`ComparatorChallenges/IndependentProducts.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

21 declarations, 4 contain sorry, 0 declared as axioms

abbrev OAI.IndependentProducts.Vertex The tree of finite sequences of positive integers line 15
Source documentation
The tree of finite sequences of positive integers.
L4DEL explanation
The tree of finite sequences of positive integers.
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/Analysis/ProductSpaces/Model.lean:15 (same declaration text).
```
abbrev Vertex := List ℕ+
```
abbrev OAI.IndependentProducts.Edge Coordinates are attached only to nonroot vertices line 17
Source documentation
Coordinates are attached only to nonroot vertices.
L4DEL explanation
Coordinates are attached only to nonroot vertices.
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/Analysis/ProductSpaces/Model.lean:17 (same declaration text).
```
abbrev Edge := {v : Vertex // v ≠ []}
```
Mentions: OAI.IndependentProducts.Vertex
abbrev OAI.IndependentProducts.Sample line 18
Also declared: OAI/Analysis/ProductSpaces/Model.lean:18 (same declaration text).
def OAI.IndependentProducts.productMeasure The product law of independent copies of the multiplier line 21
Source documentation
The product law of independent copies of the multiplier.
L4DEL explanation
The product law of independent copies of the multiplier.
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/Analysis/ProductSpaces/Model.lean:21 (same declaration text).
```
def productMeasure (μ : Measure ℝ) : Measure Sample :=
…
```
Mentions: OAI.IndependentProducts.Sample
def OAI.IndependentProducts.pathPrefix The nonroot prefix ending at position j line 25
Source documentation
The nonroot prefix ending at position j.
L4DEL explanation
The nonroot prefix ending at position j.
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/Analysis/ProductSpaces/Model.lean:25 (same declaration text).
```
def pathPrefix (v : Vertex) (j : Fin v.length) : Edge :=
…
```
Mentions: OAI.IndependentProducts.Vertex, OAI.IndependentProducts.Edge
def OAI.IndependentProducts.pathProduct Root product is the empty product 1; every other product uses the path prefixes line 34
Source documentation
Root product is the empty product 1; every other product uses the path prefixes.
L4DEL explanation
Root product is the empty product 1; every other product uses the path prefixes.
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/Analysis/ProductSpaces/Model.lean:34 (same declaration text).
```
def pathProduct (v : Vertex) (ω : Sample) : ℝ :=
…
```
Mentions: OAI.IndependentProducts.Vertex, OAI.IndependentProducts.Sample
def OAI.IndependentProducts.productSpan The closed real L1 span of the path products line 38
Source documentation
The closed real L1 span of the path products.
L4DEL explanation
The closed real L1 span of the path products.
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/Analysis/ProductSpaces/Model.lean:38 (same declaration text).
```
def productSpan (μ : Measure ℝ) : Submodule ℝ (Lp ℝ 1 (productMeasure μ)) :=
…
```
Mentions: OAI.IndependentProducts.productMeasure
structure OAI.IndependentProducts.MultiplierHypotheses Positivity, nonconstancy, mean one, and finite second moment of the multiplier law line 43
Source documentation
Positivity, nonconstancy, mean one, and finite second moment of the multiplier law.
L4DEL explanation
Positivity, nonconstancy, mean one, and finite second moment of the multiplier law.
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)
Distinct definitions of this name: OAI/Analysis/ProductSpaces/Model.lean:43 (same statement text; different proof or body). Qualify references as OAI.IndependentProducts.MultiplierHypotheses@path.
```
structure MultiplierHypotheses (μ : Measure ℝ) : Prop where
…
```
def OAI.IndependentProducts.CofiniteSubspaces Closed subspaces of finite codimension, with respect to the inherited topology line 53
Source documentation
Closed subspaces of finite codimension, with respect to the inherited topology.
L4DEL explanation
Closed subspaces of finite codimension, with respect to the inherited topology.
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/Analysis/ProductSpaces/Model.lean:55 (same declaration text).
