From this statement
I gather that the no start is the starter motor won't engage?
First, the easy part. There is no designed in low voltage cut off / interlock in the starter motor circuit. .
With absolutely no load on a 12 v lead acid battery (needs to be disconnected from the car because of the parasitic loads), a battery terminal voltage of 12.9 volts indicates 100% charged and a battery voltage of 11.4 volts indicates totally discharged and perhaps a candidate for the recycler. Close to the top, this web page has a nice graphic showing charge versus voltage.
https://batteryuniversity.com/learn/...tate_of_charge
Your 12.4 - 12.6 volts says that your battery is running at about 55% charge at best so not in great shape (I am assuming you have a lead acid battery). If your alternator is running at a nice 14 volts or a nudge higher I would expect that a good battery should be showing closer to 12.9 volts unloaded - assuming you are using a reasonable quality multimeter to measure voltage, not the dash gauge.
In order to get the starter motor to start, the battery has to have a voltage high enough to get the starter solenoid to engage which closes the main contacts on the starter motor causing it to operate. I would expect that 12.4 volts should easily be good enough to cause the solenoid to engage and that you should be getting starter action. Why there is a cut-off between 12.6 v and 12.4 v is a mystery.
That said, I suggest two tests. The first and easiest one is connect a voltmeter to the convenient jump start terminal in the engine fuse & relay box. Have someone try to crank the starter. If the starter does not start and the voltage at the jump start terminal goes in the toilet, then you likely have a bad connection at the battery post or the battery has an internal problem. I had a random starting problem after I got my NSXr and I discovered that the battery cable clamps had been stretched by somebody over tightening them and I could lift the clamp right off the post without loosening anything. If the voltage at the jump start terminal stays above 12v but the starter does not engage the battery and connections are probably OK and its time for part 2. Insert a little jumper in the connection to the starter solenoid that will allow you to measure the voltage at the solenoid terminal (this can be a bit of a hassle). Measure this voltage while you have somebody try to crank the starter. If the voltage stays above 12v and the starter does not engage then you have some kind of internal problem with the starter. If the voltage takes a nose dive when you attempt to start the car then you have an upstream problem in the solenoid circuit and the likely candidate would be the ignition switch which can be intermittent before they fail completely