That depends on the symmetry of the induced field. Let me give you my take on that paper (who appears to be a draft considering there is at least a minor error and a (?) meaning what, exactly? Do you know if this was ever published and where is the definitive version?)

Well, it's interesting but it is nonetheless amazing that to compute the emf on an open path, the area always pops out.

Well this paper is what i could find on the topic freely accessible on the internet. Flowing sources and searching on the topic shows more work on the topic but most of it is locked behind paywalls (The usual thing with scientific publications). I'm not a university student anymore to have access to those for free. Feel free to dig deeper into it if you want.

It makes sense to me that the geometric origin of the field would be important since that's the single point where nothing happens to the magnetic field lines as the flux changes. The integral of an area just happens to be a good way to capture the fact that a combination of two spatial dimensions affect the result. Since both Faradays law and a wire segment require it they both use an area (And besides its both magnetic induction so you expect something similar to happen). Area integrals pop up a lot in electronics just because how useful they are.

But yeah the question weather there is voltage on the open loop of wire depends on how you look at it. The EMF will balance out with the electric field of charge separation so technically the voltage is zero but there are more electrons on one end than the other. Circuit analysis makes use of this concept for making wire segments in the form of inductors.

What happens if your EMF is ZERO? Doesn't that ring you a bell? Where did you see ZERO before?

Ahh! KIRCHHOFF! He says that "The algebraic sum of all the voltages around any closed loop in a circuit is equal to ZERO"!!!!!!!

Kirchhoff on the other hand does talk about the voltages summing to zero. Notice that it says "all the voltages" so this implies induced EMF too as that's a voltage inside the loop.

Here is where your problem lies. When Kirchhoff says all the voltages, he is not considering any kind of EMF. In Kirchhoff's wonderful world there is absolutely NO EMF whatso-fluxing-ever!

In a circuit mesh schematic a wire has zero length, zero resistance and zero reactance. These wires are immune to all field effects. Building the two resistor circuit with such wires in the real world results in a circuit that must have a circumference of zero, this means it also must have a loop area of zero, zero area means no induced EMF and KVL works.

Philosophical question: if Kirchhoff only applies to circuits that cannot exist, why do we need his stupid theory?

That diagram i fully agree with. Kircchhoffs laws are an abstracted application of Maxwells equations and are meant to be only used on lumped circuit meshes.

You do not only have a problem with Kirchhoff, Maxwell and Lewin. You also have a problem with Georg Cantor.

Between you and me. Forget all you know about circuits. Follow the guidelines I published in a previous message. Study calculus, vector analysis and a good book on electromagnetism. You'd be better off than struggling with theories you do not master.

Well Faradays law says that with no flux change the induced EMF voltage equals zero. There are other voltages possible in circuits that are not coming from induced EMF, batteries also produce voltage that is NOT induced. So saying that the induced EMF is zero does not automatically mean the sum of all voltages is zero as induced EMF is not the only voltage possible in a circuit

Kirchhoffs laws are a circuit analysis tool, not a law that governs how the universe works. Circuit analysis is just an practical application of physics, while physics is a practical application of mathematics.

Physics has no use for Kirchhoffs law since it doesn't deal with anything physical. Maxwells equations and the things that come from them (like for example Faradays law) are completely sufficient to explain what is going on. Okay we did discover even more fundamental quantum effects a layer deeper than Maxwell explains, but Maxwell still works perfectly on a macroscopic level so that's good enough, we can't say its wrong because of that (Just slightly abstracted that's all)

So then why do we even need Kirchhoffs law anyway if it appears to be useless in physics? Heck why do we even need circuit analysis if physics can already describe anything electrical? Lets just use physics instead!

Well.. we could. The problem is that calculating all of this for a real physical circuit involving only a few components would already result in a LOT of math. Engineers regularly deal with circuits that involve 10 to 1000 components, not just 3, this causes the math complexity to skyrocket and it simply becomes unpractical to calculate. Have a look at how hard EM simulations are on computers. They take a ton of memory and take a significant time to compute even on a modern PC.

Turns out engineers often have to calculate the behavior of circuits as part of there work. When they wonder how a RC low pass filter circuit acts at audio frequencies they simply don't care what happens with magnetic and electric fields around there circuit. To solve this problem the science of circuit analysis was created. This cherry picks the effects from physics that matter and condenses them down into a simpler form of math.

The engineer can now chose if he wants to take in account the magnetic fields or not, rather than it simply being part of the process that must be included for everything to function. Our universe breaks down if you suddenly have magnetic fields disappear, circuit mesh models keep working. If the engineer wants to ignore magnetic effects they simply leave them out and the circuit continues to work as if they are not there. However the circuit now behaves as if it is made out of these mythical ideal wires. In most cases this results in identical circuit behavior. If these effects cause a significant difference in behavior(Such as Dr. Lewins circuit) then the engineer must realize this and chose to add them. In this case this is done by adding an equivalent model of the wire in the form of inductor. The inductor model then calls upon Maxwell to only solve that single wire, this is quick to do because we have generalized easybake equations already prepared from Maxwell.

So KVL only interacts with Faradays law when circuit mesh modeling deems it necessary. Its all just part of a elaborate mathematical shortcut that we call "circuit analysis". Engineers use it to calculate stuff faster and you apply it outside of that there is no grantee it will work. It might still work in special cases outside of circuit meshes but that's just a special case, not intended use.