`BookProof.ScalaronDensitized.densConfOp_quadForm_ge` (halpha : 0 < alpha) (f : densConfCore M alpha) : -(M ^ 4 / (16 * alpha)) * ‖(f : Lp ℂ 2 qgSrcMeasure)‖ ^ 2 ≤ quadForm ((densC

Proved

BookProof.ScalaronDensitized.densConfOp_quadForm_ge

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterScalaronDensitizedTransfer.

BookProof.ScalaronDensitized.densConfOp_quadForm_ge (halpha : 0 < alpha) (f : densConfCore M alpha) : -(M ^ 4 / (16 * alpha)) * ‖(f : Lp ℂ 2 qgSrcMeasure)‖ ^ 2 ≤ quadForm ((densConfCore M alpha).subtype.comp (densConfOp M alpha)) f

Formalization note: Lean 4 identifier BookProof.ScalaronDensitized.densConfOp_quadForm_ge.

Preamble

-- Generated from ChapterScalaronDensitizedTransfer.lean — theorem BookProof.ScalaronDensitized.densConfOp_quadForm_ge
import Definitions.Def_ChapterStarobinskyPotential
import Definitions.Def_ChapterQuantumGravityDensitized
import Definitions.Def_ChapterNavierStokesAffineFiberEsa
import Definitions.Def_ChapterNavierStokesFockContinuum
import Definitions.Def_ChapterFarisLavine
import Definitions.Def_ChapterStoneBridge
import Definitions.Def_ChapterSirkTrotterKato
import Definitions.Def_ChapterEsaClosureCore
import Mathlib
import Definitions.Def_ChapterScalaronDensitizedTransfer
import Definitions.Def_ChapterFarisLavineCore
import Definitions.Def_ChapterQuantumGravityHalfDensity
open BookProof.QuantumGravityHalfDensity
open BookProof.ScalaronDensitized



open MeasureTheory Set Filter Topology
open BookProof.Starobinsky BookProof.QuantumGravityDensitized
open BookProof.QuantumGravityHalfDensity
open BookProof.NavierStokesFlow BookProof.NavierStokesFlow.FockContinuum
open BookProof.FarisLavine BookProof.StoneBridge
open BookProof.ChapterStoneResolvent BookProof.EsaClosure

noncomputable section

variable (M alpha : ℝ)
variable {X : Type*} [MeasurableSpace X]

Formal statement

theorem BookProof.ScalaronDensitized.densConfOp_quadForm_ge (halpha : 0 < alpha) (f : densConfCore M alpha) :
    -(M ^ 4 / (16 * alpha)) * ‖(f : Lp ℂ 2 qgSrcMeasure)‖ ^ 2
      ≤ quadForm ((densConfCore M alpha).subtype.comp (densConfOp M alpha)) f := by sorry

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