Prove
Definition 7. A sequence of marked graph annuli D_n has admissible boundary…Lemma 6 ([The cap identity]). For every coloring of the annulus, H_ b -H_ w =…Proposition
Every compact subset of a lobe persists by Jordan convergence. The lobe joins the back across every compact subinterval of its open seam. Conversely, every point of the limiting boundary is approached by the approximating boundaries. Contacts between different lobes do not open passages: the individual Jordan pieces are attached along their specified seams, not through a point of contact. These facts give kernel convergence with the indicated accesses. The persisting hole and separating back exclude degeneration of the modulus. The conformal kernel theorem therefore gives convergence of normalized annular maps on interiors and convergence of their moduli. It remains to control the boundary parameters. Fix a nonconstant open subarc of one of the finitely many Jordan boundary pieces and a point in the persisting back. Jordan-map convergence in a lobe, or in a disk patch along an unchanged back boundary, gives a positive lower bound for the harmonic measure of a smaller subarc as seen from a point in that patch. A Harnack chain through persisting interiors gives the same kind of lower bound from the fixed back point. In round annular coordinates the harmonic-measure densities from its preimage are bounded above uniformly: the moduli and the preimage stay in compact nondegenerate ranges. Consequently no fixed nonconstant subarc of the boundary traversal can be squeezed into an angular interval of length tending to zero. The Jordan convergence of the finitely many pieces gives convergence of the full boundary traversals in curve order. Partition those traversals into finitely many small image arcs, and apply the preceding lower bound to intervening nonconstant subarcs. A failure of equicontinuity in angular parameters would traverse one such fixed subarc in a vanishing angular interval, a contradiction. This proves equicontinuity of the boundary maps. Interior convergence then identifies every uniformly convergent subsequence, giving convergence on the closure. Small crosscuts at the finitely many joins give the same conclusion there. There is no constant boundary interval in the limiting conformal map, by uniqueness for a holomorphic function continuous on a boundary arc. The boundary order and converging marked accesses thus also identify the limiting switches.