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465b10d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 | 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
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