```
def CofiniteSubspaces : Set (Submodule ℝ X) :=
…
```
def OAI.IndependentProducts.averagedMidpointModulus The averaged asymptotic midpoint modulus, with directions of norm at least 1 line 57
Source documentation
The averaged asymptotic midpoint modulus, with directions of norm at least 1.
L4DEL explanation
The averaged asymptotic midpoint modulus, with directions of norm at least 1.
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/Analysis/ProductSpaces/Model.lean:59 (same declaration text).
```
def averagedMidpointModulus (N : Seminorm ℝ X) (t : ℝ) : ℝ :=
…
```
def OAI.IndependentProducts.oneSidedModulus The usual one-sided modulus, whose directions have norm exactly 1 line 64
Source documentation
The usual one-sided modulus, whose directions have norm exactly 1.
L4DEL explanation
The usual one-sided modulus, whose directions have norm exactly 1.
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/Analysis/ProductSpaces/Model.lean:66 (same declaration text).
```
def oneSidedModulus (N : Seminorm ℝ X) (t : ℝ) : ℝ :=
…
```
def OAI.IndependentProducts.IsEquivalentNorm A seminorm with these bounds is a genuine norm equivalent to the original norm line 70
Source documentation
A seminorm with these bounds is a genuine norm equivalent to the original norm.
L4DEL explanation
A seminorm with these bounds is a genuine norm equivalent to the original norm.
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/Analysis/ProductSpaces/Model.lean:72 (same declaration text).
```
def IsEquivalentNorm (N : Seminorm ℝ X) : Prop :=
…
```
def OAI.IndependentProducts.IsAUC line 73
Also declared: OAI/Analysis/ProductSpaces/Model.lean:75 (same declaration text).
def OAI.IndependentProducts.gaussianTail Gaussian-tail probability, in real rather than extended-nonnegative-real notation line 76
Source documentation
Gaussian-tail probability, in real rather than extended-nonnegative-real notation.
L4DEL explanation
Gaussian-tail probability, in real rather than extended-nonnegative-real notation.
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/Analysis/ProductSpaces/Model.lean:80 (same declaration text).
```
def gaussianTail (r : ℝ) : ℝ :=
…
```
def OAI.IndependentProducts.excessConstant The Gaussian excess constant at threshold twice the parameter line 80
Source documentation
The Gaussian excess constant at threshold twice the parameter.
L4DEL explanation
The Gaussian excess constant at threshold twice the parameter.
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/Analysis/ProductSpaces/Model.lean:84 (same declaration text).
```
def excessConstant (K : ℝ) : ℝ :=
…
```
def OAI.IndependentProducts.exponentialLaw The density exp(-w) on the positive half-line line 84
Source documentation
The density exp(-w) on the positive half-line.
L4DEL explanation
The density exp(-w) on the positive half-line.
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/Analysis/ProductSpaces/Model.lean:88 (same declaration text).
```
def exponentialLaw : Measure ℝ := expMeasure 1
```
def OAI.IndependentProducts.gaussianSquareLaw The law of the square of a standard real Gaussian line 87
Source documentation
The law of the square of a standard real Gaussian.
L4DEL explanation
The law of the square of a standard real Gaussian.
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/Analysis/ProductSpaces/Model.lean:91 (same declaration text).
```
def gaussianSquareLaw : Measure ℝ := (gaussianReal 0 1).map (fun x : ℝ => x ^ 2)
```
theorem OAI.IndependentProducts.main_general Challenge statement: Independent-product spaces and positive midpoint moduli sorry line 89
L4DEL explanation
This declaration is an exact comparator-selected statement. Read the Lean statement below for its mathematical content.
1. The linked formalization note covers 7 distinct comparator results, so its combined Scope prose is not copied onto this declaration.
2. The repository's comparator configuration selects OAI.Analysis.ProductSpaces.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Analysis/ProductSpaces/WeakNullity.lean:203 (same statement text; different proof or body). Qualify references as OAI.IndependentProducts.main_general@path.
