAI.RecursivePotentials.FiniteRootedTree.extend Hnoncomputable abbrev head (T : FiniteRootedTree ι) := _root_.OAI.RecursivePotentials.FiniteRootedTree.head Tclass Heads : Prop where  initial : ∀ {ι : Type uIndex} [DecidableEq ι] (T : FiniteRootedTree ι) (v : ι →₀ ℝ),    T.Initial (T.head v)variable [ComparatorModel.RecursivePotentials.FiniteRootedTree.Heads]noncomputable def P (v : ι →₀ ℝ) : ι →₀ ℝ :=  extend (T.head v) (fun s => ((T.restrict (T.head v) (FiniteRootedTree.Heads.initial T v)).fields    (fun s => v s.1) s).1)noncomputable def Q (v : ι →₀ ℝ) : ι →₀ ℝ :=  extend (T.head v) (fun s => ((T.restrict (T.head v) (FiniteRootedTree.Heads.initial T v)).fields    (fun s => v s.1) s).2)noncomputable def normSum (v : ι →₀ ℝ) : ℝ := T.P v T.root + T.Q v T.rootnoncomputable abbrev projectHead (T : FiniteRootedTree ι) := _root_.OAI.RecursivePotentials.FiniteRootedTree.projectHead Tend FiniteRootedTreevariable [ComparatorModel.RecursivePotentials.FiniteRootedTree.Heads]def TreeVector {ι : Type uIndex} (_T : FiniteRootedTree ι) := ι →₀ ℝnamespace TreeVectorvariable {ι : Type uIndex} [indexDecidable : DecidableEq ι] (T : FiniteRootedTree ι)instance : AddCommGroup (TreeVector T) := inferInstanceAs (AddCommGroup (ι →₀ ℝ))instance : Module ℝ (TreeVector T) := inferInstanceAs (Module ℝ (ι →₀ ℝ))instance : CoeFun (TreeVector T) (fun _ => ι → ℝ) := ⟨fun v => (show ι →₀ ℝ from v)⟩noncomputable instance : Norm (TreeVector T) := ⟨fun v => T.normSum v⟩class Construction : Prop where  core : ∀ {ι : Type uIndex} [DecidableEq ι] (T : FiniteRootedTree ι),    NormedSpace.Core ℝ (TreeVector T)  coordinate_bound : ∀ {ι : Type uIndex} [DecidableEq ι] (T : FiniteRootedTree ι)    (v : ι →₀ ℝ) (s : ι), |v s| ≤ T.normSum v  projection_bound : ∀ {ι : Type uIndex} [DecidableEq ι] (T : FiniteRootedTree ι)    (H : Finset ι), T.Initial H → ∀ v, T.normSum (T.projectHead H v) ≤ T.normSum vvariable [ComparatorModel.RecursivePotentials.TreeVector.Construction]noncomputable instance : NormedAddCommGroup (TreeVector T) :=  NormedAddCommGroup.ofCore (TreeVector.Construction.core T)noncomputable instance : NormedSpace ℝ (TreeVector T) := NormedSpace.ofCore (TreeVector.Construction.core T)def coordinateLinear (s : ι) : TreeVector T →ₗ[ℝ] ℝ where  toFun v := v s  map_add' _v _w := rfl  map_smul' _a _v := rflnoncomputable def coordinate (s : ι) : TreeVector T →L[ℝ] ℝ :=  (coordinateLinear T s).mkContinuous 1 (fun v => by    change ‖v s‖ ≤ 1 * T.normSum v    rw [Real.norm_eq_abs, one_mul]    exact TreeVector.Construction.coordinate_bound T v s)def projectionLinear (H : Finset ι) : TreeVector T →ₗ[ℝ] TreeVector T where  toFun := T.projectHead H  map_add' v w := by    change ι →₀ ℝ at v w    apply Finsupp.ext    intro s    change T.projectHead H (v + w) s = (T.projectHead H v + T.projectHead H w) s    by_cases hs : s ∈ H <;> simp [FiniteRootedTree.projectHead, _root_.OAI.RecursivePotentials.FiniteRootedTree.projectHead, hs]  map_smul' a v := by    change ι →₀ ℝ at v    apply Finsupp.ext    intro s    change T.projectHead H (a • v) s = (a • T.projectHead H v) s    by_cases hs : s ∈ H <;> simp [FiniteRootedTree.projectHead, _root_.OAI.RecursivePotentials.FiniteRootedTree.projectHead, hs]noncomputable def projection (H : Finset ι) (hH : T.Initial H) :    TreeVector T →L[ℝ] TreeVector T :=  (projectionLinear T H).mkContinuous 1 (fun v => by    change T.normSum (T.projectHead H v) ≤ 1 * T.normSum v    rw [one_mul]    exact TreeVector.Construction.projection_bound T H hH v)end TreeVectorvariable [ComparatorModel.RecursivePotentials.TreeVector.Construction]abbrev TreeSpace {ι : Type uIndex} [DecidableEq ι] (T : FiniteRootedTree ι) :=  UniformSpace.Completion (TreeVector T)namespace TreeSpaceopen UniformSpacevariable {ι : Type uIndex} [indexDecidable : DecidableEq ι] (T : FiniteRootedTree ι)def coordinate (s : ι) : TreeSpace T →L[ℝ] ℝ := (TreeVector.coordinate T s).fromCompletiondef P (s : ι) : TreeSpace T → ℝ :=  Completion.extension (fun v : TreeVector T => T.P v s)def Q (s : ι) : TreeSpace T → ℝ :=  Completion.extension (fun v : TreeVector T => T.Q v s)def projection (H : Finset ι) (hH : T.Initial H) : TreeSpace T →L[ℝ] TreeSpace T :=  (TreeVector.projection T H hH).completiondef tail (H : Finset ι) (hH : T.Initial H) : Submodule ℝ (TreeSpace T) :=  (projection T H hH).kerend TreeSpacesection