Two-current model with unequal magnetic layers ($R_{AP}-R_P=(a_1-b_1)(a_2-b_2)/\Sigma$)

Let the first layer have channel resistances $a_1,b_1>0$ and the second $a_2,b_2>0$, with a common spacer $r\ge 0$. Define $\Sigma =a_1+b_1+a_2+b_2+2r$. The same circuit rules giveSubtracting their numerators gives $(a_1-b_1)(a_2-b_2)$. The sign therefore depends on whether the two layers favor the same global spin channel. With $(a_1,b_1)=(1,4)\Omega$, $(a_2,b_2)=(4,1)\Omega$ and $r=1\Omega$, the values reverse: $R_P=3.00\Omega$, $R_{AP}=2.25\Omega$. The model's ratio $m$ is then $-1/4$. Positive magnetoresistance was a consequence of the identical-layer assumptions, not a theorem about every two-channel network.