Numerator: hbar

The passage around this formula

Before pointing this construction at an interferometer, one methodological trap needs naming. A single measured phase $\Phi$ at one fixed loop area cannot, by itself, distinguish m from an unknown additive offset baked into the apparatus — any constant phase shift from an uncalibrated pulse timing, an unaccounted path-length difference, or a stray field looks identical to a rescaling of $\mathbf b$$\cdot$$\mathbf v$ . The estimator that survives this is a derivative, not a snapshot: $m_{\rm op}=\frac{\hbar}{c^2}\,\frac{\partial\Delta\phi}{\partial\Delta\tau},\qquad m_{\rm op}[\mathcal L_{\rm G}]=\hbar\,\frac{\partial\Phi}{\partial(\mathbf b\cdot\mathbf v)}$. The sign convention on the proper-time form follows the phase of a free relativistic particle, $\Delta$$\phi$=-m$c^2$$\Delta$$\tau$/$\hbar$ , so that m_{$\rm$…

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