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Yes, the tach can kill the battery if hooked up all the time.
You're kidding, right?
When you get a charged/new battery remove the hot cable from the solenoid.
Make sure it doesn't touch ground and connect the test light.
Take the other wires from the hot stud of the solenoid and touch them one at a time with the test light.
Tell us which wire it is.
Gary's EVTM diagram shows only the ignition circuit.
I would make a guess, but I want to hear your findings first.