| import Mathlib | |
| namespace OAI | |
| /-! | |
| # A sharp Fourier certificate for planar circle packing | |
| The normalization and Fourier convention are those of | |
| *A sharp Fourier certificate for planar circle packing* (OpenAI, 2026). | |
| -/ | |
| open MeasureTheory | |
| namespace SharpPlanar | |
| abbrev Plane := EuclideanSpace ℝ (Fin 2) | |
| /-- Radiality of a real function on the Euclidean plane. -/ | |
| def Radial (f : Plane → ℝ) : Prop := | |
| ∀ x y, ‖x‖ = ‖y‖ → f x = f y | |
| /-- Fourier transformation with kernel `exp (-2πi⟪x, ξ⟫)`. -/ | |
| noncomputable def fourier (f : Plane → ℝ) (ξ : Plane) : ℂ := | |
| ∫ x : Plane, (f x : ℂ) * | |
| Complex.exp (((-2 * Real.pi * inner ℝ x ξ : ℝ) : ℂ) * Complex.I) | |
| /-- A normalized sharp certificate for disks of radius one half. -/ | |
| def SharpCertificate (f : SchwartzMap Plane ℝ) : Prop := | |
| Radial f ∧ fourier f 0 = 1 ∧ f 0 = 2 / Real.sqrt 3 ∧ | |
| (∀ ξ, (fourier f ξ).im = 0 ∧ 0 ≤ (fourier f ξ).re) ∧ | |
| (∀ x, 1 ≤ ‖x‖ → f x ≤ 0) | |
| end SharpPlanar | |
| /-- The sharp Fourier certificate theorem. -/ | |
| theorem sharp_fourier_certificate : | |
| ∃ f : SchwartzMap SharpPlanar.Plane ℝ, SharpPlanar.SharpCertificate f := by | |
| sorry | |
| end OAI | |