Nice tangent, Dan, but X10 is more common than you might think at least around here.
I know of at least a half-dozen homes in my neighborhood with X10, and that's just from what I overheard in conversation over the past 10 years. Once of those homes is on my block (different transformer, though), and another has had X10 for over 30 years. I'm sure there are others that I DON'T know about, too. And I live in a pretty small neighborhood. Considering there are about 1500 homes in my neighborhood, 5 homes would be 1:300 ratio. not 1:5000. considering there are 28 houses on my block, and I KNOW of two with X10, that's a 1:14 ratio.
Over the years, I have seen other random codes show up from time to time, leading me believe that another neighbor was playing around with X10 at some point.
I'd like to get back to the question-at-hand.
There is a command coming in at the same time every day. Since the user says he has no timers or any controller capable of sending a timed signal, AND it seems to be received on the powerline (not via RF), the most logical answer would be that it is coming from outside his home.
SO, I would suspect there is a neighbor with *something* sending a timed signal at the same time every day. Yes, it might be unlikely that a neighbor would have X10, too. However, A1 IS the default code, so that is the one most likely to get interference from someone just buying the stuff and using it without changing the house code.
Now, the OP said that he has been pressing the button trying to annoy a neighbor (if that's the cause), but hasn't seen any reaction. It is possible that the neighbor is sending a more powerful signal than he is sending back, so his weaker signal doesn't make it to the neighbor's system.
Of course, if a neighbor does have X10, changing to a different house code might not completely correct the issue. I'm using a bunch of housecodes (C and D for devices, L for flags, M for macros), for example.
That's one of the shortcomings of the X10 design. It was never expected to be used for complex setups with lots of modules per house, and lots of houses per transformer. Had they expected it, they might have tried to provide more device addresses, to reduce the chances of overlap between homes.