At the root the theorem supplies at most Cs wires and degree at most Delta. To analyze an active depth-d call, assume only the shrinkage already proved for its ancestors. That gives its current depth and parameter bounds, hence the packing inequalities at sufficiently large s. The child-size argument then proves shrinkage and the degree cap for its children. Induction establishes the invariant at every active depth. A call below H is a leaf; a fixed finite range of root sizes failing the asymptotic requirements uses direct summation. Current packing is not justified by assuming its own unconstructed children already shrink.