Quantum Entanglement and Proper-Distance Attenuation within the Framework of Relativistic Configuration Space and Universal Quantum Energy

ABSTRACT The representation of two spatially separated particles by a single entangled quantum state raises the question of whether their laboratory coordinate separation and their relational distance in the configuration space of the joint state are the same quantity. In this study, the relative sector of a two-particle system is described by the coordinates and . While the laboratory coordinate separation is held fixed, a phenomenological model is proposed in which the effective proper distance assigned to the joint state depends on the correlation energy and the relativistic phase velocity. The constitutive geometric postulate of the model is a reciprocal choice of the temporal and spatial metric coefficients in the reduced -dimensional relative configuration-space sector: . This condition preserves the metric determinant and the two-dimensional volume element of the relative sector. The induced metric on a constant-relative-time slice yields (1) Consequently, although remains unchanged, in the limit . Under the null constraint, the coordinate characteristic speed becomes , whereas the speed measured with local physical time and length intervals remains exactly . The response amplitude is related to a correlation-energy scale , the phase velocity , and a correlation-localization factor . It is also shown explicitly that a finite-energy, Planck-suppressed ansatz does not produce in the laboratory regime. Vanishing proper distance is therefore not presented as a necessary consequence of standard quantum mechanics or finite entanglement, but as a boundary behavior of the geometry in the strong-response limit. The model quantifies relational proximity for an entangled joint state without claiming that the physical coordinate distance disappears