Trace Balance, Reachability, and Support Localization

Abstract

The note studies what retained-trace observations determine in a finite trace-balanced process. It proves an exact finite-horizon trace ledger, identifies complementary maps on reachable cones and their complex spans, and develops weighted order-domination bounds. A zero weighted complementary trace localizes a positive map to the complement of the aggregate support; full support gives global vanishing. A rank-one qubit instrument shows why reachability coverage is essential.