High contrast destroys contrast-independent conditioning

The normal corrector has derivative ${\chi }_1^{\mathrm{\prime }}=c\mathrm{/}{a}_{\kappa }-1$ with $c=({\int }_0^1{a}_{\kappa }^{-1}{)}^{-1}=2\kappa \mathrm{/}(1+\kappa )$; this derivative has zero mean, and its periodic primitive gives constant normal flux $c$. With $\lambda =\kappa$ and $\mathrm{\Lambda }=1$, [F3] supplies the one-dimensional mean-zero corrector; lifting its smooth periodic approximants in the first coordinate gives the required member of $V_{\mathrm{\#}}(Y)$. For each tangential coordinate $j\ge 2$, zero is the corrector: for every smooth periodic $v$, Tonelli [F4] and the one-dimensional FTC in $y_j$ give ${\int }_Y{a}_{\kappa }(y_1){\mathrm{\partial }}_{j}v(y)dy=0$, and density extends this to $V_{\mathrm{\#}}$. Its mean flux is ${\int }_0^1{a}_{\kappa }=(1+\kappa )\mathrm{/}2$. These correctors yield the stated diagonal entries.

Since $0<\kappa \le 1$, the tangential eigenvalue is at least the normal one: $(1+\kappa {)}^2-4\kappa =(1-\kappa {)}^2\ge 0$. The spectral condition number is therefore their ratio, $(1+\kappa {)}^2\mathrm{/}(4\kappa )$, which diverges as $\kappa \downarrow 0$. Every fixed $\kappa$ remains positive and satisfies [F1]; only a uniform-in-contrast condition number fails.