Prove that in a *nonlinear* circuit, adding resistor between equipotential terminals draws no current
If the non-linear circuit has a steady-state voltage, any resistor to the same voltage can not inject any current and thus no matter what resistance it is, it will be perceived as an open circuit.  But if some external load changes which might affect that voltage, then current will flow and affect the circuit.
Ridiculous Example:
A tiny coin cell at 3.05V with an internal resistance of 100 Ohms has a nonlinear ESR that changes with %SoC, voltage and temperature..  If you attach a 100A 3.05V power supply, there is no change to the battery voltage.  However, an external load voltage would be affected by this connection.
Practical interpretation.
Say you had a portable nonlinear circuit and wanted to find the average current load without disconnecting the batteries.  Using a programmable power supply with 4 digit V,I,P displays (like the one I just got) Set the voltage lower than what you expect , connect and turn on. Vbat is display and I=0. Now set V to the same voltage
and notice current is still 0 mA.
Cool.
Now raise the voltage 1% ( typical load regulation error) I had 3V from two AAA inside a 3 channel wireless digital thermometer) So I raised in 10 mV increments to verify battery charge raised current momentarily then dropped to read the external supply current where I assume the battery current was now 0.  I could also estimate mAh and ESR of batteries by jogging voltage and computing dV/dI and C= I*dt/dV for the asymptotic decay time after raising the voltage.  Now with my load current at 5 mA s.s.,  I could estimate if the 3 mos battery life met the spec of good batteries or not. I was getting only 2.4V and LCD was getting dim.  I had just charged the alkaline primary batteries to 3V with a digital CC limit of 100 mA and then decided to perform this experiment, while watching TV and thinking about this question.   (It's a Hanmatek 30V 3A)  n.b. charging primary cells is possible but far less efficient than secondary cells. The only risk is corrosion leakage after a long time if using safe current limit.