High-contrast homogenization: #
Provider.PolynomialHomogenization.Root.Dirichlet.WitnessErrorGeometricFactor
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The witness error bound at a supplied generation factor #
HCPoly.Provider.PolynomialHomogenization.Root.Dirichlet.WitnessErrorEccentricityBound
takes 0 ≤ j — the witness cube
index nonnegative — as its one geometric binder, to kill the 3 ^ (−r j) factor
of the λ-route.
That hypothesis is false in general. The frozen clause carries
U ⊆ ellipsoid abar 1, and
ellipsoid abar r = {x | vecDot x ((symmPart abar)⁻¹ x) ≤ specBound (symmPart abar)⁻¹ * r ^ 2}
is, in gauge coordinates x = matSqrt (symmPart abar) y, exactly the ball of
radius α r with α = √(specBound (symmPart abar)⁻¹). So the witness cube of
side 3 ^ j, translated, sits inside the ball of radius α, forcing
3 ^ j * √d ≤ 2 α. Taking abar = 1 gives α = 1 and 3 ^ j ≤ 2 / √d, which
is < 1 for every d ≥ 5: j < 0 there, necessarily.
The correct hypothesis is therefore not a sign but a bound: the factor
3 ^ (−r j) must be supplied as below a constant. It is law-free in the
frozen data: the same two-sided geometry gives 3 ^ j ≥ 2 α / (3 √d) from the inner-ellipsoid
normalization inner premise ellipsoid abar (1/(3√d)) ⊆ U, and the ball sandwich's inner
radius gives α ≥ ρ / 2, so 3 ^ (−r j) ≤ (ρ / (3 √d)) ^ (−r) — a function of
(d, ρ, r) only, and ρ is one of C₀ s₀ ρ Rad's own parameters.
This module restates the bound against that supplied factor, so the consumer may feed whichever geometric estimate it holds. The two-sided cube/ball geometry itself is not built here and is named in the residue.
21c, in its usable form. The λ-route's law-level tail is below a law-free constant — the supplied generation factor times the frame fold constant — multiplied by a fixed power of the witness eccentricity.