Lemma: Causal midpoint replacement with uniform coarse error

For every calibrated source,

Every directed edge of the replacement graph is future causal. For a report acquired at an event $r$ by a block with midpoint $p$,

A block footprint is at Hausdorff distance at most $\epsilon$ from its midpoint singleton.

The constants are independent of the number of vertices, registers, or ticks.

Proof. Suppose a logical register leaves one block at site $q$ and next enters another block at the same site $q$. If their spatial midpoints are $u$ and $v$, then

Consecutive distinct block uses are separated by at least one tick. Consequently

so their midpoint segment is future causal. The same argument for an initial or terminal half-tick gives displacement at most $\epsilon /2$ in time $\tau /2$.

During the storage interval between the two block uses, the interpolated midpoint segment is a convex combination of $u$ and $v$, both within $\epsilon /2$ of $q$. It is therefore within $\epsilon /2$ spatial distance of the retained vertical segment at the same time. During the adjacent half-block intervals, comparison with an endpoint or with the block footprint costs at most $2\epsilon$ in $d_c$.

Conversely, each retained storage point has the corresponding midpoint-segment point at the same time, except within those half-block intervals, where the same bound applies. Every point of a block footprint lies within

of its midpoint. This also covers footprint points not selected by a particular logical wire. Taking unions uses a maximum, not a sum, proving (eq:midpointHausdorff).

A report output event has time displacement $\tau /2$ and spatial displacement at most $\epsilon /2$ from the block midpoint, proving (eq:acquisitionmidpoint). The block-footprint statement follows from the same estimate.