SS-
You and GD have been going around and around here, so just so we all know we're together on this, let me try to explain. Again, a language barrier has hindered us, but that's not StudioStriver's fault, it's just a problem we have to overcome to be able to speak precisely about these things. But let me try to lay this out for SS' benefit.
Based on SS testing and descriptions, JohnH has apparently correctly identified the problem with the neck coil split switch. We can talk SS through that fix with a soldering iron if he needs us to do so.
As far as we can tell, the neck coil split is the only problem. In all other respects, the guitar seems to be working as intended.
Now, a word about HB coils in series and in parallel. Greekdude was trying to draw the distinction, correctly, between the wiring of the two coils within a HB pickup, as opposed to the wiring of two separate pickups together, neck and bridge.
Virtually every HB guitar,from the Les Paul through to the modern metal-shredder, has the 2 coils of each HB wired together
in series When the guitar is set to the center position, we get both HBs together. The neck HB (with its 2 coils in series) is connected
in parallel to the bridge HB (with its 2 coils in series).
This is the way SS scheme works when the toggle switch is set to "parallel" mode, just as with most any other HH guitar. The three-way toggle gives:
1) neck series HB
2) neck series HB
in parallel with bridge series HB
3) Bridge series HB
(Note that we use a "+" to designate a parallel connection and a "*" to mean a series connection)
Switch to "series" mode or to "series OOP", and the 3-way pickup selector is not operational- you have N HB * B HB, or the same OOP in the third switch position.
As previously set up, your guitar split coils in the center toggle position, now you have the "full HBs" in that setting, but you can individually split either pickup to single coil using the push/pulls. To maintain hum-cancelling on this scheme, we wired the coil splits differently, that's why the two pickups are not wired the same for the coil splitting.
Hopefully, that help explain this a bit better (or did I just confuse it more?)
The confusion was thus between discussing the internal wiring of the individual HBS (both HB coils in this scheme are always in series, unless they are split to a single coil) versus discussing the wiring of the neck (series) HB to the bridge (series) HB in the center position of the toggle switch- the two HB pickups, neck and bridge, are connected together in parallel, just like on most HH guitars- but this usual wiring can be changed to series or series OOP with the 12-lug switch.