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720fa25 | 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 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 | 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
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