42.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

8 declarations, 4 contain sorry, 0 declared as axioms

Green's Open Problem 42

References:

[Gr24] Green, Ben. "100 open problems." (2024).
[CoEl03] Cohn, Henry, and Noam Elkies. "New upper bounds on sphere packings I." Annals of Mathematics (2003): 689-714.
[Vi17] Viazovska, Maryna S. "The sphere packing problem in dimension 8." Annals of mathematics (2017): 991-1015.
[CKM17] Cohn, H., Kumar, A., Miller, S., Radchenko, D., & Viazovska, M. (2017). The sphere packing problem in dimension 24. Annals of mathematics, 185(3), 1017-1033.
[Sa21] Sardari, Naser Talebizadeh. "Higher Fourier interpolation on the plane." arXiv preprint arXiv:2102.08753 (2021).

Imports (1)

FormalConjecturesUtil

def Green42.fHat The real-valued Fourier transform used in the Cohn--Elkies conditions line 52 source
Source documentation
The real-valued Fourier transform used in the Cohn--Elkies conditions. For real radial admissible functions, the complex Fourier transform is expected to be real-valued; we take .re to expose the real scalar used in the inequality.
Convention: Mathlib's 𝓕 f w expands to $\int e^{-2\pi i \langle v, w \rangle} f(v) dv$, which matches [CoEl03]'s $\hat{f}(t) = \int f(x) e^{-2\pi i \langle x, t \rangle} dx$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MeasureTheory.MeasureSpace, BorelSpace, FiniteDimensional
Proof uses: Complex.re, FourierTransform.fourier, Complex
29 instances, coercions and auxiliary constants left out.
```
noncomputable def fHat (f : V → ℝ) (t : V) : ℝ :=
…
```
def Green42.CohnElkiesAdmissible Definition 2.1 from [CoEl03]: A function is admissible if it is continuous and both the function and its Fourier transform decay sufficiently fast line 60 source
Source documentation
Definition 2.1 from [CoEl03]: A function is admissible if it is continuous and both the function and its Fourier transform decay sufficiently fast. Continuity is essential: changing f at a single point leaves fHat f unchanged, so without it every positive bound would be achievable.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MeasureTheory.MeasureSpace, BorelSpace, FiniteDimensional
Proof uses: And, Continuous, Exists, GT.gt, OfNat.ofNat, LE.le, abs, HDiv.hDiv, HPow.hPow, HAdd.hAdd, Norm.norm, Module.finrank, Green42.fHat
43 instances, coercions and auxiliary constants left out.
```
def CohnElkiesAdmissible (f : V → ℝ) : Prop :=
…
```
def Green42.SatisfiesCohnElkiesScheme The structural rules a function must satisfy to successfully pass through the Cohn-Elkies scheme and generate some valid upper bound line 69 source
Source documentation
The structural rules a function must satisfy to successfully pass through the Cohn-Elkies scheme and generate some valid upper bound.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MeasureTheory.MeasureSpace, BorelSpace, FiniteDimensional
Proof uses: And, Green42.CohnElkiesAdmissible, Eq, Norm.norm, LE.le, OfNat.ofNat, Green42.fHat, LT.lt
35 instances, coercions and auxiliary constants left out.
```
def SatisfiesCohnElkiesScheme (f : V → ℝ) : Prop :=
…
```
def Green42.CohnElkiesOptimal The statement that there exists a function in dimension d satisfying the Cohn-Elkies scheme which achieves the center density bound bound line 81 source
Source documentation
The statement that there exists a function in dimension d satisfying the Cohn-Elkies scheme which achieves the center density bound bound.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: Exists, EuclideanSpace, Fin, And, Green42.SatisfiesCohnElkiesScheme, OfNat.ofNat, ENNReal, Eq, HDiv.hDiv, Green42.fHat
30 instances, coercions and auxiliary constants left out.
```
def CohnElkiesOptimal (d : ℕ) (bound : ℝ) : Prop :=
…
```
theorem Green42.green_42 Can the Cohn-Elkies scheme be used to prove the optimal bound for circle-packings in 2 dimensions? sorry line 90 source
Source documentation
Can the Cohn-Elkies scheme be used to prove the optimal bound for circle-packings in 2 dimensions?

depends on sorry what this means
Statement uses: Iff, True, Green42.CohnElkiesOptimal, OfNat.ofNat, Nat, HDiv.hDiv, Real, Real.sqrt
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_42 :
    answer(sorry) ↔ CohnElkiesOptimal 2 (Real.sqrt 3 / 6) := by
…
```
Mentions: Green42.CohnElkiesOptimal
theorem Green42.green_42.variants.dimension_one [CoEl03] proved this when $d = 1$ sorry line 96 source
Source documentation
[CoEl03] proved this when $d = 1$.

depends on sorry what this means
Statement uses: Green42.CohnElkiesOptimal, OfNat.ofNat, Nat, HDiv.hDiv, Real
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_42.variants.dimension_one :
    CohnElkiesOptimal 1 (1 / 2) := by
…
```
Mentions: Green42.CohnElkiesOptimal
theorem Green42.green_42.variants.dimension_eight [Vi17] established the case $d = 8$ sorry line 102 source
Source documentation
[Vi17] established the case $d = 8$.

depends on sorry what this means
Statement uses: Green42.CohnElkiesOptimal, OfNat.ofNat, Nat, HDiv.hDiv, Real
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_42.variants.dimension_eight :
    CohnElkiesOptimal 8 (1 / 16) := by
…
```
Mentions: Green42.CohnElkiesOptimal
theorem Green42.green_42.variants.dimension_twenty_four In [CKM17], [Vi17] was adapted to $d = 24$ sorry line 108 source
Source documentation
In [CKM17], [Vi17] was adapted to $d = 24$.

depends on sorry what this means
Statement uses: Green42.CohnElkiesOptimal, OfNat.ofNat, Nat, Real
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_42.variants.dimension_twenty_four :
    CohnElkiesOptimal 24 1 := by
…
```
Mentions: Green42.CohnElkiesOptimal