ON THE SUPPORTS IN THE HUMILIÈRE COMPLETION AND γ-COISOTROPIC SETS (WITH AN APPENDIX JOINT WITH VINCENT HUMILIÈRE)

The paper’s Main Theorem 7.12 proves: (1) γ−supp(L) is γ‑coisotropic for every L in the Humilière completion, (2) for each r there exists L with γ‑support containing Kr, and (3) non‑uniqueness of L with a given γ‑support. The candidate solution establishes exactly these three items. For (1) the paper argues by contradiction using lower semicontinuity of the γ‑support (Proposition 6.14), while the model uses equivariance of support, locality/fragmentation (Lemma 6.6), and continuity of the DHam‑action; both routes lead to the same conclusion (Definition 7.1; Theorem 7.12(1) ). For (2) the model’s “corridor” construction parallels the paper’s “tongue”/cube construction and yields L ∈ L̂c with γ‑support ⊃ Kr, in line with Theorem 7.12(2) and Theorem 7.13 (proof sketch) (Definitions 6.1, 6.20; Theorem 7.12(2); proof of Theorem 7.13) . For (3) the model’s stage‑locking argument (alter only inside one ball U_{j0} and never touch it again) matches the paper’s non‑uniqueness proof idea (Theorem 7.12(3); proof notes) . Minor omissions in the model (e.g., explicitly citing that for smooth L, every point is in its γ‑support, Proposition 6.17(1); and invoking fragmentation precisely) are standard and easily supplied (Proposition 6.17; Lemma 6.4/6.6) .