So an aeroplane with a rear-mounted horizontal stabiliser (let's just call it a tailplane, so much less of a mouthful) - like all current airliners since the grounding of Concorde - can only be stable if its tailplane produces negative lift throughout the flight envelope. And they all do.
Sorry, but that simply is not true.
The load on the tailplane is required to trim out the residual pitching moment from the wing; because most wings are significantly cambered, there is a definite nose-down pitching moment for typical cg positions in 1'g' flight. However, this is a static force balance issue, and has nothing to do with whether the aircraft is stable in the pitch axis.
Stability is to do with the tendency of the aircraft, if disturbed, to return to equilibrium. That depends on the pitching moment derivatives, not the absolute values of pitching moment.
For example:
take a stable aircraft, with a tail of 100 sq ft and a download on the tail of 10,000lbf (all arbitrary numbers)
now make the tail ten times the size, 1000 sq ft
for TRIM the tail download is identical - 10,000lbf. Because the rest of the aircraft is the same, so the force required to balance the wing/engine/etc is unchanged.
But an aircraft with a huge tail would be significantly more stable in the pitch axis.
do the same in the other direction. As the tail gets smaller you don't necessarily lose the ability to trim, but you become less stable.
Or:
take a REAL aircraft. Trim in 1 'g' flight. There's probably a small download on the tail, especially if we have a fairly aft cg.
now start to bunt. Push forward on the stick. The load on the tail reduces and eventually will be negative. The aircraft doesn't suddenly become unstable when the sign of the tail load changes.
Or:
Can a tandem wing aircraft only be stable with download on the second wing?
Can a canard only be stable with download on the wing?
Obviously the answer is no; those types can be stable with upload on the rear aerodynamic surface. Whether I call it a tailplane or a wing is irrelevant to the equations used to calculate stability.