Quantum reference frame transformations as symmetries and the paradox of the third particle

In a quantum world, reference frames are ultimately systems too -- but what does it mean to "jump into the perspective of particle"? this work, we show that frame (QRF) transformations appear naturally as symmetries simple physical systems. This allows us rederive and generalize known QRF within an alternative, operationally transparent framework, shed new light on their structure interpretation. We give explicit description observables measurable by agents constrained such symmetries, apply our results puzzle `paradox third particle'. argue can be reduced question how relationally embed fewer more particles, thorough algebraic analysis question. leads generalization partial trace (`relational trace') which arguably resolves paradox, uncovers important structures constraint quantization information setting, relational key in resolution. While restrict attention finite Abelian groups for transparency mathematical rigor, intuitive appeal makes expect they remain valid general situations.