AP1 had corroded on the bottom of the circuit board. This tiny detail was repaired by resoldering the board (something for which I definitely needed help-- and I happily recieved.) That and a bit of wiring verification and now the horn, happily, operates through the AP1. I don't fully understand how it was working at all, but it is moot.
The relays are operating as expected.
The horn is operating as expected.
So my saga is now over, except I will be buying a new AP1 (or 2).
Of interest to all of you, the oscilliscope confirmed that there is no spike coming back into the system from the horn operation.
On to the next project....driving lights...