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?