Previous work has shown that perturbation analysis in algorithmic information dynamics can uncover generative causal processes of finite objects and quantify each of its element's information contribution to computably constructing the objects. One of the challenges for defining emergence is that the dependency on the observer's previous knowledge may cause a phenomenon to present itself as emergent for one observer at the same time that reducible for another observer. Thus, in order to quantify emergence of algorithmic information in computable generative processes, perturbation analyses may inherit such a problem of the dependency on the observer's previous formal knowledge. In this sense, by formalizing the act of observing as mutual perturbations, the emergence of algorithmic information becomes invariant, minimal, and robust to information costs and distortions, while it indeed depends on the observer. Then, we demonstrate that the unbounded increase of emergent algorithmic information implies asymptotically observer-independent emergence, which eventually overcomes any formal theory that any observer might devise. In addition, we discuss weak and strong emergence and analyze the concepts of observer-dependent emergence and asymptotically observer-independent emergence found in previous definitions and models in the literature of deterministic dynamical and computable systems.