I have the answer, in two parts.
Preface: I had all the answers before, from the internet. Google away. You will find information about discovering the source of food, and trapping the outliers with funnels, cones of paper, bits of rotting fruit in glasses and bottles, using Saran wrap, etc. etc. etc.
I knew nothing.
FIRST, the quest of finding their breeding place is paramount. After eliminating all kitchen counter sources, and pouring boiling water down the drains every night for a month, I moved the fridge. I found a lot of very dry congealed fruit juice, but no larva, nothing sufficiently juicy to explain the flies in the house. But more below…
SECOND, kill them where they fly. Forget this Internet business of creating traps with rotting fruit, Saran wrap, and little holes you punched with a pencil. Fuck that, seriously. Get an open tumbler half-full of apple cider vinegar; add a drop of dish soap, swirly around, and you’re done. They will fly in there and drown, because the soap destroyed the surface tension. Seven in one blow, and again, and again, and again. While this problem raged I literally counted hundreds dead each day. Apple cider vinegar, no trap, just break the surface tension and they are dead.
BUT FINALLY … if this doesn’t cure you, after you’ve moved the fridge, keep looking. I’d resigned myself to breathing in the flies forever, until I went looking for the fire extinguisher in the garage for another reason*. It turned out, it was under some cardboard to be recycled, and under that was …
a bag of potatoes. Very, very very old potatoes, probably left there in the garage during the chill of last winter, never to be thought of again. Liquid now, of course. Not even smelling as bad as you’d imagine,. But full of Life.
At least 40 Million, I estimate, in bugs. All inside that one bag, and outside of it too, the primary weight of it. Over time they must have migrated from the garage to the house, where they obligingly died in vinegar glasses, but not at a rate fast enough to keep them out of people’s food, and mouths.
But now we’re done. About a million survived the trip of the potato bag from the garage through the kitchen to to the outdoors, and I welcome them. Come drown and die in my vinegar, one by one, over the next several days. I don’t care. I have discovered your nest and have defeated your army.
*Don’t ask. We’re on top of it now, thanks.