High-contrast homogenization: Provider.Transport.WindowTerminalNormalization #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The terminal mean, its comparison, and the bridged normalization #
The terminal clause of e.source.adapted.bound compares the annealed block
of the terminal cell with the reference block from both sides,
b_{\mathbf r}^{-1}đ_* †F_0 †b_{\mathbf r}đ with
b_{\mathbf r} = B_{\mathbf r}\E[Y_P]
(e.two.grid.source.normalization), and reads the resulting
normalization Î(F_0;đ) †Îș_đb_{\mathbf r} and its bridged form
Î(F;đ) †Îș_đb_{\mathbf r}/(1-η).
The upper comparison is the primal cell bound of the window integrated: the
expectation is monotone and the multiplier's mean is the printed factor. The
lower comparison needs no operator Jensen once the sharp involution is
available. The sharp reverses the Loewner order and is homogeneous of degree
minus one, so the upper comparison gives b_{\mathbf r}^{-1}đ_* †F_0^âŻ; and the
annealed primal-adjoint order gives F_0^⯠†F_0. The
two together are the printed lower bound, and they also give positivity of
F_0 a second time.
The normalization is then a change of reference block: the reference ratio
Îș_đ = Î(đ_*;đ) is by definition the cost of
measuring đ against its own dual, and the lower comparison converts that into
the cost of measuring đ against F_0. One more change of reference, by the
lower bridge, gives the bridged form; the upper bridge is not used.
The agreement of the primal and adjoint normalizations is an identity, not an estimate: the reflection carries the defining set of one scalar size onto the defining set of the other.
Strict positivity of the printed constant #
The witness eccentricity đą_{\mathbf q} = (|m||m^{-1}|)^{1/2} of a positive
witness is positive in positive dimension: a positive definite matrix is a unit,
hence nonzero, and so is its inverse.
The boundary constant is positive in positive dimension, at a positive dimensional constant and below the critical growth exponent.
A positive dilation of a positive definite block is positive definite.
The adjoint normalization is an identity #
The two scalar normalizations agree. The reflection commutes with scalar dilation and preserves the Loewner order in both directions, so the two scalar sizes are infima of the same set of reals.