Lemma: Two-sided fine support and labelled estimates

For every sufficiently large $n$,

The same upper bound applies to each individually labelled wire support, including its local endpoint waits.

Every actual payload acquisition is within ${\delta }_{\Gamma ,n}$ of its assigned graph vertex. The footprint of each original payload map, including a map inside a contracted macrovertex, is at Hausdorff distance at most ${\delta }_{\Gamma ,n}$ from its graph-vertex singleton.

Proof. For an edge $v\to w$, compare the original and implemented paths under the common affine parameter $\lambda \in [0,1]$. The original time is

The implemented route time is

Its time error satisfies

Lemma lem:pacing bounds the spatial error of the interpolated route by $b_n$.

An active block prism is within $2a_n$ in $d_c$ of the corresponding interpolated route. Endpoint waits, gathering, and dispersal occur within $b_n$ of the vertex position and within the frame's time bound. Payload singleton intervals have the same bound.

Conversely, every original edge point has the corresponding implemented route point, and every graph vertex is close to its hub frame. These comparisons establish both Hausdorff directions. The deliberately enlarged constant in (eq:fineerror) covers all the displayed bounds.

The argument applies edge by edge and vertex by vertex. Taking the union takes the maximum of their errors. The named wire estimates include the access and wait pieces attached to that wire. Acquisition times are actual frame output ticks, not replacement metadata, and satisfy the same estimate.