5.5151 Must the sign of a negative proposition be constructed with the sign of a positive proposition? Why should one not be able to express a negative proposition by means of a negative fact. (For instance: If “a” does not stand in a specific relation to “b”, this could express the fact that aRb is not the case.)

But even here the negative proposition is given indirectly by the positive proposition.

The positive proposition must presuppose the existence of the negative proposition and vice versa.

Yes, this should be old news within the Tractatus. Cf. 5.44.