Baffled by 3-way switch wiring

Question

I am the new owner of an old, WWII-era home. The previous owner rented the property out for many years, and upkeep seems to have always gone to the lowest bidder. Among other things this means we have encountered many "interesting" examples of electrical maintenance work. For example, since the house is old and does not have grounded wiring, every three-pronged outlet in the house was wired up with what I eventually learned is called a "bootleg ground". (I removed those.)

Currently I am trying to fix a non-functioning three-way switch in the kitchen. I have looked at what feels like every diagram of three-way switch wiring on the entire internet and still cannot make heads or tails of how ours ought to be wired up. The basic situation is two wall switches on either side of a track light. There is standard 12/3 Romex running from one switch to the other. I have verified using a the continuity tester on my multimeter that each wire in the Romex is connected to the other outlet as expected.

The continuity tester indicates that the older, braided wire shown in the photo with the white arrow connects to the hot rail on the track light. Continuity tester doesn't give any more beeps if I check any of the Romex wires against the ground, neutral or hot rails of the track light. If I connect the black wire to source, the white wire becomes hot relative to ground. If I then connect the white wire and braided wires, the lights turn on.

I'm confused for several reasons. If white becomes hot when I connect black to source, shouldn't there be continuity between white and black? If so I'm not able to measure it. Also what is the role of the braided cable here? This wiring set up seems unlike anything I have found online.

Other outlet. The bottom four wires are the Romex coming from the first outlet. The two wire nutted bundles are a separate circuit (there is an outlet on the other side of the wall for the dining room). I have no idea what the black wire in the upper left is for. Seems to be part of the other circuit as well but I have verified that it is dead.

Answer 1

Electricity travels in loops

In utility electrical, every hot must have a return wire partnered with it. They must be in the same cable. All the currents in each cable must be equal: the current coming down as hot must return up as neutral. With 3 or more wires, the currents added together must equal/cancel out.)

When a neutral is disconnected, it will light up as hot. Electrons are trying to return, and they can't. So this "floats" the neutral up to 120V. This is a really good reason not to bootleg ground - nor use 3-prong dryer/range outlets, which "legally" bootleg ground.

Get some colored tape

3-way circuits will make you ape crazy, because the physical topology of your wiring, and the equal-currents rule, forces some strange routings sometimes. Here's what is happening electrically, and note each wire has a different purpose, and in this diagram, each purpose has its own color.

There is no reason to distinguish travelers from each other; they are interchangeable.

Of course this does not match either your wiring topology or your colors. I would make it so using colored tape or shrink wrap.

So assuming your /3 cable rings out alright...

Figure out how you want to wire your 3-way circuit. I would take always-hot down to the remote 3-way and bring two messengers back. Black is a good always-hot color, so I would tape the red and white wires yellow.

Your neutral from supply is already tied to neutral up to the light. Done.

Your hot from supply is easily tied to the /3 cable black.

On both 3-way switches, the yellow wires go to the traveler (brass) screws.

Each box has only 1 wire left, that goes to the common (black) terminal.