Sharp Nonlocality of Supported Attachment Repair in Bispanning Graphs

We construct, for every rank r ≥ 5, a simple atomic 3-connected bispanning pair with an interval-exchange-supported matching whose every supported order fails a specified attachment query. No supported single-cycle matching switch of size less than r−2 can leave this matching. A single switch of size exactly r−2 reaches an explicit target-first interval-exchange order and repairs the query. Thus the minimum switch bottleneck is exactly r−2, and arbitrarily long sequences of bounded-size supported switches cannot repair every starting matching. The first tree is a path, the source tree has exactly four nonleaf vertices, and the source-tree distance from the cap to the target is always two. The proof gives a complete order grammar, two routing-forest reductions, and a constructive sharp repair at arbitrary rank. Complete finite banks corroborate the argument but are not its arbitrary-rank premise.