Volume 12, Issue 1, pages 468-480
We present the R-Framework, a quantitative formalization of Relational Quantum Mechanics (RQM) that defines a measurable quantity $R(A,B)$ representing the relational magnitude between physical systems. Drawing on information theory and the Holevo bound, we derive $R = \Delta E \cdot \tau / (\hbar \ln 2)$, where $\Delta E$ is the interaction energy and $\tau$ is the characteristic timescale. The critical threshold $R_c = 1$ bit marks the quantum-classical boundary: relationships with $R < 1$ remain quantum-indefinite, while $R \geq 1$ yields classical definiteness. We introduce a multi-mode generalization where the number of modes $M = \Delta E \cdot \tau / h$ determines the regime: for $M < 1$ (quantum), $R$ is bounded by the single-mode Holevo capacity $g(N)$; for $M \gg 1$ (classical), $R$ recovers the linear approximation. The framework provides concrete predictions for gravitational decoherence, asymmetric Bell experiments, and connects naturally to tensor network representations of quantum states.
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