High-contrast homogenization: Provider.Response.RandomAdaptedResponseDefectBridge

Imported from the Apache-2.0 HighContrastHomogenization development at commit 7a13dbcd8d6609264a713373f5c69ceeac870472.

Adjoint response-defect coordinates

The sign congruence in the adjoint recentered block turns the doubled load (-p,r) into the common positive-coordinate load (p,r) inside its quadratic form. This is the coordinate form used by response calibration.

theorem HCPolySupport.HighContrast.Response.profile_adjoint_response_defect_eq_recentered_quadratic {d : ℕ} {P : MeasureTheory.Measure (CoeffSpace d)} [MeasureTheory.IsProbabilityMeasure P] {q : Mat d} (hq : Matrix.PosDef q) {g : Mat d} (hg : IsSkewMat g) {s t : ℤ} (hints : HasFiniteAdaptedMean P q s) (hintt : HasFiniteAdaptedMean P q t) (p r : Vec d) : profileAdjointResponseDefect P hq g hg s t p r = 1 / 2 * (blockVecDot (p, r) (blockMatVecMul (profileRecenteredMean P q g s) (p, r)) - blockVecDot (p, r) (blockMatVecMul (profileRecenteredMean P q g t) (p, r)))

The adjoint response defect is the increment of the primal recentered quadratic form at the positive doubled load.