`ComparatorChallenges/HarmonicGrowth.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 120 curated AI-written annotations and 1273 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 9 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. Those 9 annotations were revised on 2026-10-14 to bind all of them to exact page-one quotations from 6 pinned OpenAI manuscripts, the matching pinned formalization notes, and named historical primary sources; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. The 3 manuscript-grounded matrix-multiplication annotations were composed from pinned OpenAI manuscripts and named historical sources in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Six additional target-specific annotations for Catalan's constant, the irrationality exponent of pi, dyadic Erdos similarity, Unique Games, quintic Lienard systems, and Yau nodal growth were composed from pinned OpenAI manuscripts, exact page-one quotations, exact Lean statements, and named historical sources in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Four additional target-specific annotations for Seymour's second-neighborhood conjecture, Cannon's conjecture, Koebe circle-domain uniformization, and circle-domain rigidity were composed from 4 pinned OpenAI manuscripts, exact page-one quotations, exact Lean statements, and named historical primary sources in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Four additional target-specific annotations for simple Lebesgue spectrum on the three-torus, Rokhlin multiple mixing, the nine-dimensional Borsuk counterexample, and Tingley's sphere-isometry problem were composed from 4 pinned OpenAI manuscripts, exact page-one quotations, exact Lean statements, and named historical primary sources in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Four additional target-specific annotations for the Abhyankar-Sathaye counterexample, the circulant Hadamard conjecture, the Eilenberg-Ganea counterexample, and short Egyptian fractions were composed from 4 pinned OpenAI manuscripts, exact page-one quotations, exact Lean statements, and quotations from Sathaye, Turyn with Storer, Eilenberg with Ganea, and Erdos in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Four additional target-specific annotations for the complete Crouzeix inequality, unrestricted Euclidean five-coloring, the general Mahler conjecture, and Thompson F nonamenability were composed on 2026-10-14 from exact Lean statements, 4 pinned OpenAI manuscripts, the matching formalization notes, and linked historical sources by Crouzeix, de Grey, Mahler, Cannon, Floyd, and Parry; no human review is claimed. Four additional target-specific annotations for algorithmic strong thin trees, Barnette's Hamiltonian-cycle conjecture, the explicit Artin CAT(0) counterexample, and generalized amplitude-damping capacity were composed on 2026-10-14 from exact Lean statements, 4 pinned OpenAI manuscripts, matching formalization notes, and linked historical sources by Anari with Goddyn, Arts with Hopman and Timmermans, Charney, and Khatri with Sharma and Wilde; no human review is claimed. Fourteen solution-level annotations for Seymour's second-neighborhood theorem were added on 2026-10-14 from the pinned 15-page manuscript, covering the exact-neighborhood model, minimal-counterexample reduction, subset deficit, transitive blowup, pruning, extremal contradiction, and proof-bearing final theorem. The subset and blowup annotations also bind exact quotations from Tyler Seacrest (2019) and Brantner, Brockman, Kay, and Snively (2009); no human review is claimed. Thirty-seven declaration-level annotations were added on 2026-10-14 for the rest of ComparatorChallenges/AlgorithmicThinTrees.lean, covering its multigraph semantics, stack-machine cost model, explicit and binary-multiplicity encodings, output certificate, and complete algorithmic proposition. They are bound to the pinned OpenAI manuscript and linked earlier thin-tree sources by Luis Goddyn, Nima Anari, and Shayan Oveis Gharan; the annotation explicitly preserves that the concluding comparator theorem still has body by sorry, and no human review is claimed. Thirty-one supporting declarations in ComparatorChallenges/UniqueGamesTheorem.lean were upgraded or added on 2026-10-14, completing curated coverage of all thirty-two declarations in that file. The annotations cover the finite game model, Boolean formula and parser, binary-vector translations, explicit output serialization, finite-alphabet machine contract, and gap theorem; they distinguish manuscript-backed mathematics from Lean-specific canonical parsing and unary game output encoding, bind exact OpenAI manuscript quotations, and credit Subhash Khot through his 2010 survey. The comparator theorem remains explicitly labeled as having body by sorry; no human review is claimed. The source-derived generator was hardened on 2026-10-14 so a formalization note covering multiple comparator results no longer copies its combined Scope prose onto each declaration; those targets now use declaration-specific source comments when available and otherwise remain explicitly bounded to the exact Lean statement. The remaining source-derived annotations were generated mechanically from pinned formalization notes and source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the source prose.

Credit: OpenAI's pinned repository says an unreleased internal model produced the manuscripts and supporting proof artifacts. The collection builds on the mathematics community's accumulated problems, results, language, and formal libraries; each manuscript's BibTeX and bibliography are the authority for individual scholarly credit.

