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Write $\Delta =a_1-a_2<0$, and let $q=t\Delta \lambda {\chi }_1$. The inverse contrast in the Hashin-Shtrikman conductivity variational principle is interpreted only in phase 1: the admissible $q$ vanishes in phase 2, where $a-a_{2}I=0$. Thus

For the nonlocal term define $K(x,x^{\prime })={\nabla }_x{\nabla }_{x^{\prime }}g(x,x^{\prime })$. The given Green representation impliesThe two-point phase correlation function therefore givesBecause the unweighted double integral of $K$ is zero, subtract $p_1^2$ inside the bracket. Set $h(r)=p_{11}(r)-p_1^2$; it vanishes for $r>\rho$, so only the microscopic near-diagonal kernel contributes. Its contact part isAt coincident points ${\chi }_1^2={\chi }_1$, hence $h(0)=p_1-p_1^2=p_1{p}_2$. The ordinary dipole part has zero angular integral by the stipulated spherical-shell identity. Since $h$ is radial, its weighting preserves this cancellation:Consequently, in the macroscopic bulk regime implicit in the supplied local Green approximation,andThis is the dipole Green tensor angular cancellation: the constant background correlation disappears and the connected contact value supplies $p_1{p}_2$.

Insert all three averaged terms in the variational upper inequality and normalize by $|\Omega ||\lambda {|}^2$. For the bulk effective conductivity one obtainsThe kernel in the question has an approximation sign. At fixed finite size, spheres around points near $\partial \Omega$ are truncated and the radial cancellation is not an exact finite-body identity. Their contribution and the regular boundary part disappear in the stated microscopic-to-macroscopic limit. The calculation derives the intended homogenized bound, rather than silently treating that Green approximation as exact for arbitrary finite $\rho$.

Let $C=1+\Delta p_2/(3a_2)$. Since $a_1>0$, $\Delta >-a_2$, so $C>0$. Also $\Delta p_1<0$ when both phases are present. Completing the square in the trial upper bound givesThus the tightest upper bound is obtained by minimizing at $t=1/C$, not maximizing. It isPure-phase cases follow directly or by continuity. These Hashin-Shtrikman bounds for conductivity are volume-fraction bounds on the statistically isotropic homogenized material; the full variational framework is discussed in Chapter 23 of www.math.utah.edu/~milton/TheoryCompositesNOPRINT.pdf .