Debuzzing a Wheaton TriPlanar problem

RFI/EMI problems at tonearm wires can tend to jump around unpredictably as you describe and vary in proximity with the human body. Grounding the tonearm or TT is sometimes not enough, and screened tonearm wire may be necessary to resolve the problem.

Does the hum continue when the cartridge pins are disconnected from the tonearm wires?

You might experiment by relocating the TT to another part of the room. The current location may be in a bad RFI field or too close to in-wall AC wiring or to the AC PS of another component.

Now this is making more sense to me, and I thank you. The TT is 6 inches from the wall which conceals all the electrical lines from the apartment's circuit breaker. And I must ask what is 'screened tonearm wire' ? I want to try this fix before relocating the equipment rack. Thanks.

Based on your responses to Dave and Ralph, I am now thinking that I got it wrong. Am I now correct in thinking that the tonearm per se is not necessarily vibrating but that the hum you hear (over the speakers, I presume) is ameliorated when you grasp the arm tube? If so, this makes much more sense, and I agree with Dave. But I will throw in another idea; the arm tube structure is grounded to the arm pivot (i.e., makes electrical contact with) the bearings only. I once had a maddening hum problem that was only cured when I took my tonearm to Herb Papier (the inventor and original manufacturer). Herb lived about 10 miles from my house. Herb took my arm, and before my eyes he tightened the bearings by a hair, and the problem was gone. That has to have been nearly 20 years ago, and I have had no recurrence of the problem. Problem is you need a Herb or a Tri to do it so as not to over-tighten the bearings.

Yes, Lewm, the hum disappears when I grasp the covered portion of the tube. Sometimes the hum is amplified when I grasp the finger rest at the cartridge end; release the grasp, and no hum. This has been a problem since day one of installation, and several well-known analog experts and reviewers have applied various fixes, including triple-grounding the whole TT setup, wrapping the pre-amp in -yes-a chicken wire cage, and finally purchasing and inserting a $3K passive-step-up device in the chain prior to pre-amplification in order to permit such minimal gain that the hum will not be heard.
All this effort, no success.
And so, with the original dealer out of business and the non-communicative Wheaton lineage located thousands of miles away from NYC, it seems that I possibly purchased a defective tonearm, one which is not all that 'world-class precise' and which probably at this point cannot be righted.
But I do appreciate your insight and experience, Lewm. There is some solace at least in identifying the problem, though no solution seems possible.

Wow! Why so glum? Send it back to Tri, and he can tighten up the bearings to restore the proper ground connection, if that really is the issue, which I am beginning to believe it is. None of the strategies you mention could be expected to cure that problem. Just explain the symptom to him, and refer him to me if he does not get it. (We have met a few times; I was at Herb's house when Tri was there to learn how to build the tonearm. I last talked to him in person at RMAF last month.) The cost of shipping to and from Tri for you would probably be less than $25 round trip, if you use US Priority Mail. Chin up, old man! Hum due to bad, inadequate, or incorrect grounding is ALWAYS fix-able, and definitely does not indicate you have a defective sample. IOW, shit happens.