Good and Bad Concatenated Rectangles

Fix an integer $q\ge 1$, a recursively replaced gadget family with disjoint child rectangles, and a fault path $F\subseteq L(C)$ in its finite physical schedule $C$, as in Fault Locations, Paths, and Rectangles. Goodness and badness below are relative to this same $q$ and $F$. A level-zero rectangle at location $\lambda$ is good exactly when $\lambda \mathrm{\notin }F$, and is bad otherwise. Recursively, a level-$(r+1)$ rectangle is $q$-good exactly when at most $q$ of its level-$r$ child rectangles are $q$-bad; it is $q$-bad otherwise. Thus the status of a rectangle depends only on the restriction of $F$ to its elementary-location set. For a random fault path these predicates define the corresponding good and bad events.

The implication from $q$-goodness to the ideal reference-preserving logical action is a property that must be proved from the gadget contracts in Fault-Tolerant Gadget and Proper-Code Contract. The definition of good and bad rectangles alone does not prove that implication. For each parent, the number of children is at most $a$, and their underlying elementary-location sets are pairwise disjoint.