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The second, $\mathcal L_{\rm P}$ , lives in spacetime itself: send a system from event A to event B along worldline $\gamma_0$ , and imagine comparing it against a return trip along the reverse of a different worldline $\gamma_1$ that also connects A to B . The loop closes in space and time — it starts and ends at A — but the two worldlines need not have the same proper time, $\tau_0$$\neq$$\tau_1$ , precisely because proper time is path dependent once gravity or acceleration is present. This is not a new observation; it is the entire content of general relativity’s twin-paradox structure, imported into a single-particle quantum phase by early clock-interferometry proposals [ 10 ] .

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