Modulivariable {E : Type uSpace} [NormedAddCommGroup E] [NormedSpace ℝ E]noncomputable abbrev averagedModulus := @_root_.OAI.RecursivePotentials.averagedModulusend Modulinoncomputable abbrev sequenceTree := @_root_.OAI.RecursivePotentials.sequenceTreenoncomputable abbrev ComponentNode := @_root_.OAI.RecursivePotentials.ComponentNodenoncomputable abbrev JoinedNode := @_root_.OAI.RecursivePotentials.JoinedNodenoncomputable abbrev joinedTree := @_root_.OAI.RecursivePotentials.joinedTreeabbrev XSigma := TreeSpace sequenceTreeabbrev XJoined := TreeSpace joinedTreedef ZeroTreeVector {ι : Type uIndex} [DecidableEq ι] (T : FiniteRootedTree ι) : Type _ :=  ↥((TreeVector.coordinate T T.root).ker)namespace ZeroTreeVectorvariable {ι : Type uIndex} [indexDecidable : DecidableEq ι] (T : FiniteRootedTree ι)instance : AddCommGroup (ZeroTreeVector T) :=  inferInstanceAs (AddCommGroup (TreeVector.coordinate T T.root).ker)instance : Module ℝ (ZeroTreeVector T) :=  inferInstanceAs (Module ℝ (TreeVector.coordinate T T.root).ker)def toTree : ZeroTreeVector T →ₗ[ℝ] TreeVector T :=  (TreeVector.coordinate T T.root).ker.subtypeinstance : Norm (ZeroTreeVector T) := ⟨fun v => T.P (toTree T v) T.root⟩class Construction : Prop where  core : ∀ {ι : Type uIndex} [DecidableEq ι] (T : FiniteRootedTree ι),    NormedSpace.Core ℝ (ZeroTreeVector T)  half : ∀ {ι : Type uIndex} [DecidableEq ι] (T : FiniteRootedTree ι)    (v : ZeroTreeVector T), ‖v‖ = ‖toTree T v‖ / 2variable [ComparatorModel.RecursivePotentials.ZeroTreeVector.Construction]instance : NormedAddCommGroup (ZeroTreeVector T) := NormedAddCommGroup.ofCore (ZeroTreeVector.Construction.core T)instance : NormedSpace ℝ (ZeroTreeVector T) := NormedSpace.ofCore (ZeroTreeVector.Construction.core T)def inclusion : ZeroTreeVector T →L[ℝ] TreeVector T :=  (toTree T).mkContinuous 2 (fun v => by rw [ZeroTreeVector.Construction.half]; exact le_of_eq (by ring))def projectionLinear (H : Finset ι) (hH : T.Initial H) : ZeroTreeVector T →ₗ[ℝ] ZeroTreeVector T :=  (((TreeVector.projection T H hH).toLinearMap).comp (toTree T)).codRestrict    (TreeVector.coordinate T T.root).ker (fun v => by      change TreeVector.coordinate T T.root (TreeVector.projection T H hH (toTree T v)) = 0      change T.projectHead H (toTree T v) T.root = 0      change (if T.root ∈ H then toTree T v T.root else 0) = 0      rw [ite_eq_left hH.1]      exact v.2)def projection (H : Finset ι) (hH : T.Initial H) : ZeroTreeVector T →L[ℝ] ZeroTreeVector T :=  (projectionLinear T H hH).mkContinuous 1 (fun v => by    rw [one_mul, ZeroTreeVector.Construction.half, ZeroTreeVector.Construction.half]    exact (div_le_div_iff_of_pos_right (by norm_num : (0 : ℝ) < 2)).mpr      (TreeVector.Construction.projection_bound T H hH (toTree T v)))end ZeroTreeVectorvariable [ComparatorModel.RecursivePotentials.ZeroTreeVector.Construction]abbrev ZeroTreeSpace {ι : Type uIndex} [DecidableEq ι] (T : FiniteRootedTree ι) :=  UniformSpace.Completion (ZeroTreeVector T)namespace ZeroTreeSpaceopen UniformSpacevariable {ι : Type uIndex} [indexDecidable : DecidableEq ι] (T : FiniteRootedTree ι)def inclusion : ZeroTreeSpace T →L[ℝ] TreeSpace T := (ZeroTreeVector.inclusion T).completiondef projection (H : Finset ι) (hH : T.Initial H) : ZeroTreeSpace T →L[ℝ] ZeroTreeSpace T :=  (ZeroTreeVector.projection T H hH).completiondef tail (H : Finset ι) (hH : T.Initial H) : Submodule ℝ (ZeroTreeSpace T) :=  (projection T H hH).kerend ZeroTreeSpaceabbrev XZero := ZeroTreeSpace joinedTreesectionopen UniformSpacevariable {ι : Type uIndex} [indexDecidable : DecidableEq ι] (T : FiniteRootedTree ι)def ZeroTreeSpace.coordinate (s : ι) : ZeroTreeSpace T →L[ℝ] ℝ :=  (TreeSpace.coordinate T s).comp (inclusion T)def ZeroTreeSpace.P (s : ι) (x : ZeroTreeSpace T) : ℝ := TreeSpace.P T s (inclusion T x)def ZeroTreeSpace.Q (s : ι) (x : ZeroTreeSpace T) : ℝ := TreeSpace.Q T s (inclusion T x)endopen Filter Topologynoncomputable abbrev WeaklyNull := @_root_.OAI.RecursivePotentials.WeaklyNullnoncomputable abbrev EquivalentNorm := @_root_.OAI.RecursivePotentials.EquivalentNormnamespace EquivalentNormvariable {E : Type uSpace} [NormedAddCommGroup E] [NormedSpace ℝ E] (A : EquivalentNorm E)noncomputable abbrev modulus (A : EquivalentNorm E) := _root_.OAI.RecursivePotentials.EquivalentNorm.modulus Anoncomputable abbrev AUC (A : EquivalentNorm