Can every pair of compatible states on a quantum logic be simultaneously extended to a single state on the full algebra, and if not, what is the minimal algebraic obstruction?

This problem is open because the failure of distributivity in orthomodular lattices destroys the measure-theoretic machinery that resolves the analogous classical question. Partial results exist for specific classes such as complete Boolean algebras or Hilbert lattices arising from finite-dimensional Hilbert spaces, but the general orthomodular setting resists these techniques. The obstruction is not merely topological or measure-theoretic but is deeply tied to the combinatorial structure of the compatibility relation among propositions, and no complete characterization of when simultaneous extension fails is known. Constructive or computable variants of the problem add further difficulty because one cannot freely invoke nonconstructive choice principles that some existence proofs rely on.