Project sources: Pinned openai/math documentation, Pinned formalization catalogue

20 declarations, 0 contain sorry, 1 declared as axioms

abbrev OAI.HarmonicCounterexample.Space line 11
Also declared: OAI/Geometry/HarmonicGrowth/Basic.lean:11 (same declaration text).
abbrev OAI.HarmonicCounterexample.Mat line 12
Also declared: OAI/Geometry/HarmonicGrowth/Basic.lean:12 (same declaration text).
structure OAI.HarmonicCounterexample.SmoothMetric A genuine smooth positive definite covariant metric tensor on R^n line 15
Source documentation
A genuine smooth positive definite covariant metric tensor on R^n.
L4DEL explanation
A genuine smooth positive definite covariant metric tensor on R^n.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Geometry/HarmonicGrowth/Basic.lean:15 (same declaration text).
```
structure SmoothMetric (n : ℕ) where
…
```
def OAI.HarmonicCounterexample.coordDeriv The coordinate directional derivative line 21
Source documentation
The coordinate directional derivative.
L4DEL explanation
The coordinate directional derivative.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Geometry/HarmonicGrowth/Basic.lean:21 (same declaration text).
```
def coordDeriv {n : ℕ} (i : Fin n) (f : Space n → ℝ) (x : Space n) : ℝ :=
…
```
Mentions: OAI.HarmonicCounterexample.Space
def OAI.HarmonicCounterexample.christoffel Levi-Civita connection in the single global Cartesian chart line 25
Source documentation
Levi-Civita connection in the single global Cartesian chart.
L4DEL explanation
Levi-Civita connection in the single global Cartesian chart.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Geometry/HarmonicGrowth/Basic.lean:25 (same declaration text).
```
def christoffel {n : ℕ} (g : SmoothMetric n) (x : Space n)
    (k i j : Fin n) : ℝ :=
…
```
Mentions: OAI.HarmonicCounterexample.SmoothMetric, OAI.HarmonicCounterexample.Space
def OAI.HarmonicCounterexample.ricci Ricci tensor, with the convention in which the round sphere has positive Ricci line 33
Source documentation
Ricci tensor, with the convention in which the round sphere has positive Ricci.
L4DEL explanation
Ricci tensor, with the convention in which the round sphere has positive Ricci.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Geometry/HarmonicGrowth/Basic.lean:33 (same declaration text).
```
def ricci {n : ℕ} (g : SmoothMetric n) (x : Space n) (i j : Fin n) : ℝ :=
…
```
Mentions: OAI.HarmonicCounterexample.SmoothMetric, OAI.HarmonicCounterexample.Space
def OAI.HarmonicCounterexample.RicciNonnegative line 39
Also declared: OAI/Geometry/HarmonicGrowth/Basic.lean:39 (same declaration text).
def OAI.HarmonicCounterexample.laplaceBeltrami Trace of the covariant Hessian, i.e line 43
Source documentation
Trace of the covariant Hessian, i.e. the Laplace-Beltrami operator.
L4DEL explanation
Trace of the covariant Hessian, i.e. the Laplace-Beltrami operator.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Geometry/HarmonicGrowth/Basic.lean:43 (same declaration text).
```
def laplaceBeltrami {n : ℕ} (g : SmoothMetric n) (u : Space n → ℝ)
    (x : Space n) : ℝ :=
…
```
Mentions: OAI.HarmonicCounterexample.SmoothMetric, OAI.HarmonicCounterexample.Space
def OAI.HarmonicCounterexample.pathLength Riemannian length of a path on the unit parameter interval line 49
Source documentation
Riemannian length of a path on the unit parameter interval.
L4DEL explanation
Riemannian length of a path on the unit parameter interval.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Geometry/HarmonicGrowth/Basic.lean:49 (same declaration text).
```
def pathLength {n : ℕ} (g : SmoothMetric n) (γ : ℝ → Space n) : ℝ :=
…
```
Mentions: OAI.HarmonicCounterexample.SmoothMetric, OAI.HarmonicCounterexample.Space
def OAI.HarmonicCounterexample.distance Infimum of smooth path lengths, the intrinsic Riemannian distance line 54
Source documentation
Infimum of smooth path lengths, the intrinsic Riemannian distance.
L4DEL explanation
Infimum of smooth path lengths, the intrinsic Riemannian distance.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Geometry/HarmonicGrowth/Basic.lean:54 (same declaration text).
```
def distance {n : ℕ} (g : SmoothMetric n) (x y : Space n) : ℝ :=
…
```
Mentions: OAI.HarmonicCounterexample.SmoothMetric, OAI.HarmonicCounterexample.Space
def OAI.HarmonicCounterexample.Complete Sequential metric completeness, for the intrinsic distance, not the background one line 59
Source documentation
Sequential metric completeness, for the intrinsic distance, not the background one.
