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Source. Theorem M and Section 7, printed pp. 127–128 (published original).

Statement

Suppose a planted tree in the tight current quotient acquires an edge between two outer vertices. Contract the resulting odd blossom $B$, set its new weight to $m_B$, and use the remembered edge-weight rule. The feasible weighted hierarchy and tight current matching are preserved. The planted tree contracts to a planted tree with the new node outer. If its root lies in $B$, a compatible minimum-base lift may change the original matching inside $B$, but cannot decrease its weight or increase the number of positive exposed current nodes.

Proof

The flowered-tree lemma gives an odd circuit with a near-perfect circuit matching and at most one matching edge crossing its boundary. All of its edges are tight, because the tree and the new edge lie in the tight graph. Record all edges between its children as well: their inequalities were feasible in the old current graph. Thus the stored requirements of a weighted hierarchy hold for $B$.

Set $w(B)=m_B=\min_Aw(A)$ over its children. Then $d_B=0$. For a crossing edge attached at child $A$, old feasibility gives

$$\bar c_e-w(A)+m_B\le w(D)+m_B=w(D)+w(B),$$

where $D$ is its other current endpoint. This is the new edge inequality. If the edge belongs to the surviving matching, the old inequality was equality, and so is the new one. Other quotient inequalities and matching edges are unchanged. The same equality calculation applies to every surviving tree edge, so the contracted tree still lies in the tight-edge graph.

The planted-tree contraction lemma gives the new tree and makes its pseudovertex outer. A matching edge crossing the blossom specifies the omitted circuit child, so the compatible lift then preserves the former matching. If no matching edge crosses, the blossom contains the old exposed root. Choose a minimum-weight child for its omitted circuit child and lift recursively, as in the certificate lemma. This explicitly enforces condition (e), even if the old root was not a minimum-weight child.

Creating $B$ changes no original node weight or older $d$ and adds $d_B=0$. Thus it changes neither the dual variables nor $U$. The exposed-node sum in the certificate gap replaces the old root weight by $m_B$, which is no larger. Formula (2) of that lemma shows that the new compatible original matching has no smaller weight. The number of positive exposed current nodes is unchanged, or drops by one if $m_B=0$. All other exposures are unaffected. $\square$

The source's abbreviated contraction description is expanded here by stating the minimum-base rotation explicitly. This is compatible with its deferred lifting rule, not an assumption that distinct child weights are equal.