LOWER SEMI-CONTINUITY OF LAGRANGIAN VOLUME

The paper proves γ–lower semi-continuity of Lagrangian volume in two settings (Kähler/homogeneous and torus) via Lagrangian tomographs, Crofton’s formula, and stability of Floer barcodes, culminating in Theorems 3.1, 1.2, and 1.3. The candidate solution reproduces this strategy: tomographs + Crofton, control of intersection counts by barcode lengths, γ–stability of barcodes, and the homogeneous/torus specializations. However, the model glosses over two technical points handled carefully in the paper: (i) the exact relation between intersection numbers and bar counts requires the formula N(s) = 2 b_{ε}(L,L_s) − h (with h = dim HF(L)) on suitable parameter sets, not simply “number of long bars equals intersections” (paper’s (2.2)–(2.3) and (3.2)); and (ii) the local torus tomograph density satisfies d_T ≤ (1+η) g and equals g along L_0, not pointwise d_T ≤ g. These do not change the final conclusions but are material details for correctness. Overall, the methods and results align closely; the model’s proof outline is essentially the same as the paper’s, with minor but fixable inaccuracies.