High-contrast homogenization: #

Provider.PolynomialHomogenization.Root.EnergyPrice.GaugeFrameEccentricity

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

The gauge frame factor is a power of the witness eccentricity #

The frozen boundaryEnergy is pinned in the physical frame (hsNormSq U s (fun x ↦ matVecMul (matSqrt (symmPart abar)) (g₀.grad x)), 168:68-74), while the law-free cube price of HCPoly.Provider.PolynomialHomogenization.Root.EnergyPrice.GaugeCubeEnergyPriceWitness lives in the gauge frame on matImage (matSqrt (symmPart abar))⁻¹ U. The change of variables is the hsNormSq_matImage_le, whose constant is

hsAffineFactor L⁻¹ s N = max (|det L⁻¹| ^ (-(2s)/d)) (|det L⁻¹| · N ^ (d + 2s)).

This file proves that this constant, multiplied by the gauge cube side, collapses to a dimension-only constant times a power of witnessEccentricity (symmPart abar):

hsAffineFactor L⁻¹ s ‖L‖ · ell ^ (2s)
  ≤ d ^ (d/2) · 2 ^ (2s) · witnessEccentricity (symmPart abar) ^ (d + 2s)

whenever ell ≤ 2 ‖L⁻¹‖, which the inner-ellipsoid normalization premise U ⊆ ellipsoid abar 1 supplies (the gauge image of ellipsoid abar 1 is the ball of radius ‖L⁻¹‖, because ellipsoid is normalized by specBound (symmPart abar)⁻¹).

So the exponent is p = d + 2s, and no factor of ‖L‖ alone survives: the absolute scale of abar cancels between the frame distortion (‖L‖ ^ 2s) and the gauge cube side (ell ^ 2s ≤ (2‖L⁻¹‖) ^ 2s). This is paid by the eccentricity fold.