What would the voltage across the resistor be? What would its polarity be? I don't see anyway for KVL to not be violated.
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An ideal circuit diagram has an associated set of equations. In this particular case, the resistor is irrelevant to the KVL equation which is $$1V = -1V$$
which is nonsense.
Just as one can write inconsistent mathematical equations, one can draw inconsistent ideal circuit diagrams (which are simply a different representation of a system of equations).
Here's another example which is, in fact, the dual of the given circuit:
The KCL equation for this circuit is
$$1A = -1A$$
which is, again, nonsense.
So, there are rules for ideal circuit diagrams including but not limited to
don't place ideal voltage sources in parallel (or short-circuit a voltage source)
don't place ideal current sources in series (or open-circuit a
If you assumed a resistor was in series with each voltage source the resultant voltage across R1 is zero. The series resistor could be nano ohms or tera ohms and the voltage across R1 would still be zero.
This configuration is incorrect. You can not put two different voltages sources in parallel. It violates the Kirchhoff Voltage Law.