I checked the air pressure in my tires last night. Front showed 32 psi and the rear showed 0 psi. I added air and it still showed 0, so I figured I had a puncture. Today I removed the rear tire from the bike and removed the valve to prepare the tire for removal. As I removed the valve I thought I felt a slight bit of air movement across my fingers. With 0 psi this shouldn't happen.

I grabbed a tooth pick and stuck it into the valve stem (valve is out, on the workbench). As the toothpick makes its way past the base of the stem I got a blast of air. Clearly, the tube is holding air.

I reinserted the valve and I was briefly able to get a reading at 58 psi - so the tube held all of the extra air I added. As I was removing air the flow stopped and I again was showing 0 psi. I removed the valve again and reinserted the toothpick and removed all remaining air.

I refilled the tire and was able to get readings along the way, but when I reached 35psi I stopped getting reading again. I have 35 psi, yet the gauge shows 0 psi. I tried several different gauges. They all work on the front but none work on the rear.

Is it possible that I have an obstruction in the valve stem. How can the valve be removed, with 58 psi, yet I have no outward flow?

Good guess, but I installed this tube two years ago.

Gary,You might try loosening the nut on the valve stem first as this is used to hold the stem in place when putting the tire on ,but then should be backed off to hand tight .If that doesn't work ,instead of a tooth pick,I'd add air to the tube to keep it rounded and run a very short piece of coat hanger just through the valve stem after heating it fairly hot with a torch,any blockage should melt away.It may sound a bit odd but I think the tube is made then the stem added later and there is a small flap of rubber in the base that allows air in but acts as a check valve not allowing air out and that's what you'll be melting away.I also looked at your build pic's ,the right side shows your new front tire ,on the side wall there is a yellow dot that tells the light side of the tire and should be lined up with the valve stem.

If you are using a double head tire gauge, like used for truck tires, push the chuck's core stem in on the back side with a small bolt/screwdriver/pen and at the same time check  you check the tires air pressure.

Some of these Chinese made double head air chucks and gauges have the stem cores too short, or the springs too stiff. The Schrader stem core and the air chuck/gauge stem core should open each other up. The weaker sprung Schrader core opens easy, the stiffer spring in the chucks core is harder too move and needs to bottom out against the Schrader core to open. If the chuck core stem is a little too short, or spring too stiff, it's hard to get it to bottom out or open to let air to pass.

By pushing on the back side of a double head air chuck/gauge, you can get the chuck core to open.

You can get cores that have a long stem/spring on the core......this has the potential to hold the flapper away from whatever it is seating against.

Gary ,If you end up having to use my method ,once the blockage is gone the air should rush out cooling the wire quickly,but the wire should be pulled out just as quickly so the sagging tube doesn't have contact with it, the whole process taking only a couple of seconds.The stem must have a metal lining with threads to except the valve,so the blockage must be at or near the base,the part of the wire you insert should only extend 1/4 inch beyond, and you only need to heat the last 1/4 inch.It's a kill or cure ,but if it doesn't work you have to change the tube anyway.

Right now there's only good stuff in that tube - air. Once I get that flapper cleared I should be good. As Batman said, I have nothing to lose.