Quantifying Relational Structure in Quantum Measurement: An Axiomatic Account of Recursive Observation Depth

We formalize the relational pair (O, M), observer and observed system, as a primitive mathematical object governed by three operational axioms. From this structure we define Recursive Observation Depth (n^k), the length of the longest informative measurement chain an observer O can perform on a system M through a Positive Operator-Valued Measure (POVM). Five propositions about n^k are proved: monotonicity, relational dependence, a characterization of superposition, an exact value for a compatible (classical) observer, and, for a generic informationally complete observer, the impossibility of a terminal state, from which the divergence of n^k follows. Rank appears only as a special case, attained when the observer's measurements are mutually compatible; once they become informationally complete and incompatible, n^k diverges. We test the framework against four no-go theorems. It is consistent with Bell, Kochen–Specker, and No-Cloning on independent structural grounds, and consistent with PBR conditionally, provided the quantum state is taken to be a property of the relation (O, M) and not of the system alone. Wigner's Friend is resolved within this framework without further postulates. The Born rule is not derived, and no new experimental predictions are offered.