Fell's Thm and Haag's Thm: How do we know there are interactions?

I've been thinking recently about a weird puzzle in algebraic and
constructive QFT. Namely, when one combines the conclusions of two
central theorems, one arrives at a seemingly absurd result: that free
and interacting quantum fields are experimentally indistinguishable.

Fell's theorem implies that any state can be approximated arbitrarily
closely (in the weak*-topology) by Fock states. Measuring the
expectation values of any finite set of observables in the C*-algebra
of observables, up to some finite degree of accuracy, will only
determine the state to within a weak*-neighborhood.

This would seem to imply that no finite number of experiments can
determine whether a given state is a Fock state or not.

Haag's theorem implies that no interacting state is in the same folium
as any free state, and therefore that no interacting state is a Fock
state.

So, if (by Fell's Thm) we can't rule out the possibility that any
given state is a Fock state, then (by Haag's Thm) we can't rule out
the possibility that any given state is a free/noninteracting state.

Something has to be wrong with this reasoning, but what?

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