List of symptoms:
1. Car will not start. The key indicator (the "ding ding" sound when you insert the key) sounds low pitched, sad, as if the voltage is low.
2. Measuring the battery with a multimeter shows 12.4 volts, basically a good charge.
2b. When I first put the leads on, the voltage was 4.5 volts. Then I removed the leads and put them back, and got 12.4 V. WTF?
3. It is possible to jump start the car, at which point it runs fine until the engine is turned off.
4. The positive battery terminal connector is very badly corroded (copper is green).
So my first theory, that the battery is drained, is incorrect, since the voltage is high (and yes, I did sanity-check my multimeter)
My second theory, that there is a bad terminal connection, is also incorrect, since the car is able to start when jump-started (and a car should not run at all if the battery connection is bad, since the battery is part of the overall circuit).
My third theory, that there is a problem with the starter motor, is also wrong, since that doesn't explain the weird sounding indicator beep, and the car, as I stated, will start just fine if jump-started.
I don't know what the hell is going on. The only theory I have left is that the resistance of the corroded terminal connector is through the roof, and I can't draw enough current to turn the starter. But the "ding ding" indicator doesn't require that much current, and yet even that sounds like it's not getting the power it needs. And besides, if the terminal resistance was super high, that terminal should be getting ultra-hot, and that's not happening.
Any ideas? I'm going to pull the terminal connector off and clean it with a wire brush just to see what happens.