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