```
theorem main_general (μ : Measure ℝ) (hμ : MultiplierHypotheses μ) :
    ¬ FiniteDimensional ℝ (productSpan μ) ∧
    ∀ N : Seminorm ℝ (productSpan μ),
      IsEquivalentNorm (productSpan μ) N → ¬ IsAUC (productSpan μ) N := by
…
```
Mentions: OAI.IndependentProducts.productSpan, OAI.IndependentProducts.IsEquivalentNorm, OAI.IndependentProducts.IsAUC
theorem OAI.IndependentProducts.main_exponential Challenge statement: Independent-product spaces and positive midpoint moduli sorry line 95
L4DEL explanation
This declaration is an exact comparator-selected statement. Read the Lean statement below for its mathematical content.
1. The linked formalization note covers 7 distinct comparator results, so its combined Scope prose is not copied onto this declaration.
2. The repository's comparator configuration selects OAI.Analysis.ProductSpaces.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Analysis/ProductSpaces/Main.lean:54 (same statement text; different proof or body). Qualify references as OAI.IndependentProducts.main_exponential@path.
Checked implications (found by the pipeline and checked by Lean's kernel when it ran; a recorded result, not checked again in this build: what these are)
Gives
- together with OAI.PlanarAnderson.anderson_spectrum_ae, gives ∀ (t : ℝ), 0 < t → ∀ᵐ (v : OAI.PlanarAnderson.Configuration) ∂OAI.PlanarAnderson.disorderLaw (7 * t / 30 * OAI.IndependentProducts.excessConstant (10 / t)), ∃ H, OAI.PlanarAnderson.IsAndersonOperator v H ∧ IsSelfAdjoint… (short composition) checked derivation
- together with OAI.SKGap.sk_main, gives ∀ (t : ℝ), 0 < t → 7 * t / 30 * OAI.IndependentProducts.excessConstant (10 / t) < 1 → ∃ C, 0 < C ∧ Filter.Tendsto (fun n => (OAI.SKGap.disorderLaw (7 * t / 30 * OAI.IndependentProducts.excessConstant (10 / t)) n) (OAI.S… (short composition) checked derivation
4 more checked entries are routine (the same statement, a restatement, a generated lemma, one challenge file, a field of a structure, a route the released proof already takes, a duplicate): in the catalogue.
```
theorem main_exponential (t : ℝ) (ht : 0 < t) :
    (7 * t / 30) * excessConstant (10 / t) ≤
      averagedMidpointModulus (productSpan exponentialLaw)
        (normSeminorm ℝ (productSpan exponentialLaw)) t ∧
    0 < (7 * t / 30) * excessConstant (10 / t) := by
…
```
Mentions: OAI.IndependentProducts.excessConstant, OAI.IndependentProducts.averagedMidpointModulus, OAI.IndependentProducts.productSpan, OAI.IndependentProducts.exponentialLaw
theorem OAI.IndependentProducts.main_gaussian_24 Challenge statement: Independent-product spaces and positive midpoint moduli sorry line 102
L4DEL explanation
This declaration is an exact comparator-selected statement. Read the Lean statement below for its mathematical content.
1. The linked formalization note covers 7 distinct comparator results, so its combined Scope prose is not copied onto this declaration.
2. The repository's comparator configuration selects OAI.Analysis.ProductSpaces.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Analysis/ProductSpaces/GaussianBounds.lean:140 (same statement text; different proof or body). Qualify references as OAI.IndependentProducts.main_gaussian_24@path.
