The exact memory boundary remains partially open

The exact memory boundary remains partially open

Let 1] be the noise parameters admitting a stationary qutrit source with the full word law. This set is compact: three positive 9 9 Choi matrices with summed partial trace form a compact set, as do initial density matrices. Stationarity and every finite-word equality are continuous closed constraints. Their intersection is closed in that compact parameter space, and its projection to is compact. The explicit high-noise family makes it nonempty; the low-noise exclusion shows that its attained minimum lies in (3]. Below, four dimensions are necessary and sufficient; at the rank-nine cut makes the minimum three. This does not prove is an interval. Adding noise to emitted symbols need not preserve the delayed-process family, so global monotonicity cannot be inferred that way.