I decided to build a new machine - here's what I cooked up: 
EVGA X299 FTW K
Nvidia GEForce RTX 2070 Super
Intel Core i7-9800X 3.8 GHz
Coolermaster ML240L RGB
Corsair DDR4 4133 4x8GB RAM in slots 1,3,5,7 per the sticker that came on the mother board. 
Corsair MP400 2TB NVMe PCIe M.2 SSD
Corsair HX1200 Full Modular PS
Antec P101 case
Pioneer Blue Ray Writer
 
I built the machine and booted it up. Worked fine. Checked everything in BIOS and everything lined up - the right amount of memory and everything. I installed Windows 10 from a USB stick / the internet. I then ran the included CD and updated all of the drivers. Rebooted a few times in the process, no problems. Played around with the RGB thing, colored fans on the front look cool. I downloaded and installed AutoCAD, played around with it a bit. Then installed Google remote Desktop. I'm going to run this machine remotely, from my other computer, so I gave it a name and logged in from the other room. Worked fine. Shut it down for the night. This morning, I tied the cables in a few places but I'm not super fastidious about that, I just got them tidy enough to get the back cover on. Put the other cover on, and brought it into my office. Plugged in Ethernet and power only, booted up. Doesn't show up on Google remote desktop. Well, it's an untested Ethernet cable, so I swapped that. Noticed that it beeped five times, made a note of it - turns out that means it has no KVM, no problem. But still nothing on remote desktop.
 
Remedial measures taken, with tests in between:
1. took the covers off of both sides, checked that the cable ties weren't too tight then removed them completely. At this point the machine was running with the RGB cycling on the mother board and on the CPU cooler and fans.
....Shut off the computer.
 
2. swapped memory 1<->3 and 5<->7. Tested, the motherboard quickly cycles through POST numbers, displays "36", pauses, displays "22" and then shuts down! 
3. Removed 24-pin power cable, rerouted it and plugged it back in. Tested, results the same.
4. removed both CPU power cables, checked them and reattached. 
5. Removed power cables to Nvidia card, rerouted and plugged them back in. 
6. Removed 12v connectors to Blue Ray and fans RGB. Tested, same results. 
7. Swapped memory 1<->5 and 3<->7. Tested, same results. 
8. moved computer back into the Den, where it had worked, plugged into the same monitor (4K TV via HDMI), mouse, and even the same Ethernet cable. Tested, Same results. 
 
What's my move? 
 
 

Thank you Arestavo. I'm an old fart and I've built machines (computers, motorcycles, etc) over the last 40 years, so I am sure there is no problem with the CPU and its heat sink. I even took pictures during the installation process, so I have photographic evidence that everything in there is hunky-dory! 
 
Good point about the memory, and thanks for the pointers on troubleshooting method. I started with that. I had read the codes in the install guide and here on the site, but I'm not familiar with them by my own experience, so that's why I'm here. 
 
So, further testing results, test no. 9: With only one memory stick in socket #1 (rechecked the manual to make sure I have the right slot) it runs through a bunch of post codes, then reads "36" and shuts down. As you said, 36 is "CPU Post Memory Initialization. System Management Mode (SMM) initialization". I tried all four memory sticks, same result each time. Then I tried: 
 
10. remove power supply, reinstall, reseating all power connectors on P.S. Same test result. 
11. OK, just because you said, I removed the CPU and checked everything. I took more photos, and visually inspected using my biggest magnifying glass. No visible evidence of damage. The heat sink grease had made a bit of a mess around the edges of the CPU, not too bad, no grease getting close the socket anywhere, but I cleaned it up anyway. Reinstalled CPU and cooler. Same test results. 
12. What else is there to try? Well, there's the graphics card. I removed it...
13. The M.2 drive is under the graphics card, so I removed and reinstalled it, then reinstalled the graphics card. Same test results. 
 
So... next steps? 
 

Thanks again A. Modular power supply: test 10. Graphics card PCIE: tests 5 and 12. 
 
I have my old machine I can swap parts with - except the CPU. I'll swap memory and graphics card. Thanks again. 
 
Anyone else have suggestions? 
 
Thanks 
TZF 
 

OK, thanks again. I took a couple days off, but here are the results: no change. 
The BIOS switch has two positions; BIOS1 is the removable one.
14: When I switch to BIOS2, the result is the same (blinks through a bunch of codes, then shuts down; it takes about the same amount of time, maybe 3 seconds), but the last POST code is 6d: North Bridge DXE initialization. 
My old machine turns out to be a DDR3 machine, so no memory swap between machines. 
15: Swap graphics card: my old machine has an NVIDIA Quadro K1200, which works fine in this old machine, but makes no difference in the new machine.  
I downloaded the photos of the CPU socket and zoomed in all the way. It looks good. I don't see what else I can do... ritual seppuku? 
 
Thanks 
TZF 

Well, removing and reinstalling the motherboard is not a bad idea. When I was installing everything the first time, a piece of metal came loose from the case - I determined it was probably a piece of a rivet, about 2mm diameter. So maybe there is something in there that came loose and is wreaking havoc under the motherboard. 
 
(four hours later...) 
O...........
M..........
G!!!!!!!!!
 
Found the problem...
 
I removed the motherboard, looked carefully for anything wrong. Then, proper troubleshooting procedure requires us to plug in the minimum number of components and cables, and test, and then put one more in, and test, etc. So I connected up the 24-pin power and the two CPU power cables, and the CPU cooler pump. And tested it. And it booted! Of course it complains that there is no KVM, but it didn't shut itself down. So I turned it off, and plugged in the front panel USB 2.0, and it still worked, and then the front panel USB 3.0, and it still worked, until I got to the front panel power, reset, and HDD LED, and THEN it went back to shutting itself down. It turned out to be that the power switch is SHORTED. So the motherboard was seeing that and she says to herself "SOMEONE'S HOLDING THE POWER SWITCH DOWN! I'D BETTER SHUT DOWN RIGHT NOW!" ... as for the different POST codes that were showing up when it shut down, well, it was just whatever stage the POST happened to be at when it got around to shutting down. 
 
Needless to say, I'm NEVER buying another ANTEC product for as long as I live! 
 
Arestavo, thanks for sticking with me on this - if you had not suggesting isolating the motherboard I never would have thought of removing it, and I would have ended up sending a perfectly good motherboard back to EVGA at great expense (i.e. not just the shipping/RMA fee but not having access to my machine on the day I start my new job). So THANK YOU again! 
 
TZF