= Phase polarization comparison bound
{title2=$W^{\mathrm{eff}}(E)\le\tfrac12 E:C^0E+\overline\tau:E-\tfrac12\sum_{r,s}\tau^r:A_{rs}\tau^s-\sum_r c_rF_{r*}(\tau^r)$}
Take a positive comparison <elastic stiffness tensor> $C^0$, excess energies $F_r(e)=W_r(e)-e:C^0e/2$, and a phasewise constant <elastic polarization>. The <lower conjugate> inequality majorizes the local energy by a quadratic-plus-linear function. Minimizing that function over compatible <strains> gives the displayed bound, where $A_{rs}=\int\int(\chi_r-c_r)\Gamma(\chi_s-c_s)$. Stationarity in $\tau^r$ gives $c_re^r+\sum_sA_{rs}\tau^s=c_rE$, with $e^r=DF_{r*}(\tau^r)$. Without this stationarity, dual <strains> must not be identified with the comparison solution's phase means.
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