`BookProof.QgMultiHalfDensity.exists_field_halfDensity_unitary` (n : ℕ) : ∃ _W : Lp ℂ 2 (BookProof.ScalaronDensitized.physMeasure.prod (volume : Measure (Fin n → ℝ))) ≃ₗᵢ[ℂ] Lp ℂ 2

Proved

BookProof.QgMultiHalfDensity.exists_field_halfDensity_unitary

by leonardopedro · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterQgMultiHalfDensity.

BookProof.QgMultiHalfDensity.exists_field_halfDensity_unitary (n : ℕ) : ∃ _W : Lp ℂ 2 (BookProof.ScalaronDensitized.physMeasure.prod (volume : Measure (Fin n → ℝ))) ≃ₗᵢ[ℂ] Lp ℂ 2 (qgSrcMeasure.prod (volume : Measure (Fin n → ℝ))), True

Formalization note: Lean 4 identifier BookProof.QgMultiHalfDensity.exists_field_halfDensity_unitary.

Preamble

-- Generated from ChapterQgMultiHalfDensity.lean — theorem BookProof.QgMultiHalfDensity.exists_field_halfDensity_unitary
import Definitions.Def_ChapterNavierStokesFockContinuum
import Mathlib
import Definitions.Def_ChapterQgMultiHalfDensity
import Definitions.Def_ChapterQuantumGravityHalfDensity
import Definitions.Def_ChapterScalaronDensitizedTransfer
open BookProof.QuantumGravityHalfDensity
open BookProof.ScalaronDensitized
open BookProof.QgMultiHalfDensity



open MeasureTheory Set
open BookProof.NavierStokesFlow.FockContinuum
open BookProof.QuantumGravityHalfDensity

noncomputable section

variable {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y]
  {mu : Measure X} {nu : Measure Y} {Phi : X → Y} {Psi : Y → X}
variable {g : Y → ℝ}
variable {X : Type*} [MeasurableSpace X] (mu : Measure X)
variable [SFinite mu]

Formal statement

theorem BookProof.QgMultiHalfDensity.exists_field_halfDensity_unitary (n : ℕ) :
    ∃ _W : Lp ℂ 2 (BookProof.ScalaronDensitized.physMeasure.prod
        (volume : Measure (Fin n → ℝ)))
      ≃ₗᵢ[ℂ] Lp ℂ 2 (qgSrcMeasure.prod (volume : Measure (Fin n → ℝ))), True := by sorry

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