Multiple decays vanish over fixed total time with fast checks

The mutually orthogonal block projectors in [F2] sum to the identity. Hence $0\le Q\le \gamma I$. In particular, for every positive trace-one conditional state $\sigma$, including any state produced by a measurement or recovery, $\mathrm{Tr}(Q\sigma )\le \gamma$. The bound therefore holds throughout every interval, not merely at its start or for a stationary state.

The initial state of each interval is supported in the host block: this holds initially, after a detected outcome by the assumed recovery, and after a null outcome by the projector in [F3]. Write $P={\mathrm{\Pi }}_{J_0}$ and let $P\sigma P=\sigma$ be that initial state. Since $H$ and $Q$ are diagonal in the block basis, $PH=HP$ and $PQ=QP$. Hence $P$ commutes with $B=-iH-Q\mathrm{/}2$ and its diagonal exponential $V_t=e^{tB}$. It follows that $P{S}_t(\sigma )P=S_t(\sigma )$, where $S_t$ is the exact no-jump map in [F6]. Its strictly positive trace permits normalization, so the zero-event conditional state is still supported in $P$. Therefore $(I-P)S_{\tau }(\sigma )(I-P)=0$, and [F3] gives zero detection probability conditional on no events. Thus, if no jump occurs in an interval, the $I-P$ outcome cannot occur. A detected outcome therefore implies at least one jump in that interval; the converse is not needed for the bound.

Let $A_k$ be the measurable event that check $k$ detects a decay, put $X_k={\mathbf{1}}_{A_k}$, and let ${\mathcal{G}}_{k-1}$ be the preceding full-history sigma-algebra. Then $D_M={\sum }_{k=1}^M{X}_k$. By [F7], $\mathbb{E}{X}_k=\mathbb{P}(A_k)\le 1$; since $\mathrm{\mid }{X}_k\mathrm{\mid }=X_k$, each indicator is integrable. The same indicator identity for the whole probability space makes the constant one integrable with expectation one. These statements apply on the full record probability space, regardless of whether it has finitely many outcomes.

Apply [F4] with $r=\gamma$. The per-interval conditional bound is ${\gamma }^2{\tau }^2\mathrm{/}2$, and summing over the $M=T\mathrm{/}\tau$ intervals bounds the probability of at least one multiple-decay interval by ${\gamma }^{2}T\tau \mathrm{/}2$. This tends to zero for fixed $T$ and $\gamma$ as $M\to \mathrm{\infty }$. No independence of intervals or Poisson assumption is needed.

Conditional on the preceding history, [F5] and the hazard bound give a probability of at least one jump no greater than $1-e^{-\gamma \tau }\le \gamma \tau$. By step 1.2 the same upper bound applies to a detection. The actual conditional detection law supplies a version $p_k=\mathbb{E}[X_k\mid {\mathcal{G}}_{k-1}]$ in the sense of [F8], with $0\le p_k\le \gamma \tau$ almost surely. Taking the whole probability space as the event in [F8] gives $\mathbb{E}{X}_k=\mathbb{E}{p}_k$; monotonicity and scalar linearity in [F7] therefore give $\mathbb{P}(A_k)\le \gamma \tau$. Finite iteration of the two-variable linearity in [F7], starting from the integrable indicators in step 1.3, makes every partial sum integrable and yields $\mathbb{E}{D}_M={\sum }_{k=1}^M\mathbb{E}{X}_k\le M\gamma \tau =\gamma T$. This proves the stated rate-controlled bound without asserting that every jump is detected or that recovery is exact on logical information.