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$\Delta\phi\to0$

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Two degenerate limits of the proper-time form confirm the derivative is doing the right job rather than hiding a division problem. As m $\to$ 0 , $\Delta$ $\phi$ $\to$ 0 at every fixed $\Delta$ $\tau$ , so the slope — and therefore $m_{\rm op}$ — vanishes identically: a massless worldline is exactly the case with no proper time to speak of in the first place, and the formula correctly returns nothing to divide. And if the two branches are held at equal proper time, $\Delta$ $\tau$ =0 , then $\Delta$ $\phi$ =0 regardless of m ; a single measurement at that one setting is a 0/0 statement about mass, not a small number. Both limits are trivial once stated, and both are exactly why the estimator was built as a slope across a…

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\Delta\phi

Symbol Δphi

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\Delta\phi\to0

Equation 76 · Evolutionary Physics

The Clock That Comes Back Wrong by Exactly Its Mass

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

Two degenerate limits of the proper-time form confirm the derivative is doing the right job rather than hiding a division problem. As m $\to$ 0 , $\Delta$ $\phi$ $\to$ 0 at every fixed $\Delta$ $\tau$ , so the slope — and therefore $m_{\rm op}$ — vanishes identically: a massless worldline is exactly the case with no proper time to speak of in the first place, and the formula correctly returns nothing to divide. And if the two branches are held at equal proper time, $\Delta$ $\tau$ =0 , then $\Delta$ $\phi$ =0 regardless of m ; a single measurement at that one setting is a 0/0 statement about mass, not a small number. Both limits are trivial once stated, and both are exactly why the estimator was built as a slope across a…

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Δϕ→0\Delta\phi\to0Δϕ→0