L4DEL explanation
Sequential metric completeness, for the intrinsic distance, not the background one.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Geometry/HarmonicGrowth/Basic.lean:59 (same declaration text).
```
def Complete {n : ℕ} (g : SmoothMetric n) : Prop :=
…
```
Mentions: OAI.HarmonicCounterexample.SmoothMetric
def OAI.HarmonicCounterexample.EuclideanNearOrigin line 64
Also declared: OAI/Geometry/HarmonicGrowth/Basic.lean:64 (same declaration text).
def OAI.HarmonicCounterexample.ballVolume Riemannian measure of the intrinsic open ball, retaining infinite values line 68
Source documentation
Riemannian measure of the intrinsic open ball, retaining infinite values.
L4DEL explanation
Riemannian measure of the intrinsic open ball, retaining infinite values.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Geometry/HarmonicGrowth/Basic.lean:68 (same declaration text).
```
def ballVolume {n : ℕ} (g : SmoothMetric n) (R : ℝ) : ℝ≥0∞ :=
…
```
Mentions: OAI.HarmonicCounterexample.SmoothMetric
def OAI.HarmonicCounterexample.omega The ordinary Euclidean unit ball volume, in Cartesian Lebesgue measure line 73
Source documentation
The ordinary Euclidean unit ball volume, in Cartesian Lebesgue measure.
L4DEL explanation
The ordinary Euclidean unit ball volume, in Cartesian Lebesgue measure.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Geometry/HarmonicGrowth/Basic.lean:73 (same declaration text).
```
def omega (n : ℕ) : ℝ≥0∞ :=
…
```
def OAI.HarmonicCounterexample.PositiveSubunitAVR The volume-ratio limit is finite, positive, and strictly below one line 77
Source documentation
The volume-ratio limit is finite, positive, and strictly below one.
L4DEL explanation
The volume-ratio limit is finite, positive, and strictly below one.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Geometry/HarmonicGrowth/Basic.lean:77 (same declaration text).
```
def PositiveSubunitAVR {n : ℕ} (g : SmoothMetric n) : Prop :=
…
```
Mentions: OAI.HarmonicCounterexample.SmoothMetric
def OAI.HarmonicCounterexample.HarmonicGrowth Membership in the actual space H_k, with the bound required at every point line 83
Source documentation
Membership in the actual space H_k, with the bound required at every point.
L4DEL explanation
Membership in the actual space H_k, with the bound required at every point.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Geometry/HarmonicGrowth/Basic.lean:83 (same declaration text).
```
def HarmonicGrowth {n : ℕ} (g : SmoothMetric n) (k : ℕ) (u : Space n → ℝ) : Prop :=
…
```
Mentions: OAI.HarmonicCounterexample.SmoothMetric, OAI.HarmonicCounterexample.Space
def OAI.HarmonicCounterexample.euclideanDimension line 88
Also declared: OAI/Geometry/HarmonicGrowth/Basic.lean:88 (same declaration text).
def OAI.HarmonicCounterexample.MainClaim The manuscript's unconditional main existence claim line 92
Source documentation
The manuscript's unconditional main existence claim.
L4DEL explanation
The manuscript's unconditional main existence claim.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Geometry/HarmonicGrowth/Basic.lean:92 (same declaration text).
```
def MainClaim : Prop :=
…
```
axiom OAI.HarmonicCounterexample.mainStatement line 100
theorem OAI.HarmonicCounterexample.main Challenge statement: Failure of integer-degree harmonic dimension comparison line 102
L4DEL explanation
The formalized result disproves the proposed Euclidean upper comparison for dimensions of harmonic functions with integer polynomial growth. It constructs one complete smooth metric with nonnegative Ricci curvature on an even-dimensional Euclidean space, Euclidean near the origin and with asymptotic volume ratio strictly between zero and one, admitting more independent harmonic functions of the prescribed growth degree than the Euclidean count. The construction uses dimension $16$ and degree $50000$. Separate tangent-cone and cone-spectrum conclusions are not included.
1. This is an exact comparator target linked by the pinned repository's formalization note for this result.
2. The repository's comparator configuration selects OAI.Geometry.HarmonicGrowth.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Geometry/HarmonicGrowth/Main.lean:475 (same statement text; different proof or body). Qualify references as OAI.HarmonicCounterexample.main@path.
```
theorem main : MainClaim := mainStatement
…
```
Mentions: OAI.HarmonicCounterexample.MainClaim, OAI.HarmonicCounterexample.mainStatement