TwoCenterAffineRecovery.lean`

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Recovering both coefficients of the two-center commutator residual

theorem OAI.RegularPointCoordinates.two_center_affine_recovery line 18

Source documentation

Exact normalization and the two published harmonic interior estimates recover the matrix difference and constant term from the SAME residual.

theorem two_center_affine_recovery {K : ℝ} {n : ℕ} {hX : NoncollapsedRCD K n X}
    (hgradient : LocalHarmonicL2GradientInput hX)
    (hmean : LocalHarmonicL2MeanValueInput hX) (hn : 2 ≤ n)
    {κ : ℝ} (hκ : 0 ≤ κ) :
    ∃ Cg Cm : ℝ, 1 ≤ Cg ∧ 1 ≤ Cm ∧
      ∀ (p : X) (R : ℝ), 0 < R → -κ ≤ K * R ^ 2 →
      ∀ (u : Fin n → SobolevSpace (hausdorff n X) hX.quadratic)
        (V : X → Euclidean n) (N : Euclidean n ≃L[ℝ] Euclidean n),
      IsExactMeanNormalization (hausdorff n X) hX.quadratic
        (sobolevColumnMap (hausdorff n X) hX.quadratic u)
        (ball p (R / 2)) measurableSet_ball N →
      (∀ j, IsSobolevHarmonicOn (hausdorff n X) hX.quadratic (ball p R) (u j)) →
…

Mentions: OAI.RegularPointCoordinates.NoncollapsedRCD, OAI.RegularPointCoordinates.LocalHarmonicL2GradientInput, OAI.RegularPointCoordinates.LocalHarmonicL2MeanValueInput, OAI.RegularPointCoordinates.SobolevSpace, OAI.RegularPointCoordinates.hausdorff, OAI.RegularPointCoordinates.Euclidean, OAI.RegularPointCoordinates.IsExactMeanNormalization, OAI.RegularPointCoordinates.sobolevColumnMap, OAI.RegularPointCoordinates.IsSobolevHarmonicOn