Symmetry of the Indexed Weak Hessian

Proved

FunctionalInequalities.KLS.LogConcave.PoissonIterate.gradField_gradField_comp_swap_readable

by Lazyfill · Oct 26, 2026 · Mathlib 0df444a (Lean v4.33.1)

functional-inequalitieskls-conjecturekls-modularkls-readablelog-concave

This statement proves symmetry of the first two indices of an indexed weak Hessian.

Let $V$ be finite-dimensional, let $\nu=m\mathbin{.}\mathrm{tilted}(-U)$ with $U$ of temperate growth and all exponential moments, and let $h$ be a level- $k$ field whose components lie in the domain of $A=\bar T^\dagger\bar T$ . For every index map $J:\mathrm{Fin}(k+2)\to\mathrm{Fin}(\dim V)$ , swapping the first two positions leaves the iterated gradient field unchanged:

\operatorname{gradField}(\operatorname{gradField}(h))(J\circ(0\;1)) =\operatorname{gradField}(\operatorname{gradField}(h))(J).

This is the coordinate symmetry needed to construct symmetric approximations to centered gradient fields.

Original Lean declaration header

Within namespace FunctionalInequalities.KLS.LogConcave.PoissonIterate. Shown in its original module context. Proof omitted.

theorem gradField_gradField_comp_swap {k : ℕ} (h : Field ν k)
    (hh : ∀ I, h I ∈ (adjointComp T̄).domain)
    (J : Fin (k + 2) → Fin (Module.finrank ℝ V)) :
    gradField T̄ (gradField T̄ h) (J ∘ Equiv.swap 0 1) = gradField T̄ (gradField T̄ h) J

Module navigation

Module: KLS/BKL/PoissonIterate/Defect.

Definition modules: KLS/LogConcave/WeightedGradient/Closure, KLS/BKL/PoissonIterate/Basic, Analysis/Semigroup/AdjointComp, Probability/LogConcave.

Supporting theorems: compLp_innerSL_closedWeightedOperator_comm, convexOn_of_eq_tilted, exists_seq_smooth_tendsto_weakHessian.

Previous: normSq_gradField_gradField_step · Next: exists_symm_normSq_grad0_step_sub_le.

Source excerpt, lines 252–255 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_FunctionalInequalities_KLS_BKL_PoissonIterate_Defect

open FunctionalInequalities.Analysis.Semigroup
variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [CompleteSpace V]
  [MeasurableSpace V] [BorelSpace V] [SecondCountableTopology V] [FiniteDimensional ℝ V]
  {ν : MeasureTheory.ProbabilityMeasure V} (m : MeasureTheory.Measure V) [m.IsAddHaarMeasure] (hν : FunctionalInequalities.KLS.LogConcave ν)
  (hi : (interior (ν : MeasureTheory.Measure V).support).Nonempty) {U : V → ℝ}
  (hνU : (ν : MeasureTheory.Measure V) = m.tilted (-U)) (hUt : U.HasTemperateGrowth)
  (hexp : ∀ r : ℝ, MeasureTheory.Integrable (fun x => Real.exp (r * ‖x‖)) (ν : MeasureTheory.Measure V))
  {C : NNReal} (hC : FunctionalInequalities.Poincare.LocalSlope.Intrinsic.constant ν ≤ (C : ENNReal))
local notation "T̄" => hν.closedWeightedOperator m (ContinuousLinearMap.id ℝ V) hi
include hνU hUt hexp

Formal statement


theorem FunctionalInequalities.KLS.LogConcave.PoissonIterate.gradField_gradField_comp_swap_readable {k : ℕ} (h : Field ν k)
    (hh : ∀ I, h I ∈ (adjointComp T̄).domain)
    (J : Fin (k + 2) → Fin (Module.finrank ℝ V)) :
    gradField T̄ (gradField T̄ h) (J ∘ Equiv.swap 0 1) = gradField T̄ (gradField T̄ h) J := by sorry

Source

Lean source: https://github.com/human-ai-mathematics/kls-lean/blob/408915f/formalization/FunctionalInequalities/KLS/BKL/PoissonIterate/Defect.lean#L249-L267 (repository human-ai-mathematics/kls-lean, commit 408915f, module `FunctionalInequalities.KLS.BKL.PoissonIterate.Defect`, declaration `FunctionalInequalities.KLS.LogConcave.PoissonIterate.gradField_gradField_comp_swap`; the repository is private at upload time).

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