$$m_{\rm op}=-(\hbar/c^2)\,\partial\Delta\phi/\partial\Delta\tau$$

Equation 72 · Evolutionary Physics

The Clock That Comes Back Wrong by Exactly Its Mass

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

The sign convention on the proper-time form follows the phase of a free relativistic particle, $\Delta$$\phi$=-m$c^2$$\Delta$$\tau$/$\hbar$ , so that $m_{\rm op}$=-($\hbar$/$c^2$)\,$\partial$$\Delta$$\phi$/$\partial$$\Delta$$\tau$ recovers m identically when $\Delta$$\phi$ has no other dependence. Both forms require sweeping the loop’s own parameter — hold time, translation distance, boost velocity — across at least two settings and fitting the slope, exactly as a spectroscopist extracts an energy gap from a swept detuning rather than from one photon count. A single unswept measurement cannot report a mass gauge at all; it can only report a phase.

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