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

This article axiomatically formalizes the relational structure of quantum observation and introduces a quantity that arises from this structure: Recursive Observation Depth (nᵏ). The main contribution of this article is the formalization of the relational pair (O, M) as a self-standing mathematical object equipped with its own operational axioms. From this structure, nᵏ is defined as a measure of the length of the informative measurement chain that observer O can perform on system M, formalized through a Positive Operator-Valued Measure (POVM). Five formal propositions are proven: monotonicity, relationality, superposition characterization, boundedness, and a lower bound. The rank of the Hilbert space appears only as an upper bound and special case of nᵏ, not as the basis of its definition. The consistency of this framework is tested against four no-go theorems (Bell, Kochen–Specker, PBR, No-Cloning) and is found to be consistent under one mandatory ontological commitment: the quantum state must be regarded as a property of the relation (O, M), not as a property of M alone. The Wigner’s Friend paradox is resolved without additional postulates. The Born rule cannot be derived from this framework, and no new experimental predictions are claimed.