Quantum Corrections to the Lorentz Transformation from the Page-Wootters Constraint: The k-Factor as Operator, Simultaneity Decoherence, and Experimental Predictions

This paper extends the derivation of the Lorentz transformation from count-based observables to the quantum level. It maps each ingredient of the classical Bondi k-calculus derivation onto the Page-Wootters (PaW) framework and extracts the k-factor as a ratio of clock POVM eigenvalues, completing the derivational chain from Ĥ|Ψ⟩ = 0 to the full Lorentz transformation. Promoting the k-factor to an operator k̂ = (Ĥcm + p̂c) / (mc²) reveals quantum corrections for a detector in momentum superposition. The corrections are asymmetric: the time dilation sector receives mean corrections at order (σₚ/ (mc) ) ², while the simultaneity sector is protected to leading order. Both sectors decohere at rates proportional to σₚ/ (mc). The paper derives a concrete experimental prediction: fringe visibility in an atom interferometer decays as V (L) = exp (− (ω₀σₚL/ (mc²) ) ²/2), with a coherence horizon LC = √2 mc²/ (ω₀σₚ). For cold cesium at the D2 optical transition (852 nm), LC ≈ 7. 3 km at 1 μK. The effect is absent for macroscopic detectors and present only when the detector itself is a quantum system. The primary experimental discriminator is the temperature dependence: predicted visibility tracks the atom's thermal momentum spread σₚ = √ (mkBT), a property of the detector atom that no environmental noise source can mimic. All corrections vanish in the classical limit σₚ → 0, recovering the companion paper's Bondi derivation exactly. The prediction provides testable consequences of the surplus structure framework in atom interferometry and quantum clock networks.