Constitutive Closure and the Cost of Observable Attribution in Quantum Measurement

The spectral measure of a Hamiltonian and the POVM implemented by an apparatus need not coincide. We formulate observable attribution relative to an apparatus class defined by its preparations, controls, ancillary couplings, memory, and readable records. For a nondegenerate qutrit coupled to a binary threshold pointer and an amplifying environment, the explicit dynamics induces a classical observer channel that preserves only ground-versus-excited weight. We prove complete factorization of every allowed primitive through this channel and lift it to all finite adaptive protocols. Thus repetition, coherent controls within the excited sector, sector-only ancillas, calibration, and feed-forward cannot distinguish the two excited eigenstates. The full energy PVM is not attributable to the base class, and its optimal worst-case total-variation error is exactly one half. Adding a selective sector-changing control realizes the full PVM. Within the specified finite-protocol resource model, its exact minimum integrated operator-norm action is ; the optimal one-pulse approximate cost is also obtained in closed form. For imperfect devices we derive an exact minimax decoder for arbitrary calibrated binary responses, explicit error floors from record overlap, leakage, detuning, and dephasing, and a diamond-norm stability theorem for finite adaptive protocols. Observable attribution is therefore determined by physically available channels and resources, although the outcome statistics of every fixed experiment remain objective. This is a preprint version of a manuscript currently under journal review.