Certified shortcuts are precisely the ones constructible in near-linear time, and no almost-linear-size certified shortcut can reduce diameter below n^{1/4-o(1)}.
In the former case, ⟨e1(v′ 1),v ′⟩>0 andv ′ 1 =v 1 is the unique maximizer due to strict convexity
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Better Diameter Bounds for Efficient Shortcuts and a Structural Criterion for Constructiveness
Certified shortcuts are precisely the ones constructible in near-linear time, and no almost-linear-size certified shortcut can reduce diameter below n^{1/4-o(1)}.