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Write $a=\min (C_1,C_2)>0$, $b=\max (C_1,C_2)$ and $c=C^0>0$. At crystal orientation $\theta$, the conductivity tensor is $Q_{\theta }\mathrm{diag}(a,b)Q_{\theta }^T$. Because the comparison tensor is scalar, its resolvent rotates in the same way:Uniform angular averaging uses $⟨{\cos }^2\theta ⟩=⟨{\sin }^2\theta ⟩=1/2$ and $⟨\sin \theta \cos \theta ⟩=0$. It therefore sends any such rotated diagonal tensor to half its trace times $I$. In particular,Dividing these scalar coefficients in the stated Hashin-Shtrikman conductivity variational principle givesThis averaging calculation uses the uniform orientation model in the question; a nonuniform distribution need not give isotropic effective conductivity.

The lower Hashin-Shtrikman conductivity bounds use a comparison no larger than every local principal conductivity, so $0<c\le a$. The upper variational bound uses $c\ge b$. These opposite directions come from the sign of the contrast $C-cI$ in the polarization functional: positive contrast gives a maximizing lower principle, and negative contrast a minimizing upper principle. The endpoint values are understood as limits when a contrast eigenvalue vanishes. Sincethe strongest lower and upper values in their admissible comparison ranges are, respectively,Thus $C_{l}I⪯C^{\mathrm{eff}}⪯C_{u}I$ in the statistically isotropic polycrystal setting of these Hashin-Shtrikman conductivity bounds.

The self-consistent conductivity approximation chooses the comparison conductivity equal to the predicted conductivity, $c=C_{\mathrm{SC}}=H(c)$. Multiplication by the positive denominator cancels the terms $c(a+b)$ and leaves $2c^2=2ab$. Only the positive root is an admissible conductivity:For a direct proof of its ordering relative to the variational bounds, set $t=\sqrt{b/a}\ge 1$. ThenBoth quantities are nonnegative, proving $C_l\le C_{\mathrm{SC}}\le C_u$, with equality throughout when $a=b$. The self-consistent conductivity approximation is a closure calculation; no unrequested claim of exactness for every spatial arrangement follows just from this fixed-point computation.

A self-consistent conductivity approximation sets the homogeneous comparison effective conductivity equal to the predicted response. For a uniformly oriented two-dimensional polycrystal with principal conductivities $a,b>0$, the comparison formula gives $C_{\mathrm{SC}}^2=ab$, so its positive solution is $\sqrt{ab}$. This is an approximation defined by a fixed-point closure; exactness requires additional geometric or statistical hypotheses.