Local gaps between stable looptrees and jump-unwrapped real trees

We compare the looptree metric encoded by an α-stable Lévy excursion of index α ∈ (1,2) with the jump-unwrapped metric obtained from the same excursion by keeping each jump contribution linear. Both metrics are built from the same jump configuration, but they incorporate the discontinuities in distinct ways. A jump of size Δ is represented by a line segment of length Δ in the real tree, whereas it corresponds to a loop of perimeter Δ in the looptree. If a path traverses this jump at a position q ∈ [0,Δ], its contribution to the distance is q in the tree metric and min{q,Δ−q} in the looptree metric. The distance saved at this jump is therefore (2q−Δ)₊. We call the cumulative distance saved by replacing each jump segment with the shorter circular arc the gap functional between the two metrics. For the unconditioned spectrally positive α-stable process X, we establish the exact first-moment identity E_X[Γ_X^+(h)] = [2^(1−α)/Γ(1+1/α)] h^(1/α). The identity is derived from the Lévy compensation formula and the law of the running infimum; no spinal decomposition is needed. We also explain its relation to existing jumpwise constructions of stable looptrees. Its transfer to the normalized excursion is stated only conditionally, under an explicit local absolute-continuity assumption and an unproved Lᵖ moment bound. The contribution of ancestral jumps and the resulting full local-gap asymptotic remain conjectural.