Wow, so many things wrong with... ok, let me guess. You have some experience in low voltage DC wiring.
Did you ever follow the War of the Currents? Edison vs Tesla, DC vs AC, and all that stuff? Did you follow the technical side of how Tesla won?
Tesla relied on Electro-Magnetic Fields, or the radiated energy coming off any electrical wire. Movement of the electrons causes a magnet-like effect. On DC power, the EMFs are static/unchanging - as innocuous as a refrigerator magnet or the Earth's magnetic field. However, with AC power, the magnetism reverses every 1/120 of a second -
this force is considerable. In fact, this is how Tesla won the war of the currents - this great force could be used in a transformer to do entirely passive power conversion! Edison needed to spin a shaft to do the same thing, to create the animated moving fields that allowed power conversion. AC power animates itself!
However that also means all aspects of AC power design must contend with this forceful, animated magnetic field. So it's NOT simple-like-electronics at all.
There are many rules which follow from AC's violent magnetism, but the first rule is that currents must be equal and opposite in each cable or conduit. That way, these strong magnetic fields
cancel each other out.
There's a separate issue: we actually wire our current return wires. These are called neutral wires. since we don't have a big PCB groundplane or the chassis of an automobile to return current on, we can't just ignore the capacity of those neutral wires. They are the same size as the hot wires. They can't return any more currentthan the hot wires! As a cost-cutting measure, we don't fuse neutrals. If 2 hots returned current on the same neutral, they would overload it,
and nothing would detect this. So neutrals must only return current for their partner hot(s) - no borrowing neutrals from other circuits.
Grounds, on the other hand, only carry current during a fault condition, so promiscuous grounds are fine.
One last thing, we have protective devices such as GFCI and AFCI that depend on monitoring (and comparing) the activity on both hot and neutral wire. This depends on the "currents must be equal" rule, because under normal conditions all current that goes out a protected hot must come back a protected neutral, so the device can see it!
To accomplish this electrically all I need to do is pigtail into the hot wire of one of outlets on the load side of the AFCI outlet.
So you want to grab "just the hot" off the load side of the AFCI, and not the neutral. That ticks all the boxes that I discussed above: unequal currents, neutral being returned via a different path (which might overload it), and protective devices seeing unequal currents.
Is it to code to just clip off the unused neutral and ground wires at both ends where the sheath ends, or do I need to wire them in normally (they would be redundant)?
You can't do the first option because then, you would not be able to do the mandatory third option: leave them intact but cap them off.
Because of the above issues, you are not allowed to "parallel" i.e. have 2 wires do the job of 1 wire.
Doing the latter would bring the allowable conductor volume just up to the limit and make stuffing all of the wires into the box even more of a struggle.
Well, as discussed elsewhere you need a different approach altogether. But as an answer to the "doing work to Code would fill my box", the Code answer is use a box extension.
Could you elaborate on the heating?
Well, anytime you have current following a circular path (i.e. not returning on the same cable from whence it came), you create the core of a transformer, with the energies found therein. The least of the effects is eddy current heating, or inductive heating. This is not inconsequential - 10 ampere-turns can operate a reed switch, after all!
Here is a rule-of-thumb that can help a lot: The physical layout of cables and conduits should always be a "tree topology". You notice tree branches split many places, but they never rejoin each other into a loop. It should be that same way for all your wiring; except; grounds are allowed to loop/mesh/web because service current never travels on them.
And point me to the code section with the conditions?
That is all over the codebook. Once you know what to look for, the "don't loop" / currents must be equal rule shows up in dozens of places.
Right, I should have mentioned that I'm also running a new circuit for the living room, which shares a wall w/the kitchen.
The ceiling fan will be on that circuit.
Well, that's easy, then. Wherever you're getting hot from, get neutral from there too. It's really that simple. (and certainly ground, since cable these days is sold with ground wires already in it).
You said something about wanting AFCI protection for the fan. Did you mean GFCI? Totally different thing. I don't see a strong reason to have either one, honestly. AFCI, given what it is, seems very appropriate for a living room circuit,
double especially where older wiring is involved... but as said, it is not anything like GFCI, and so you'd provision it quite differently. AFCI protection is best done at the breaker, since AFCI mostly protects in-wall wiring from fire.
GFCI on a hardwired installed appliance doesn't make a lot of sense if it's grounded.
I believe it's the same principle, which I don't understand, for why the building department said I couldn't add a ground to my old 2-wire kitchen circuit by running a ground wire for it along with the new circuit wires, which I run in conduit on my roof.
As long as that's coming from the same service panel I don't see a problem.
Either you got confused and were really asking them to steal neutral (nope), or *they* were confused by the Retrofit Ground rules. These rules are quite new (NEC 2014, but NEC 2014 got adopted as late as 2018 in many states because it was controversial; it had crushing requirements for GFCI/AFCI which drove up the cost of new construction considerably, so many states spent *years* arguing about it). As such, your request may have predated your state's adoption of NEC 2014, or it was just Super New at the time you asked and the inspectors had no experience with it.
It can't be run with the old circuit wiring because it's buried under the tar and gravel roof (under flashing).
Retrofit ground rules don't require it be anywhere near the old wiring. It can go to anywhere that has metal conduit or Grounding Electrode conductor going back to the same panel, or anywhere that has a large enough ground wire going to the same panel. If the wiring comes in through the roof, the retrofit ground can go down into the basement to the water heater and back its #10 ground wire.
