Which simulator observables have size-independent error bounds?

Local Hamiltonian $H$ on $N$ sites; the device implements $H+\sum_i \delta h_i$ with $\mid\mid \delta h_i\mid\mid \le \epsilon$. Characterize the classes of states (Gibbs states at temperature T, quench dynamics to time t) and local observables O for which $\left|\langle O\rangle_{\mathrm{device}}-\langle O\rangle_{\mathrm{ideal}}\right|\le f(\epsilon,T,t)$ independent of $N$. Answer: theorems with explicit $f$, and counterexamples outside these classes.

Proofs of size-independent bounds for broad classes (for example all gapped ground states or Gibbs states above a threshold temperature) and explicit counterexamples elsewhere.