Hi Simon,
1) this test should prove the 5-pin cable and the jumpers ok.
2) with the selector in standby, you shouldn't get any output or hum regardless of the settings of the vol knobs. but the fact that you do, makes the 5-pin cable or internal wiring in the bass more likely than power supply problems.
3) normally quiet with the batteries would suggest that the power supply is the problem, but here we've got the unusual behavior in line 2.
4) in rereading, i see my wording was ambiguous. i was suggesting continuing to get the signal down the 5-pin cable and from the jack on the power supply to your amplifier, and seeing if turning the power supply on and off made a difference. but in reading the results of your test, i think regardless it doesn't affect this situation.
5) i wasn't clear here either, it was just a suggestion following up on Nate's ground loop theory, so that turning on the batteries with a 1/4 inch plug but listening thru the 5-pin cable and jumper between the power supply and the amplifier, that you could hear the effect of a ground loop caused by the safety wire in the AC power cable because you can listen for any difference when the safety ground is present or not.
i can't say for sure where the problem is now. my biggest concern is that you can still hear hum and possibly some signals even with the selector switch in standby. it would be really hard for any failure in the power supply to cause this. so i'm leaning to thinking that the problem is in the 5-pin cable or in the instrument.
should it be in the cable, it would be the shield is failing (usually this happens within 10 inches of the connectors since this is where flexing causes the most strain), or pin 1 on either end is not connecting to ground firmly. usually a failing shield is pretty obvious as it makes a big change when you flex the cable even slightly, but have a listen with the rig set up so you're hearing the hum with the selector switch in standby and see if wiggling the cable causes any difference. if it doesn't, then i would be suspicious of the central ground point in the instrument. this point is the sleeve terminal of the 1/4 inch jack, which has a small flag terminal which we extend with a large buss wire. a black wire from pin 1 of the 5-pin connector goes here as does the black wire from the selector switch. a yellow wire from the ribbon cable supplies the ground for the circuit card, and it's also the common for the batteries - with one black and one red battery wire soldered there. if these ground signals are not well connected to this point then this could cause the symptoms observed.
if you have access to an ohmmeter, you could connect the 5-pin cable to the power supply and with the pickup selector switch in standby and the mono/stereo switch on the power supply in the stereo position, insert a plug in the bass 1/4 inch jack on the power supply. you should measure 0 ohms (a dead short) from the tip terminal to the sleeve. the measurement with the plug moved to the treble jack should be the same.
if that shows more than one or two ohms resistance, then you could connect one terminal of the ohmmeter to the sleeve of the 1/4 inch jack on the instrument and probe the pins in the 5-pin connector on the instrument. you should observe 0 ohms on pins 1, 2, and 3.
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and finally, i'm not completely able to rule out a power supply problem. if one of the two large capacitors in the power supply failed, then instead of having approx 2000 microfarads of storage, we would have only the 1.5 microfarads provided by the small bypass capacitors in the instrument. this would cause a very big ripple current in the 5-pin cable and the wiring in the instrument, and might cause some amount of hum due to the small but finite resistance of the conductors.
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one way to check the ground connection thru the 5-pin cable and it's connectors is to connect a wire from the sleeve on the 1/4 inch jack on the instrument to one of the 1/4 inch sleeves on the power supply while listening to the hum, to see if there is any difference when the wire is touched to the jacks. if the 5-pin cord and connectors are ok, then there should be no difference when you add the independent ground connection between the instrument and the power supply.
let us know of any other observations or symptoms that might help definitively diagnose the problem.
-ron