Checked implications (found by the pipeline and checked by Lean's kernel when it ran; a recorded result, not checked again in this build: what these are)
Gives
- together with OAI.PlanarAnderson.anderson_spectrum_ae, gives ∀ (t : ℝ), 0 < t → ∀ᵐ (v : OAI.PlanarAnderson.Configuration) ∂OAI.PlanarAnderson.disorderLaw (OAI.IndependentProducts.gaussianTail (4 * √2 * (2 + √2) / t) / 8), ∃ H, OAI.PlanarAnderson.IsAndersonOperator v H ∧ IsSelfAdj… (short composition) checked derivation
- together with OAI.SKGap.sk_main, gives ∀ (t : ℝ), 0 < t → OAI.IndependentProducts.gaussianTail (4 * √2 * (2 + √2) / t) / 8 < 1 → ∃ C, 0 < C ∧ Filter.Tendsto (fun n => (OAI.SKGap.disorderLaw (OAI.IndependentProducts.gaussianTail (4 * √2 * (2 + √2) / t) / 8) n… (short composition) checked derivation
4 more checked entries are routine (the same statement, a restatement, a generated lemma, one challenge file, a field of a structure, a route the released proof already takes, a duplicate): in the catalogue.
```
theorem main_gaussian_24 (t : ℝ) (ht : 0 < t) :
    gaussianTail (4 * Real.sqrt 2 * (2 + Real.sqrt 2) / t) / 8 ≤
      averagedMidpointModulus (productSpan gaussianSquareLaw)
        (normSeminorm ℝ (productSpan gaussianSquareLaw)) t ∧
    0 < gaussianTail (4 * Real.sqrt 2 * (2 + Real.sqrt 2) / t) / 8 := by
…
```
Mentions: OAI.IndependentProducts.gaussianTail, OAI.IndependentProducts.averagedMidpointModulus, OAI.IndependentProducts.productSpan, OAI.IndependentProducts.gaussianSquareLaw
theorem OAI.IndependentProducts.main_gaussian_12 Challenge statement: Independent-product spaces and positive midpoint moduli sorry line 109
L4DEL explanation
This declaration is an exact comparator-selected statement. Read the Lean statement below for its mathematical content.
1. The linked formalization note covers 7 distinct comparator results, so its combined Scope prose is not copied onto this declaration.
2. The repository's comparator configuration selects OAI.Analysis.ProductSpaces.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Analysis/ProductSpaces/GaussianBounds.lean:121 (same statement text; different proof or body). Qualify references as OAI.IndependentProducts.main_gaussian_12@path.
Checked implications (found by the pipeline and checked by Lean's kernel when it ran; a recorded result, not checked again in this build: what these are)
Gives
- together with OAI.SKGap.sk_main, gives ∀ (t : ℝ), 0 < t → OAI.IndependentProducts.gaussianTail (8 * √6 / t) / 4 < 1 → ∃ C, 0 < C ∧ Filter.Tendsto (fun n => (OAI.SKGap.disorderLaw (OAI.IndependentProducts.gaussianTail (8 * √6 / t) / 4) n) (OAI.SKGap.poincareE… (short composition) checked derivation
- together with OAI.PlanarAnderson.anderson_spectrum_ae, gives ∀ (t : ℝ), 0 < t → ∀ᵐ (v : OAI.PlanarAnderson.Configuration) ∂OAI.PlanarAnderson.disorderLaw (OAI.IndependentProducts.gaussianTail (8 * √6 / t) / 4), ∃ H, OAI.PlanarAnderson.IsAndersonOperator v H ∧ IsSelfAdjoint H ∧ sp… (short composition) checked derivation
4 more checked entries are routine (the same statement, a restatement, a generated lemma, one challenge file, a field of a structure, a route the released proof already takes, a duplicate): in the catalogue.
```
theorem main_gaussian_12 (t : ℝ) (ht : 0 < t) :
    gaussianTail (8 * Real.sqrt 6 / t) / 4 ≤
      averagedMidpointModulus (productSpan gaussianSquareLaw)
        (normSeminorm ℝ (productSpan gaussianSquareLaw)) t ∧
    0 < gaussianTail (8 * Real.sqrt 6 / t) / 4 := by
…
```
Mentions: OAI.IndependentProducts.gaussianTail, OAI.IndependentProducts.averagedMidpointModulus, OAI.IndependentProducts.productSpan, OAI.IndependentProducts.gaussianSquareLaw