E) := _root_.OAI.RecursivePotentials.EquivalentNorm.AUC Aend EquivalentNormopen scoped ENNRealnoncomputable abbrev IsReflexive := @_root_.OAI.RecursivePotentials.IsReflexivenamespace SquareSumvariable {ι : Type uIndex} [indexDecidable : DecidableEq ι] (E : ι → Type uFiber)    [∀ i, NormedAddCommGroup (E i)] [fiberNormedSpace : ∀ i, NormedSpace ℝ (E i)]noncomputable abbrev Space := @_root_.OAI.RecursivePotentials.SquareSum.Spaceend SquareSumopen UniformSpacenoncomputable abbrev completionEquivOfDense := @_root_.OAI.RecursivePotentials.completionEquivOfDensenoncomputable abbrev componentTree := @_root_.OAI.RecursivePotentials.componentTreenoncomputable abbrev logarithmicGamma := @_root_.OAI.RecursivePotentials.logarithmicGammaopen Filteropen scoped Topologydef XJoined_separated_familiesStatement : Prop := ∀ {ε : ℝ} (_hε : 0 < ε),    ∃ η : ℝ, 0 < η ∧ η < 1 ∧      (ε ≤ 2 → η = min (1/2) (logarithmicGamma 2 (ε/16))) ∧      ∀ (x : XJoined) (z : ℕ → XJoined),        (∀ n, ‖x+z n‖ ≤ 1) → (∀ n, ‖x-z n‖ ≤ 1) →        (∀ i j, i ≠ j → ε ≤ ‖z i-z j‖) → ‖x‖ ≤ 1-ηnoncomputable abbrev ChildExponent := @_root_.OAI.RecursivePotentials.ChildExponentnamespace VariableExponentnoncomputable abbrev heightExponent := @_root_.OAI.RecursivePotentials.VariableExponent.heightExponentnoncomputable abbrev cubicConstant := @_root_.OAI.RecursivePotentials.VariableExponent.cubicConstantnoncomputable abbrev sixthConstant := @_root_.OAI.RecursivePotentials.VariableExponent.sixthConstantend VariableExponentnamespace VariableTreeopen FiniteRootedTreevariable {ι : Type uIndex} [indexDecidable : DecidableEq ι] (T : FiniteRootedTree ι)    (e : ChildExponent)noncomputable def P (v : ι →₀ ℝ) : ι →₀ ℝ :=  extend (T.head v) (fun s => ((T.restrict (T.head v) (FiniteRootedTree.Heads.initial T v)).pfields e    (fun s => v s.1) s).1)noncomputable def Q (v : ι →₀ ℝ) : ι →₀ ℝ :=  extend (T.head v) (fun s => ((T.restrict (T.head v) (FiniteRootedTree.Heads.initial T v)).pfields e    (fun s => v s.1) s).2)noncomputable def normSum (v : ι →₀ ℝ) : ℝ := P T e v T.root + Q T e v T.rootend VariableTreedef AggregationVector {ι : Type uIndex} (_T : FiniteRootedTree ι) (_e : ChildExponent) := ι →₀ ℝnamespace AggregationVectorvariable {ι : Type uIndex} [indexDecidable : DecidableEq ι] (T : FiniteRootedTree ι)    (e : ChildExponent)instance : AddCommGroup (AggregationVector T e) := inferInstanceAs (AddCommGroup (ι →₀ ℝ))instance : Module ℝ (AggregationVector T e) := inferInstanceAs (Module ℝ (ι →₀ ℝ))instance : CoeFun (AggregationVector T e) (fun _ => ι → ℝ) := ⟨fun v => (show ι →₀ ℝ from v)⟩noncomputable instance : Norm (AggregationVector T e) := ⟨fun v => VariableTree.normSum T e v⟩class Construction : Prop where  core : ∀ {ι : Type uIndex} [DecidableEq ι] (T : FiniteRootedTree ι) (e : ChildExponent),    NormedSpace.Core ℝ (AggregationVector T e)  coordinate_bound : ∀ {ι : Type uIndex} [DecidableEq ι] (T : FiniteRootedTree ι)    (e : ChildExponent) (v : ι →₀ ℝ) (s : ι), |v s| ≤ VariableTree.normSum T e vvariable [ComparatorModel.RecursivePotentials.AggregationVector.Construction]noncomputable instance : NormedAddCommGroup (AggregationVector T e) :=  NormedAddCommGroup.ofCore (AggregationVector.Construction.core T e)noncomputable instance : NormedSpace ℝ (AggregationVector T e) := NormedSpace.ofCore (AggregationVector.Construction.core T e)def coordinateLinear (s : ι) : AggregationVector T e →ₗ[ℝ] ℝ where  toFun v := v s  map_add' _v _w := rfl  map_smul' _a _v := rflnoncomputable def coordinate (s : ι) : AggregationVector T e →L[ℝ] ℝ :=  (coordinateLinear T e s).mkContinuous 1 (fun v => by    change ‖v s‖ ≤ 1 * VariableTree.normSum T e v    rw [Real.norm_eq_abs, one_mul]    exact AggregationVector.Construction.coordinate_bound T e v s)end AggregationVectorvariable [ComparatorModel.RecursivePotentials.AggregationVector.Construction]abbrev AggregationSpace {ι : Type uIndex} [DecidableEq ι] (T : FiniteRootedTree ι)    (e : ChildExponent) :=  UniformSpace.Completion (AggregationVector T e)namespace AggregationSpaceopen UniformSpacevariable {ι : Type uIndex} [indexDecidable : DecidableEq ι] (T : FiniteRootedTree ι)    (e : ChildExponent)def coordinate (s : ι) :    AggregationSpace T e →L[ℝ] ℝ := (AggregationVector.coordinate T e s).fromCompletiondef P (s : ι) : AggregationSpace T e → ℝ :=  Completion.extension (fun v : AggregationVector T e => VariableTree.P T e v s)def Q (s : ι) : AggregationSpace T e → ℝ :=  Completion.extension (fun v : AggregationVector T e => VariableTree.Q T e v s)end AggregationSpacesectionvariable {ι : Type uIndex} [indexDecidable : DecidableEq ι] (T : FiniteRootedTree ι)    (e : ChildExponent)def AggregationVector.basisVector (s : ι) : AggregationVector T e := Finsupp.single s 1endopen Filter Topology TopologicalSpacenamespace ChildExponentinstance (e : ChildExponent) : Fact (1 ≤ ENNReal.ofReal e.val) :=  ⟨by simpa using ENNReal.ofReal_le_ofReal e.one_le⟩end ChildExponentsectionopen scoped ENNRealvariable {ι : Type uIndex} [indexDecidable : DecidableEq ι] (T : FiniteRootedTree ι)    (e : ChildExponent)abbrev AggregationVector.CoordinateLp (ι : Type uIndex) (e : ChildExponent) := ↥(lp (fun _ : ι => ℝ) (ENNReal.ofReal e.val))def AggregationVector.toLpLinear : AggregationVector T e →ₗ[ℝ] CoordinateLp ι e :=  Finsupp.lsum ℝ (fun s => lp.lsingle (𝕜 := ℝ) (E := fun _ : ι => ℝ) (ENNReal.ofReal e.val) s)class AggregationVector.LpConstruction : Prop where  bounds : ∀ {ι : Type uIndex} [DecidableEq ι] (T : FiniteRootedTree ι)    (e : ChildExponent), 1 < e.val →    (∀ c, c ≠ T.root → T.depth c = T.depth (T.parent c) + 1) →    ∀ h : ℕ, (∀ s, T.depth s ≤ h) → ∀ v : AggregationVector T e,    ‖v‖ ≤ (2*(h+1:ℝ)^(1-1/e.val))*‖toLpLinear T e v‖ ∧      ‖toLpLinear T e v‖ ≤ (h+1:ℝ)^(1/e.val)*‖v‖  dense : ∀ {ι : Type uIndex} [DecidableEq ι] (T : FiniteRootedTree ι)    (e : ChildExponent), DenseRange (toLpLinear T e)variable [ComparatorModel.RecursivePotentials.AggregationVector.LpConstruction]def AggregationVector.toLp (hp : 1 < e.val)    (hstep : ∀ c, c ≠ T.root → T.depth c = T.depth (T.parent c) + 1)    (h : ℕ) (hh : ∀ s, T.depth s ≤ h) : AggregationVector T e →L[ℝ] CoordinateLp ι e :=  (toLpLinear T e).mkContinuous ((h+1:ℝ)^(1/e.val))    (fun v => (LpConstruction.bounds T e hp hstep h hh v).2)endsectionopen AggregationVectorvariable {ι : Type uIndex} [indexDecidable : DecidableEq ι] (T : FiniteRootedTree ι)    (e : ChildExponent)variable [ComparatorModel.RecursivePotentials.AggregationVector.LpConstruction]def AggregationSpace.lpEquiv (hp : 1 < e.val)    (hstep : ∀ c, c ≠ T.root → T.depth c = T.depth (T.parent c) + 1)    (h : ℕ) (hh : ∀ s, T.depth s ≤ h) : AggregationSpace T e ≃L[ℝ] CoordinateLp ι e :=  completionEquivOfDense (AggregationVector.toLp T e hp hstep h hh)    ⟨2*(h+1:ℝ)^(1-1/e.val),by positivity⟩    (fun v => (AggregationVector.LpConstruction.bounds T e hp hstep h hh v).1)    (AggregationVector.LpConstruction.dense T e)endnamespace Variablesectionopen UniformSpace VariableExponent Filter Topology TopologicalSpacevariable [ComparatorModel.RecursivePotentials.AggregationVector.LpConstruction]def exponent (h : ℕ+) : ChildExponent := ⟨heightExponent h,(by  dsimp [heightExponent]  have hh : (0 : ℝ) < h.val := by exact_mod_cast h.pos  exact le_add_of_nonneg_right (le_of_lt (one_div_pos.mpr hh)))⟩abbrev Component (h : ℕ+) := AggregationSpace (componentTree h) (exponent h)abbrev Xv := SquareSum.Space Componentabbrev Node := Σ h : ℕ+, ComponentNode hdef componentEquiv (h : ℕ+) :    Component h ≃L[ℝ] AggregationVector.CoordinateLp (ComponentNode h) (exponent h) :=  AggregationSpace.lpEquiv (componentTree h) (exponent h) (by      dsimp [exponent, heightExponent]      have hh : (0 : ℝ) < h.val := by exact_mod_cast h.pos      exact lt_add_of_pos_right _ (one_div_pos.mpr hh))    (by      intro c hc      change c.1.length = c.1.dropLast.length + 1      rw [List.length_dropLast]      have hp := List.length_pos_iff_ne_nil.mpr (show c.1 ≠ [] from fun he => hc (Subtype.ext he))      omega) h.val (fun c => c.2)def componentCoordinate (h : ℕ+) (s : ComponentNode h) : Component h →L[ℝ] ℝ :=  AggregationSpace.coordinate (componentTree h) (exponent h) sdef coordinate (s : Node) : Xv →L[ℝ] ℝ :=  (componentCoordinate s.1 s.2).comp (lp.evalCLM ℝ Component 2 s.1)def basis (s : Node) : Xv := lp.single 2 s.1  ((AggregationVector.basisVector (componentTree s.1) (exponent s.1) s.2 : AggregationVector (componentTree s.1) (exponent s.1)) : Component s.1)def FiniteTotalSupport (x : Xv) : Prop := ∃ S : Finset Node, ∀ s ∉ S, coordinate s x = 0def projection (S : Finset Node) : Xv →L[ℝ] Xv := ∑ s ∈ S, (coordinate s).smulRight (basis s)def AdmissibleHeads (x y : Xv) (A : ∀ h, Finset (ComponentNode h)) : Prop :=  ∀ h, x h ≠ 0 → (componentTree h).Initial (A h) ∧    (∀ s ∉ A h, componentCoordinate h s (x h) = 0) ∧    (∀ s ∈ A h, componentCoordinate h s (y h) = 0)def cubic_stabilityStatement : Prop := ∀ (x y : Xv) (_hx : FiniteTotalSupport x) (A : ∀ h, Finset (ComponentNode h))    (_hA : AdmissibleHeads x y A) {R : ℝ} (_hR : 0 < R) (_hp : ‖x+y‖ ≤ R) (_hm : ‖x-y‖ ≤ R),    ‖x‖^2+cubicConstant/(2*Real.sqrt 2*R)*‖y‖^3 ≤ R^2def sixth_stabilityStatement : Prop := ∀ (x y : Xv) (_hx : FiniteTotalSupport x) (A : ∀ h, Finset (ComponentNode h))    (_hA : AdmissibleHeads x y A),    sixthConstant*‖y‖^6/(‖x+y‖^2+‖x-y‖^2)^2 ≤ ‖x+y‖^2+‖x-y‖^2-2*‖x‖^2open UniformSpace VariableExponent Filter Topology TopologicalSpacedef weakTailConstant : ℝ := cubicConstant/(2*Real.sqrt 2)def theta (ε : ℝ) : ℝ := Real.sqrt (1-weakTailConstant*ε^3)def weak_tailStatement : Prop := ∀ (x : Xv) {ε : ℝ} (_hε : 0 < ε) (_hε1 : ε ≤ 1)    (y : ℕ → Xv) (_hy : WeaklyNull y) (_hyε : ∀ j, ε ≤ ‖y j‖)    (_hp : ∀ j, ‖x+y j‖ ≤ 1) (_hm : ∀ j, ‖x-y j‖ ≤ 1),    ‖x‖ ≤ theta ε ∧ theta ε < 1endopen UniformSpace VariableExponent Filter Topologyvariable [ComparatorModel.RecursivePotentials.AggregationVector.LpConstruction]def one_sided_modulus_zeroStatement : Prop := ∀ (A : EquivalentNorm Xv),    A.modulus (A.lower/A.upper) = 0def no_equivalent_AUCStatement : Prop := ∀ (A : EquivalentNorm Xv),    ¬A.AUCend Variablevariable [ComparatorModel.RecursivePotentials.AggregationVector.LpConstruction]structure RootSumCompletionStatement {ι : Type uIndex} [DecidableEq ι]    (T : FiniteRootedTree ι) : Prop where  complete : CompleteSpace (TreeSpace T)  infinite_dimensional : ¬Module.Finite ℝ (TreeSpace T)  coordinate_continuous : ∀ s, Continuous (TreeSpace.coordinate T s)  coordinate_extends : ∀ s (v : TreeVector T), TreeSpace.coordinate T s (v : TreeSpace T) = v s  fields_continuous : ∀ s, Continuous (TreeSpace.P T s) ∧ Continuous (TreeSpace.Q T s)  fields_extend : ∀ s (v : TreeVector T),    TreeSpace.P T s (v : TreeSpace T) = T.P v s ∧ TreeSpace.Q T s (v : TreeSpace T) = T.Q v s  root_formula : ∀ x : TreeSpace T, ‖x‖ = TreeSpace.P T T.root x + TreeSpace.Q T T.root x  projections : ∀ (H : Finset ι) (hH : T.Initial H),    (∀ x : TreeSpace T, ‖TreeSpace.projection T H hH x‖ ≤ ‖x‖) ∧    (∀ x : TreeSpace T, TreeSpace.projection T H hH (TreeSpace.projection T H hH x) = TreeSpace.projection T H hH x) ∧    (∀ (s : ι) (x : TreeSpace T), TreeSpace.coordinate T s (TreeSpace.projection T H hH x) =      if s ∈ H then TreeSpace.coordinate T s x else 0) ∧    FiniteDimensional ℝ (TreeSpace.projection T H hH).range ∧    IsClosed (TreeSpace.tail T H hH : Set (TreeSpace T)) ∧    Module.Finite ℝ (TreeSpace T ⧸ TreeSpace.tail T H hH) ∧    ∀ x ∈ TreeSpace.tail T H hH, ∀ ε : ℝ, 0 < ε →      ∃ v : TreeVector T, TreeVector.projection T H hH v = 0 ∧ ‖(v : TreeSpace T)-x‖ < εstructure ZeroRootCompletionStatement {ι : Type uIndex} [DecidableEq ι]    (T : FiniteRootedTree ι) : Prop where  complete : CompleteSpace (ZeroTreeSpace T)  infinite_dimensional : ¬Module.Finite ℝ (ZeroTreeSpace T)  coordinate_continuous : ∀ s, Continuous (ZeroTreeSpace.coordinate T s)  coordinate_extends : ∀ s (v : ZeroTreeVector T),    ZeroTreeSpace.coordinate T s (v : ZeroTreeSpace T) = ZeroTreeVector.toTree T v s  fields_continuous : ∀ s, Continuous (ZeroTreeSpace.P T s) ∧ Continuous (ZeroTreeSpace.Q T s)  fields_extend : ∀ s (v : ZeroTreeVector T),    ZeroTreeSpace.P T s (v : ZeroTreeSpace T) = T.P (ZeroTreeVector.toTree T v) s ∧    ZeroTreeSpace.Q T s (v : ZeroTreeSpace T) = T.Q (ZeroTreeVector.toTree T v) s  root_fixed : ∀ x : ZeroTreeSpace T, ZeroTreeSpace.coordinate T T.root x = 0  root_formula : ∀ x : ZeroTreeSpace T, ‖x‖ = ZeroTreeSpace.P T T.root x ∧ ‖x‖ = ZeroTreeSpace.Q T T.root x  projections : ∀ (H : Finset ι) (hH : T.Initial H),    (∀ x : ZeroTreeSpace T, ‖ZeroTreeSpace.projection T H hH x‖ ≤ ‖x‖) ∧    (∀ x : ZeroTreeSpace T, ZeroTreeSpace.projection T H hH (ZeroTreeSpace.projection T H hH x) = ZeroTreeSpace.projection T H hH x) ∧    (∀ (s : ι) (x : ZeroTreeSpace T), ZeroTreeSpace.coordinate T s (ZeroTreeSpace.projection T H hH x) =      if s ∈ H then ZeroTreeSpace.coordinate T s x else 0) ∧    FiniteDimensional ℝ (ZeroTreeSpace.projection T H hH).range ∧    IsClosed (ZeroTreeSpace.tail T H hH : Set (ZeroTreeSpace