ULE residual bound and the modified-state distinction

Under the independent ULE hypotheses define $\mathrm{\Gamma }=4\gamma (\int \mathrm{\parallel }g(t){\mathrm{\parallel }}_{2,1}dt{)}^2$ and ${\tau }_B=\int \mathrm{\mid }t\mathrm{\mid }\mathrm{\parallel }g(t){\mathrm{\parallel }}_{2,1}dt\mathrm{/}\int \mathrm{\parallel }g(t){\mathrm{\parallel }}_{2,1}dt$. The source constructs a modified interaction-picture state within spectral-norm distance $\mathrm{\Gamma }{\tau }_B$ of the exact reduced state; its ULE differential equation has residual bounded by $2{\mathrm{\Gamma }}^2{\tau }_B$. Small $\mathrm{\Gamma }{\tau }_B$ controls this approximation. A residual bound is not an all-time uniform bound between two trajectories obtained by dropping the residual.

Imported contract 1: Under the same stationary Gaussian, finite normalized coupling, finite kernel-moment and remote-past preparation model as the jump supplier, Appendix C constructs $\sigma =(I+M)\rho$ with $M(t)={\int }_t^{\mathrm{\infty }}ds{\int }_{-\mathrm{\infty }}^{t}du[\mathcal{G}(s,u)-\mathcal{G}(u,s)]$, using the map kernel in (C2),(C4). It proves $\mathrm{\parallel }\sigma -\rho \mathrm{\parallel }\le \mathrm{\Gamma }\tau$ and $\dot{\sigma }=L\sigma +\eta$, $\mathrm{\parallel }\eta \mathrm{\parallel }\le 2{\mathrm{\Gamma }}^2\tau$, where $L=\int \mathcal{G}(s,t)ds$ is the ULE generator. The source envelope is $q(v)=4\gamma k(-v){\int }_{-\mathrm{\infty }}^{v}k(-w)dw$, with $k(v)=\mathrm{\parallel }g(v){\mathrm{\parallel }}_{2,1}$ and $\mathrm{\parallel }\mathcal{G}(t,s)\mathrm{\parallel }\le q(t-s)$. Exact source: Nathan–Rudner, PDF p.23 opening and (C1)–(C8); p.24 (C9)–(C22); p.25 residual conclusion; pp.25–26 (C23)–(C31). The microscopic bath-to-map interface and the remote-past BR input are imported rather than reconstructed.

Imported contract 2: The erratum, PDF p.1 opening paragraph, states that its corrections do not affect the ULE derivation or error bounds. Corrections (4)–(6) fix the adjoint jump and Lamb conventions used for the generator identification.

The approved Statement is preserved verbatim. The five local analytical lemmas explicate the finite-preparation budget, envelope moments, modification norm, derivative identity and residual composition; they do not assert that any arbitrary preparation family converges. In particular, the source's remote-past idealization is retained as the imported contract's applicability condition. Appendix A6's tail estimate alone is not a proof of existence or uniqueness of a limiting exact state, and no additional $C^1$ convergence premise is silently imposed on the imported statement. The finite-preparation composition instead retains a residual budget $2{\mathrm{\Gamma }}^2\tau +\mathrm{\Gamma }\tau \mathrm{/}(t-t_0)$ when the same analytical interface applies; the smaller constant is not asserted unchanged at finite $t_0$. The source calls its auxiliary operator a modified density matrix; these estimates use only its operator distance and residual and do not add positivity or trace-normalization claims. No defect in the article under study is asserted.