A companion program developed an exact framework for quantum reference frames, beginning with phase and clock reference frames. This note completes the canonical trio with the third reference frame: position. The work studies ideal and interacting ruler frames in exactly solvable models. A position reference frame is shown to realize the same frame-deficit structure as charge and time: under a total-momentum constraint, translation becomes a gauge symmetry, while the ruler-less observer has a locked spatial information deficit determined by the momentum distribution entropy. The matching of accessible spatial information is governed by the available translation-sector content of the ruler. The note then studies a Newtonian-form position coupling between a system and a ruler. For pure ruler states, the information recorded by the ruler is exactly equal to the remaining visibility of the system coherence. This establishes an exact record–decoherence relation: which-path information acquisition and coherence loss are two descriptions of the same process. The result realizes the visibility–distinguishability duality in a gravitationally motivated measurement model. For mixed rulers, the record fidelity is strictly weaker than the surviving coherence, showing that imperfect reference frames cannot extract information faster than the decoherence they cause. The paper further reveals a structural difference between clock and ruler reference frames: clocks couple through their generators and obey a fixed resolution–disturbance price, while rulers couple through conjugate variables and allow squeezing to reduce alignment cost. However, squeezing improves alignment only and cannot bypass the fundamental limit on information recording. The work also connects gravitationally induced entanglement, measurement records, and decoherence into a single function within the solvable model. The results are obtained in finite measurement-limit models with declared assumptions: linearized position coupling, ideal readout, and no claim of describing full dynamical gravity. Verification and reproducibility materials are included.
A companion program developed an exact framework for quantum reference frames, beginning with phase and clock reference frames. This note completes the canonical trio with the third reference frame: position. The work studies ideal and interacting ruler frames in exactly solvable models. A position reference frame is shown to realize the same frame-deficit structure as charge and time: under a total-momentum constraint, translation becomes a gauge symmetry, while the ruler-less observer has a locked spatial information deficit determined by the momentum distribution entropy. The matching of accessible spatial information is governed by the available translation-sector content of the ruler. The note then studies a Newtonian-form position coupling between a system and a ruler. For pure ruler states, the information recorded by the ruler is exactly equal to the remaining visibility of the system coherence. This establishes an exact record–decoherence relation: which-path information acquisition and coherence loss are two descriptions of the same process. The result realizes the visibility–distinguishability duality in a gravitationally motivated measurement model. For mixed rulers, the record fidelity is strictly weaker than the surviving coherence, showing that imperfect reference frames cannot extract information faster than the decoherence they cause. The paper further reveals a structural difference between clock and ruler reference frames: clocks couple through their generators and obey a fixed resolution–disturbance price, while rulers couple through conjugate variables and allow squeezing to reduce alignment cost. However, squeezing improves alignment only and cannot bypass the fundamental limit on information recording. The work also connects gravitationally induced entanglement, measurement records, and decoherence into a single function within the solvable model. The results are obtained in finite measurement-limit models with declared assumptions: linearized position coupling, ideal readout, and no claim of describing full dynamical gravity. Verification and reproducibility materials are included.
This note shows that several apparently different problems in quantum physics share one underlying structure: the role of physical reference frames and the information that becomes inaccessible when the required reference frame is unavailable. A companion program studied exact frame-dependent information deficits in solvable models, including charge-locked information in Hawking-type radiation, the loss of timing information in constrained quantum systems, and the quantum cost of interacting clocks. This work identifies these results as instances of the resource theory of asymmetry, where inaccessible information is quantified by the relative entropy of asymmetry. The note proves that the previously computed deficits are evaluations of the same asymmetry resource, including the information deficit in charge sectors, the timing deficit in constrained systems, and related entropy quantities. It also shows that the matching conditions for recovering hidden information are unified: a physical reference frame must contain the appropriate symmetry content of the locked coherence. A further result is a general resolution-disturbance tradeoff for physical reference frames. For pure reference frames, the product of measurement resolution and induced decoherence is fixed by the coupling strength and the energy or generator gap. Mixed reference frames perform strictly worse, showing that imperfect reference frames pay an additional quantum cost. The work establishes a unified framework connecting superselection rules, quantum reference frames, relational time, information accessibility, and decoherence. The results are derived and verified in finite exactly solvable models, with declared limits: no claim of a complete theory of dynamical gravity is made. Verification and reproducibility materials are included.
No physical clock is free: every clock is a quantum system that interacts with what it measures. This note studies the exact relationship between clock precision and the decoherence induced by universal time-dilation coupling in solvable constrained quantum systems. Using a clock-system interaction determined by energy coupling, the work derives the exact coherent-overlap decay law associated with time-dilation decoherence and shows how relational quantum dynamics emerges from constrained systems. The global time translation remains a gauge symmetry, while the time evolution recovered by conditioning on the clock exhibits the corresponding deformed relational dynamics. The main result is an exact resolution-decoherence tradeoff: the product of the clock's time resolution and the induced decoherence rate is independent of the clock state and depends only on the coupling strength and the system energy difference. This shows that improving the precision of a quantum clock necessarily increases the disturbance it causes when universally coupled to the system it measures. The result provides an exact model-level realization of a fundamental limitation of physical reference frames: a reference system that allows more precise access to information must pay an unavoidable quantum cost. The work is formulated within finite constrained quantum models with declared assumptions: idealized clocks, a specified universal dilation coupling, and no dynamical gravitational field. The results are accompanied by verification and reproducibility materials.