T)) ∧    Module.Finite ℝ (ZeroTreeSpace T ⧸ ZeroTreeSpace.tail T H hH) ∧    ∀ x ∈ ZeroTreeSpace.tail T H hH, ∀ ε : ℝ, 0 < ε →      ∃ v : ZeroTreeVector T, ZeroTreeVector.projection T H hH v = 0 ∧ ‖(v : ZeroTreeSpace T)-x‖ < εdef rec_completionStatement : Prop := RootSumCompletionStatement sequenceTree ∧    ZeroRootCompletionStatement joinedTree ∧ RootSumCompletionStatement joinedTree ∧    IsReflexive XZero ∧ IsReflexive XJoineddef rec_mainStatement : Prop := (∀ t : ℝ, 0 < t → t < 1 → t^3/128 ≤ averagedModulus XSigma t ∧ t^3/128 ≤ averagedModulus XJoined t) ∧    (∀ (H : Finset JoinedNode), joinedTree.Initial H → ∀ x : XZero,      (∀ s ∉ H, ZeroTreeSpace.coordinate joinedTree s x = 0) → ∀ z : XZero,      (∀ s ∈ H, ZeroTreeSpace.coordinate joinedTree s z = 0) → z ≠ 0 →      (‖x+z‖+‖x-z‖)/2 ≥ ‖x‖+‖z‖^3/(8*(2*‖x‖+‖z‖)^2)) ∧    (∀ A : EquivalentNorm XSigma, ¬A.AUC) ∧    (∀ A : EquivalentNorm XZero, ¬A.AUC) ∧    (∀ A : EquivalentNorm XJoined, ¬A.AUC)def stop_all_radiiStatement : Prop := ∀ {t : ℝ} (_ht : 0 < t),    0 < averagedModulus XSigma topen VariableExponentdef Variable.nu (h : ℕ+) (v : AggregationVector.CoordinateLp (ComponentNode h) (exponent h)) : ℝ :=  AggregationSpace.P (componentTree h) (exponent h) (componentTree h).root ((componentEquiv h).symm v) +  AggregationSpace.Q (componentTree h) (exponent h) (componentTree h).root ((componentEquiv h).symm v)structure Variable.SourceNormStatement (h : ℕ+) : Prop where  nonneg : ∀ v : AggregationVector.CoordinateLp (ComponentNode h) (exponent h), 0 ≤ nu h v  definite : ∀ v : AggregationVector.CoordinateLp (ComponentNode h) (exponent h), nu h v = 0 ↔ v = 0  triangle : ∀ v w : AggregationVector.CoordinateLp (ComponentNode h) (exponent h),    nu h (v+w) ≤ nu h v + nu h w  homogeneous : ∀ (a : ℝ) (v : AggregationVector.CoordinateLp (ComponentNode h) (exponent h)),    nu h (a • v) = |a| * nu h vdef Variable.spaceStatement : Prop := (∀ h : ℕ+, CompleteSpace (Component h) ∧ IsReflexive (Component h) ∧      SeparableSpace (Component h →L[ℝ] ℝ) ∧ SourceNormStatement h ∧      ∀ v : AggregationVector.CoordinateLp (ComponentNode h) (exponent h),        (h.val+1:ℝ)^(-1/heightExponent h)*‖v‖ ≤ nu h v ∧        nu h v ≤ 2*(h.val+1:ℝ)^(1-1/heightExponent h)*‖v‖) ∧    CompleteSpace Xv ∧ IsReflexive Xv ∧ SeparableSpace (Xv →L[ℝ] ℝ) ∧    Dense {x : Xv | FiniteTotalSupport x} ∧    ∀ S : Finset Node, Continuous (projection S) ∧ FiniteDimensional ℝ (projection S).range ∧      (∀ x : Xv, projection S (projection S x) = projection S x) ∧      ∀ (s : Node) (x : Xv), coordinate s (projection S x) = if s ∈ S then coordinate s x else 0def Variable.aggregateChildren (h : ℕ+) (s : ComponentNode h) (f : ComponentNode h → ℝ) : ℝ :=  (∑' c, if c ≠ (componentTree h).root ∧ (componentTree h).parent c = s then    f c ^ heightExponent h else 0) ^ (1/heightExponent h)def Variable.lpPotentials (h : ℕ+) (v : AggregationVector.CoordinateLp (ComponentNode h) (exponent h))    (s : ComponentNode h) : ℝ × ℝ :=  (AggregationSpace.P (componentTree h) (exponent h) s ((componentEquiv h).symm v),    AggregationSpace.Q (componentTree h) (exponent h) s ((componentEquiv h).symm v))def Variable.lp_model_exactStatement : Prop := ∀ (h : ℕ+)    (v : AggregationVector.CoordinateLp (ComponentNode h) (exponent h))    (f : ComponentNode h → ℝ × ℝ)    (_hf : ∀ s, f s = leastPair (v s)      (aggregateChildren h s (fun c => (f c).1))      (aggregateChildren h s (fun c => (f c).2))),    f = lpPotentials h v ∧ nu h v = (f (componentTree h).root).1+(f (componentTree h).root).2enddef MainClaim : Prop :=  ∃ h0 : FiniteRootedTree.Heads.{0}, letI := h0;  ∃ h1 : TreeVector.Construction.{0}, letI := h1;  ∃ h2 : ZeroTreeVector.Construction.{0}, letI := h2;  ∃ h3 : AggregationVector.Construction.{0}, letI := h3;  ∃ h4 : AggregationVector.LpConstruction.{0}, letI := h4;  rec_mainStatement ∧  rec_completionStatement ∧  stop_all_radiiStatement ∧  XJoined_separated_familiesStatement ∧  Variable.spaceStatement ∧  Variable.lp_model_exactStatement ∧  Variable.one_sided_modulus_zeroStatement ∧  Variable.no_equivalent_AUCStatement ∧  Variable.sixth_stabilityStatement ∧  Variable.cubic_stabilityStatement ∧  Variable.weak_tailStatementtheorem main : MainClaim := by  sorryend RecursivePotentialsend ComparatorModelendend OAI
Every extracted statement of the accompanying papers

