Varying the boundary period and filling it by the corresponding smooth Euclidean saddle computes the same canonical partition function. Because each bulk metric obeys the Einstein equation, the implicit first-order metric variation of the on-shell action reduces to boundary terms; regularity relates the varied horizon radius to the varied period. The thermodynamic identitythen yields the same Bekenstein-Hawking entropy. The conical method is an off-shell way to isolate the local horizon term, while the smooth-saddle method packages that term into the variation of the entire solution.
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