Odd LED operation

That looks like a plug in receptacle to me...we've had this discussion a few times on here about induced voltage generated by long wire runs next to live wires or power sources, one end has to be disconnected for it to work, so I would say that picture is the end that's disconnected, and there is some induced voltage causing this effect...ghost lights, but more properly parasitic drain....

You have done some good research and measurement that might have a clue to the observation.

If I read correctly, you still have 30mV across the line that is apparent AC. Remember that LEDs are diodes and so would be rectifying any AC signal that is passing through them. It could be that the combined "capacitive" effect of the strings between the switch and the LEDs that are apparently alight is enough for you to see the emitted light.

On one phase of the AC, the string "charges" this capacitor (through the LEDs) and then on the other phase of the AC the discharge current has enough power to be visible on those 25 LEDs.

Your observation that when unplugged there is no light seems to support this.

<...inverter...>

If this thing has no mechanical switch on it then it is quite possible that the electronics inside it consider that <...30mV...> is "off", especially when there is such a mismatch between its rating and the lighting power demand. <...LED lights...> are current-operated light sources and all they are illustrating that there is a leak through the <...inverter...> that is not noticeable at <...3500W...> loads, and is at the single-watt levels drawn by the lights.

Try moving the <...timeswitch...> contact downstream of the <...inverter...> instead, perhaps? It might need an interposing relay contact, perhaps?

Here is just a wild guess...

I'm guessing that since your inverter is mobile that you don't have the neutral side grounded. If that is the case, when the timer switch opens the hot side, the neutral will float up to 120 vac.

If there is leakage from one of the 25 light substrings, either through insulation capacitance or insulation damage, that could account for what you're seeing.

If you can use a double pole timer to turn the leds off. You could ensure that the timer switches the active wire and not the neutral of the inverter. If the active is still on the leds and there is capacitive leakage from part of the string then they may turn on as you describe. Are the strings of lights fed from the supply or through a controller to reduce the voltage to the required level for the leds?

I'm assuming that the LEDs are designed to run on US mains, so, the absolute maximum voltage they would ever expect to see would be about 170V. Typical white LED forward voltage is between 3 and 5 Volts: so a string of 100 in series would need to see at least 300V to turn on.

I suspect that the "strings" are really 25 in series times 4 in parallel. You might want to examine the wiring especially those odd connectors more closely with that in mind.

Good observations by everyone. The Inverter neutral is not tied to ground and the timer (shown below) does only switch the hot side of the line. This timer is plugged directly into one of 4 receptacles. The only problem I have had there is this timer runs fast (gains 15 to 20 minutes a day) induced by the fact the inverter is not a true sine wave output.

the second photo is another view of the little plugs that are on the string about every 25 - 26 LEDs. I've been tempted to open one up but afraid it will get destroyed in the process, and it is the last string I have. They are sealed up and whatever potential creates the lighting effect on this last string is contained between two of these. Does anyone know what is in these?

So I think there is something coming through the neutral leg and these plug devices prevent the whole string to light. The night I saw that section lit we had rain earlier so surfaces were damp to wet.

The "plugs" are just the common ends of the parallel strings.

If you are planning to take the lights down after the Christmas season, it would be interesting to see if they behave in the same way not mounted on the house. It could be that the one string that lights with the timer off is in closer proximity to a grounded object. Also, you might notice if the insulation was damaged when hanging the lights.