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

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

Theorem

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

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

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

Section 1, Introduction Read it in the paper TeX source, line 99
Show the TeX
The space $X=J^*$ defined by \eqref{eq:segment-norm} is infinite
dimensional, separable, and reflexive, and has no equivalent AUC norm.
Its given norm satisfies
\[
 \widehat\delta_X(t)\ge\sqrt{1+t^2/12}-1\qquad(t>0).
\]
For every integer $k\ge0$, every embedding of $D_k$ into $X$ has
distortion at least $\sqrt{1+k/12}$.

Midpoint lenses in segment spaces

Theorem

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

\|Q_Hy\|\le2\sqrt{R^2-\|x\|^2}.
Section 1, Introduction Read it in the paper TeX source, line 54
Show the TeX
Let $Y$ be either the finite-height forest dual $X=J^*$ or the
infinite-height coordinate predual $B_\infty$. For every finite
ancestral $H$, every $x,y\in Y$ with $x=P_Hx$, and every $R\ge0$
with $\|x+y\|,\|x-y\|\le R$,
\begin{equation}
 \|Q_Hy\|\le2\sqrt{R^2-\|x\|^2}.
\end{equation}

Distortion of countably branching diamonds from midpoint and tree energies

Theorem

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

D^2\ge1+k/4.
Section 1, Introduction Read it in the paper TeX source, line 53
Show the TeX
For every integer $k\ge0$, every distortion-$D$ embedding of
$\mathcal D_k$ into either $X$ or $B_\infty$ satisfies
\[
 D^2\ge1+k/4.
\]

Exact asymptotic moduli in a Daugavet subspace of L1

Theorem

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

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

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

Section 1, Introduction Read it in the paper TeX source, line 63
Show the TeX
Let $E$ be an infinite-dimensional closed real subspace of
$L^1(\Omega,\mathbb P)$, where $\mathbb P$ is a probability measure.
Suppose its closed unit ball is totally bounded for convergence in
measure and its inherited norm has the Daugavet property.  Then every
$x\in S_E$ and $t>0$ satisfy
\[
 H_E(x,t)=\max\{t/2,t-1\},\qquad
 D_E(x,t)=\max\{0,t-2\}.
\]
Consequently these are also the formulas for
$\widehat\delta_E(t)$ and $\overline\delta_E(t)$.  The space $E$ admits
no equivalent AUC norm.

Midpoint convexity from bounded tree potentials and path costs

Theorem

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

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

None of these spaces admits an equivalent AUC norm.

Section 2, Four potential norms, one pair of inward tests Read it in the paper TeX source, line 63
Show the TeX
Each of the four completed spaces $X_K$ is a separable infinite-dimensional
real Banach space. For $K\in\{K_{\rm s},K_{\rm p},K_{\rm a},K_{\rm g}\}$
and $0<t<1$,
\begin{equation}
 \widehat\delta_{X_K}(t)\ge\sqrt{1+t^2/4}-1.
\end{equation}
None of these spaces admits an equivalent AUC norm.
Theorem

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

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

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

Section 4, A free root and a planar perturbation Read it in the paper TeX source, line 67
Show the TeX
The real Banach space $X_{\mathrm e}$ defined by \eqref{fr:tests} and
\eqref{fr:norm} satisfies
\begin{equation}
 \widehat\delta_{\|\cdot\|_{\mathrm e}}(t)
 \ge\frac{\sqrt{1+t^2/576}-1}{4}\qquad(0<t<1).
 
\end{equation}
It admits no equivalent AUC norm.  More precisely, if
$\alpha\|\cdot\|_{\mathrm e}\le N\le\beta\|\cdot\|_{\mathrm e}$,
where $0<\alpha\le\beta<\infty$, then
$\overline\delta_N(\alpha/(4\beta))=0$.
Theorem (Clipping estimate on a head and its complement)

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

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

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

Section 5, Linear costs of root paths Read it in the paper TeX source, line 287
Show the TeX
If $x,y\in X$, $P_Ax=x$ and $P_Ay=0$ for a finite head $A$, then,
for every $c>0$,
\begin{equation}
 \max\{\|x+y\|,\|x-y\|\}
 \ge\frac{\|x\|+c\|y\|/2}{\sqrt{1+4c^2}}.
 
