High-contrast homogenization: Provider.SourceControl.ReferenceIntermediate #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The reference intermediate κ_𝐄 ≤ 1 + 6(Θ - 1) ≤ 6 Π #
The factor-six reference-block comparison states the chain
κ_𝐄 ≤ 1 + 6(Θ - 1) ≤ 6 Π for a reference block dominating its own sharp. The
endpoint κ_𝐄 ≤ 6 Π is proved in
HCPoly.Provider.Initialization.ReferenceComparison; this file proves the two
remaining links, at the intrinsic contrast Θ = refContrast 𝐄.
The printed derivation of the first link routes through the display preceding
(2.98) of Armstrong–Kuusi, which fails at large contrast (a counterexample exists at
contrast nine, where both of its displays are exceeded). The route
taken here is different, and elementary. Write (σ, σ_*, k) for the Schur data
of 𝐄 and T = σ_*⁻¹ for its lower-right block, so that at (x, y)
- the quadratic form of
𝐄is⟨σ x, x⟩ + ⟨T (y - k x), y - k x⟩; - the quadratic form of
𝐄^♯is⟨σ_* x, x⟩ + ⟨σ⁻¹ (y + kᵗ x), y + kᵗ x⟩.
Fix a skew h with σ + mᵗ T m ≤ t σ_*, put m = k - h and
K = k + kᵗ = m + mᵗ, and write w = y + kᵗ x, so that y - k x = w - K x.
Exactly as in the endpoint proof, the ordering 𝐄^♯ ≤ 𝐄 gives σ_* ≤ σ, hence
1 ≤ t, and then the three normalized bounds
σ ≤ t σ_*, Kᵗ T K ≤ 4(t - 1) σ_*, T ≤ t σ⁻¹.
The cross term ⟨T w, K x⟩ is not discarded by Young's inequality here. Doing
so — the endpoint's route — bounds the reading of the w coordinate by 6 t,
and the target 1 + 6(t - 1) = 6t - 5 is strictly below that; no admissible
Young weight repairs it below t = 5, since a weight 1 + ε on ⟨T w, w⟩ costs
4(1 + 1/ε)(t-1) on ⟨σ_* x, x⟩ and the two constraints
(1+ε) t ≤ 6t - 5 and t + 4(1 + 1/ε)(t-1) ≤ 6t - 5 are compatible only for
ε ≥ 4, hence only for t ≥ 5. The cross term is bounded instead by the sharp
ordering itself, read at the rescaled vector whose second Schur coordinate is
-5 w:
⟨σ_* x, x⟩ + 25 ⟨σ⁻¹ w, w⟩ ≤ ⟨σ x, x⟩ + 25 ⟨T w, w⟩ + 10 ⟨T w, K x⟩ + ⟨T K x, K x⟩.
The weight five is the unique one at which the resulting linear combination of
the four bounds reproduces 1 + 6(t - 1) on both coordinates at once: with
weight 1/γ the two readings are t + (t-1)/γ and (1+γ)(5t - 4) - γ, and
γ = 1/5 is the only value making both equal 6t - 5. The comparison is
therefore exact, with no slack anywhere; at t = 1 it degenerates to
𝐄 ≤ 𝐄^♯, which is the sharp ordering read backwards.
The second link is the printed chain for Π: at the Λ_0-realizing skew,
σ + mᵗ T m ≤ Λ_0 · I ≤ Λ_0 |σ_*⁻¹| σ_* = Π σ_*, so Π is admissible for the
infimum defining Θ, whence Θ ≤ Π.
There are no definitions in this file.
The intrinsic contrast as an infimum #
The intrinsic contrast is realized in Loewner form. On a symmetric
positive definite doubled block some skew matrix attains the infimum defining
Θ, as a Loewner bound against σ_*.
The Π chain. At the Λ_0-realizing skew the skew-corrected Schur form
is below Π σ_*: this is σ + mᵗ σ_*⁻¹ m ≤ Λ_0 I ≤ Λ_0 |σ_*⁻¹| σ_*.
The intrinsic contrast is below the aspect ratio, Θ ≤ Π: the second
link of the factor-six reference-block comparison. The aspect ratio is an admissible
Loewner scaling for the infimum defining the contrast.
The Schur ordering #
The Schur ordering from the sharp ordering. A symmetric positive
doubled block dominating its own sharp has ordered symmetric Schur blocks,
σ_* ≤ σ: the lower-right blocks of the two Schur forms are σ⁻¹ ≤ σ_*⁻¹.
An admissible scaling is at least one under the Schur ordering: from
σ_* ≤ σ ≤ σ + mᵗ T m ≤ t σ_* and the positivity of σ_*.
The intermediate comparison #
The reference intermediate, at an arbitrary admissible scaling. For a
symmetric positive doubled block dominating its own sharp, every skew matrix
whose corrected Schur form is below t σ_* bounds the reference ratio by
1 + 6(t - 1).
The reference intermediate κ_𝐄 ≤ 1 + 6(Θ - 1) of the factor-six
reference-block comparison, for a symmetric positive doubled block dominating
its own sharp.
The intrinsic contrast is at least one under the sharp ordering: the first conjunct of the printed reference row.
The full printed chain κ_𝐄 ≤ 1 + 6(Θ - 1) ≤ 6 Π.
The forms consumed under the coarse ellipticity assumption #
Θ ≤ Π at a reference block of e.coarse.ellipticity.
1 ≤ Θ at a reference block of e.coarse.ellipticity.
The reference intermediate under the coarse ellipticity assumption. At a
probability law the reference block of e.coarse.ellipticity dominates its
own sharp, so the intermediate applies to it.
The printed tail 1 + 6(Θ - 1) ≤ 6 Π at a reference block of
e.coarse.ellipticity.