Multiplicity one for wildly ramified representations

Let F be a totally real field in which p is unramified. Let (r) over bar: G(F) -> GL(2) ((F) over bar (p)) be a modular Galois representation which satisfies the Taylor-Wiles hypotheses and is generic at a place v above p. Let m be the corresponding Hecke eigensystem. We show that the m-torsion in the mod p cohomology of Shimura curves with full congruence level at v coincides with the GL(2)(kv)-representation D-0((r) over bar vertical bar(GFv)) constructed by Breuil and Paskunas. In particular, it depends only on the local representation (r) over bar vertical bar(GFv), and its Jordan-Holder factors appear with multiplicity one. This builds on and extends work of the author with Morra and Schraen and, independently, Hu-Wang, which proved these results when (r) over bar vertical bar(GFv) was additionally assumed to be tamely ramified. The main new tool is a method for computing Taylor-Wiles patched modules of integral projective envelopes using multitype tamely potentially Barsotti-Tate deformation rings and their intersection theory.