| import Mathlib |
|
|
| namespace OAI |
|
|
| /-! Genuine torus, smooth local quotient lifts, Haar probability, full mean-zero complex L2, and unconditional conclusion. -/ |
|
|
| section TorusInterface |
| noncomputable section |
| open MeasureTheory Set |
| open scoped ENNReal Topology |
|
|
| namespace ThreeTorus |
|
|
| abbrev Torus := Fin 3 → UnitAddCircle |
| abbrev Cover := Fin 3 → ℝ |
|
|
| def project (x : Cover) : Torus := fun i => (x i : UnitAddCircle) |
|
|
| def Smooth (T : Torus → Torus) : Prop := |
| ∀ x : Cover, ∃ U : Set Cover, IsOpen U ∧ x ∈ U ∧ |
| ∃ F : Cover → Cover, ContDiffOn ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) F U ∧ |
| ∀ y ∈ U, T (project y) = project (F y) |
|
|
| def μ : Measure Torus := Measure.pi (fun _ : Fin 3 => AddCircle.haarAddCircle) |
|
|
| instance : IsProbabilityMeasure μ := by unfold μ; infer_instance |
|
|
| abbrev H : Type := Lp ℂ 2 μ |
|
|
| /-- The full complex subspace defined by zero Lebesgue integral. -/ |
| def meanZero : Submodule ℂ H where |
| carrier := {f | ∫ x, f x ∂μ = 0} |
| zero_mem' := by |
| change ∫ x, (0 : H) x ∂μ = 0 |
| rw [integral_congr_ae (Lp.coeFn_zero ℂ 2 μ)] |
| simp |
| add_mem' := by |
| intro f g hf hg |
| change ∫ x, (f + g) x ∂μ = 0 |
| rw [integral_congr_ae (Lp.coeFn_add f g)] |
| change ∫ x, f x + g x ∂μ = 0 |
| rw [integral_add (MemLp.integrable (by norm_num) (Lp.memLp f)) |
| (MemLp.integrable (by norm_num) (Lp.memLp g))] |
| change (∫ x, f x ∂μ) + (∫ x, g x ∂μ) = 0 |
| rw [show ∫ x, f x ∂μ = 0 from hf, show ∫ x, g x ∂μ = 0 from hg, add_zero] |
| smul_mem' := by |
| intro c f hf |
| change ∫ x, (c • f) x ∂μ = 0 |
| rw [integral_congr_ae (Lp.coeFn_smul c f)] |
| change ∫ x, c • f x ∂μ = 0 |
| rw [integral_smul] |
| rw [show ∫ x, f x ∂μ = 0 from hf, smul_zero] |
|
|
| abbrev H₀ := ↥meanZero |
| abbrev CircleH : Type := Lp ℂ 2 (AddCircle.haarAddCircle (T := (1 : ℝ))) |
|
|
| def IsReal (f : H₀) : Prop := ∀ᵐ x ∂μ, (f.1 x).im = 0 |
|
|
| /-- Complete simple Lebesgue spectrum, including the stated consequences. -/ |
| def MainConclusion : Prop := |
| ∃ T : Equiv.Perm Torus, |
| Smooth T ∧ Smooth T.symm ∧ MeasurePreserving T μ μ ∧ |
| ∃ f : H₀, IsReal f ∧ |
| ∃ v : ℤ → H₀, |
| (∀ n : ℤ, ∀ᵐ x ∂μ, (v n).1 x = f.1 ((T ^ n) x)) ∧ |
| Orthonormal ℂ v ∧ |
| Dense (Submodule.span ℂ (Set.range v) : Set H₀) ∧ |
| Ergodic T μ ∧ |
| ∃ W : H₀ ≃ₗᵢ[ℂ] CircleH, |
| ∀ g : H₀, ∃ h : H₀, |
| (∀ᵐ x ∂μ, h.1 x = g.1 (T x)) ∧ |
| (∀ᵐ z ∂(AddCircle.haarAddCircle (T := (1 : ℝ))), |
| W h z = fourier 1 z * W g z) |
|
|
| end ThreeTorus |
| end |
| end TorusInterface |
|
|
| end OAI |
|
|
