Corollary: Scar invariants and repair capacity

Starting from $M_0\cap S_0=\varnothing$, admissible repair, ordinary deletion, episode scarring, and initial scarring preserve

Writing $A_t=\text{abs}{M}_t$ and $C_t=\text{abs}{S}_t$,

At state $t$, exactly $N_W-A_t-C_t$ coordinates are inactive and unscarred.

For (eq:scarupdate),

For general new scars $D\subseteq S^c$, active cardinality instead decreases by $\text{abs}D\cap M$.

Proof. Fixed parameter masks commute and multiply coordinatewise, as recorded in Psy01-A thm:nesting and Psy03-A prop:counts CheungPsy01A,CheungPsy03A. The additional eligibility restriction prevents repair from intersecting $S$. Deletion only removes active coordinates; adding a scar also removes that coordinate from $M$. Induction proves disjointness and monotonicity. Cardinality and capacity follow from these disjoint subsets.