This note extends the quantum reference-frame framework from charge, time, and position to geometry itself. It studies the consequences of treating spacetime geometry as a quantum variable in exactly solvable models. An observer without access to a superposed geometry register is shown to possess a geometric information deficit: the inaccessible geometry information is quantified by the entropy of the geometry sectors. Conditioning on the geometry register recovers the individual geometric branches exactly, establishing geometry as a quantum reference frame with the same structure as charge, time, and position reference frames. The work studies clocks evolving in superposed gravitational redshifts and derives exact proper-time interference. Different geometric branches generate which-geometry records in the clock, producing decoherence and periodic recoherence according to the same information-transfer structure found throughout the reference-frame framework. The note further investigates the emergence of classical spacetime. Multiple quantum systems coupled to the same geometry register collectively decohere the geometric superposition faster than a single system, while non-universal coupling does not produce the same amplification. This provides an exactly solvable realization of the idea that spacetime classicality can emerge through continuous measurement by matter. Finally, the paper derives the effective energy-product interaction used in gravitationally induced entanglement models. By introducing a dynamical geometry mode sourced by matter energy and eliminating the mediator, the familiar energy-product coupling emerges directly from the quantum geometry model rather than being assumed. The results connect quantum reference frames, proper-time superposition, gravitationally induced decoherence, gravitational entanglement, and the emergence of classical geometry within controlled finite models. The declared limits are finite geometry registers, simplified sourced geometry modes, and no claim of a complete theory of dynamical quantum gravity. Verification and reproducibility materials are included.
This note studies the role of a quantum mediator in gravitationally induced entanglement and decoherence using an exactly solvable dynamical mediator model. Two masses in spatial superposition interact through a mediator mode described by a displaced-oscillator Hamiltonian. The model provides a unified description of two regimes: when the mediator returns to its original state, it leaves behind pure entanglement between the masses through virtual excitation; when the mediator carries which-path information, it becomes a physical record that reduces coherence and suppresses entanglement. The work proves an exact record–decoherence relation: the loss of coherence of each subsystem equals the information stored in the rest of the universe, combining entanglement generation and decoherence into a single information budget. The paper also establishes a classical-channel dichotomy. A measure-and-feedforward channel calibrated to reproduce the same interaction produces no entanglement, while introducing additional noise. This provides an exact solvable distinction between a genuinely quantum mediator and a classical communication channel. The causal structure of mediation is studied in a multi-mode chain, where entanglement appears only after the mediator excitation reaches the relevant region, establishing a finite propagation constraint in the model. The work also proves analytically the separability boundary for partially coherent sources, including the condition relating source coherences required for entanglement generation. The results provide an exactly solvable framework connecting gravitationally induced entanglement, mediator quantum information, which-path records, decoherence, and quantum versus classical mediation. The declared limits are finite models, linear mediator coupling, and no claim of a complete theory of dynamical quantum gravity. Verification and reproducibility materials are included.
This note applies the frame-deficit dictionary to a genuinely dynamical quantum geometry: the exactly solvable flat Friedmann–Robertson–Walker universe with a massless scalar field, where the Wheeler–DeWitt equation becomes a 1+1 wave equation. The model is studied in a minisuperspace lattice, with expanding and contracting cosmological branches and the scalar field serving as an internal clock. Unlike previous toy models where geometry was treated as an external label, the geometry here is a quantum degree of freedom with its own conjugate momentum. The paper establishes four exact results. First, the timeless Wheeler–DeWitt kernel admits relational evolution: conditioning on the scalar clock recovers unitary evolution with no dispersion. Second, an observer without access to the clock carries an exact information deficit determined by the clock-momentum distribution. Third, a superposition of expanding and contracting universes decoheres through matter records, with the interference of cosmological histories controlled exactly by record overlap. Fourth, an entropic trade-off is established between the sharpness of the universe's history and the information hidden behind the clock: a more classical size history requires a larger information deficit. The result extends the frame-deficit framework from quantum reference frames to quantum cosmology, showing how missing reference information, decoherence, and classicality costs arise in a model with genuinely quantized geometry. A bibliographic correction concerning the attribution of previous quantum cosmology decoherence work is recorded in the paper. The scope is limited to the exactly solvable minisuperspace model and its lattice realization. No claim is made for full Einstein dynamics or a complete theory of quantum gravity. All results are based on pre-registered predictions, symbolic derivations, and computational verification. Reproducibility materials are included.
Gravity can both generate quantum entanglement and cause decoherence through time-dilation effects. This note studies the competition between these two effects by treating them as consequences of the same Newtonian mass–energy interaction. Using a solvable two-body model with a coupling between energy and geometry, the work shows that the position-dependent part of the interaction produces the entangling phases associated with gravitationally induced entanglement, while the energy-dependent part produces proper-time decoherence. The paper computes this race exactly and derives several results. First, the outcome is independent of the strength of gravity itself: the dynamics depends only on the combination of coupling strength and interaction time, meaning that gravity sets the timescale but does not determine the final verdict. The result is controlled by the structure of the interaction and the geometry of the quantum state. Second, a registered prediction that internal energy spread would eventually destroy gravitational entanglement is disproved by the exact calculation. Instead, the entanglement decreases but approaches a nonzero geometry-dependent floor. The mechanism is traced to the energy-product coupling, which produces a non-Gaussian characteristic function and prevents complete suppression of the entangling effect. Third, the work derives an exact separability boundary: two sources with spatial coherences v1 and v2 become entangled at the optimal phase precisely when (1 + v1)(1 + v2) > 2. Fully decohered sources cannot generate entanglement, while partial coherence can be sufficient. Finally, the paper verifies that the record–decoherence relation holds inside the same gravitational interaction: coherence lost by a system equals the information recorded by the rest of the universe, combining entanglement and proper-time decoherence into a single information budget. The results are obtained within finite exactly solvable models with declared limitations: static branches, Newtonian-form coupling, no graviton exchange, and no dynamical quantum gravity. The work provides a controlled theoretical framework for understanding the competition between gravitationally induced entanglement and gravitational decoherence.