Common Singularity Witnesses for Multiplication Extensions

We study common singularity witnesses for unital completely positive extensions of multiplication from the minimal to the spatial von Neumann tensor product of a factor and its commutant. The question is whether singularity after composition with normal positive output functionals can be witnessed by one decreasing projection sequence for the entire fibre and output space. For a nonempty fibre, a decreasing net of positive contractions converging strongly to zero and mapped to the identity by every extension exists exactly when some nonzero positive element is annihilated by every extension. If the spatial tensor product is σ-finite, these conditions are also equivalent to a common projection sequence. The witness properties are invariant under faithful normal representations and nonzero factor corners. For factors with separable predual, the common projection property holds exactly for injective non-type-I factors. Without separability, we construct an injective type-II1 factor whose spatial tensor product has a faithful normal trace but whose fibre admits no full-output net. Its fibre contains a faithful, completely isometric extension whose scalarizations by normal positive output functionals are all singular. Nevertheless, every normal input state agrees on any prescribed countable input set with a vector scalarization of this one fixed extension. For the wedge multiplication fibre of the massive free scalar field in 1 + 1 dimensions, explicit local finite-bin projections yield a common full-output sequence in bounded double cones with touching boundaries. A complementary obstruction shows that every full-output sequence has asymptotic operator-norm distance 1/2 from each fixed positive-collar input algebra.