\end{equation}
In particular the left side is at least
$\sqrt{\|x\|^2+\|y\|^2/16}$.

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

Theorem

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

\begin{aligned}\whdelta_{X_{\mathrm{exp}}}(t) &\ge \frac{7t}{30}c(10/t)>0 &&(t>0),\\ \whdelta_{X_{\mathrm{sq}}}(t) &\ge\frac{\PP\bigl(|G|\ge4\sqrt2(2+\sqrt2)/t\bigr)}{8}>0 &&(t>0),\\ \whdelta_{X_{\mathrm{sq}}}(t) &\ge\frac{\PP\bigl(|G|\ge8\sqrt6/t\bigr)}{4}>0 &&(t>0).\end{aligned}
Section 1, Introduction Read it in the paper TeX source, line 57
Show the TeX
For every multiplier $W$ as above, the Banach space $X_W$ is infinite
dimensional and has no equivalent AUC norm.
For the exponential and Gaussian-square laws, the inherited norms satisfy
\begin{align}
 \whdelta_{X_{\mathrm{exp}}}(t)
 &\ge \frac{7t}{30}c(10/t)>0 &&(t>0),\\
 \whdelta_{X_{\mathrm{sq}}}(t)
 &\ge\frac{\PP\bigl(|G|\ge4\sqrt2(2+\sqrt2)/t\bigr)}{8}>0
 &&(t>0),\\
 \whdelta_{X_{\mathrm{sq}}}(t)
 &\ge\frac{\PP\bigl(|G|\ge8\sqrt6/t\bigr)}{4}>0
 &&(t>0).
\end{align}

Midpoint convexity from two recursive potentials

Theorem

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

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

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

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

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

Section 2, The fields, the roots, and the completed spaces Read it in the paper TeX source, line 65
Show the TeX
For $N=N_\Sigma$ on $X_\Sigma$, or $N=N_J$ on $X_J$, one has
\begin{equation}
 \widehat\delta_N(t)\ge t^3/128\qquad(0<t<1).
\end{equation}
For $X_0$, a finite initial set $H$, a vector $x$ supported in $H$, and
$z\in F_H\setminus\{0\}$, one has
\begin{equation}
 \frac{N_0(x+z)+N_0(x-z)}2\ge N_0(x)
 +\frac{N_0(z)^3}{8(2N_0(x)+N_0(z))^2}.
\end{equation}
Each of $X_\Sigma,X_0,X_J$ admits no equivalent AUC norm.
Theorem (Separated midpoint families)

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

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

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

Section 6, Logarithmic propagation on the joined tree Read it in the paper TeX source, line 480
Show the TeX
For every $\varepsilon>0$ there is $\eta(\varepsilon)\in(0,1)$ such
that, for $x,z_1,z_2,\ldots\in X_J$,
\[
 \left.
 \begin{gathered}
 N_J(x+z_j),N_J(x-z_j)\le1\quad(j\ge1),\\
 N_J(z_i-z_j)\ge\varepsilon\quad(i\ne j)
 \end{gathered}
 \right\}
 \quad\Longrightarrow\quad N_J(x)\le1-\eta(\varepsilon).
\]
For $0<\varepsilon\le2$, one can take
$\eta(\varepsilon)=\min\{1/2,\gamma(2,\varepsilon/16)\}$.
Theorem

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

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

The right side is zero when S=0.

Section 7, Changing the child exponent with the height Read it in the paper TeX source, line 294
Show the TeX
Let $x\in X_{\mathrm v}$ have finite total support.  In each component
where $x^{(h)}\ne0$, choose a finite ancestral set $A_h$ containing
that support.  Suppose $y\in X_{\mathrm v}$ vanishes on every such
$A_h$.  With
$S=\|x+y\|_{\mathrm v}^2+\|x-y\|_{\mathrm v}^2$, one has
\begin{equation}
 S-2\|x\|_{\mathrm v}^2
 \ge c_6 S^{-2}\|y\|_{\mathrm v}^6,
 \qquad c_6=2\exp(-(\mathrm e-1))4^{-6}.
\end{equation}
The right side is zero when $S=0$.
Configuration and file facts
OpenAI’s Lean catalogue
OpenAI lists this as a main result in its Lean catalogue.
Allowed axioms
propext, Quot.sound, Classical.choiceThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports
Mathlib
Imports from outside Mathlib
None
Local vocabulary
159 local vocabulary declarations
Lines of Lean, without comments or blank lines
538
Second proof-verification program (nanoda)
Off in OpenAI's configuration.

Statement comparison

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Plectis Atlas

Recursive-potential spaces and midpoint convexity

Family 331: Reflexive midpoint convexity and diamond distortion

Candidate pairing

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.

Midpoint convexity from two recursive potentials

TeX label rec:main, build/sections/foundations.tex, line 65

The Lean statement says

def MainClaim : Prop :=  ∃ h0 : FiniteRootedTree.Heads.{0}, letI := h0;  ∃ h1 : TreeVector.Construction.{0}, letI := h1;  ∃ h2 : ZeroTreeVector.Construction.{0}, letI := h2;  ∃ h3 : AggregationVector.Construction.{0}, letI := h3;  ∃ h4 : AggregationVector.LpConstruction.{0}, letI := h4;  rec_mainStatement ∧  rec_completionStatement ∧  stop_all_radiiStatement ∧  XJoined_separated_familiesStatement ∧  Variable.spaceStatement ∧  Variable.lp_model_exactStatement ∧  Variable.one_sided_modulus_zeroStatement ∧  Variable.no_equivalent_AUCStatement ∧  Variable.sixth_stabilityStatement ∧  Variable.cubic_stabilityStatement ∧  Variable.weak_tailStatementtheorem main : MainClaim := by

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

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

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