Replication, thermodynamics and the limits of science
Is there a way to prove within an observer–environment system that the structure of knowing is asymptotic?
Reality
Replication may be fundamental to long-term complexity because it is the implementation of memory in a computational sense. Perhaps extending Wheeler’s argument, any sufficiently complex universe may allow for local pockets of memory (through chemical replication) to form, and once memory forms in a resource-constrained environment it is pushed toward prediction, i.e. asking yes/no questions about the environment that it itself has arisen from. This closes the loop between us and the universe. But there is a tension, because maintaining memory uses free energy and therefore cannot go on forever. Could we somehow predict to what level of prediction this self-observing universe will ever get before running out of entropy and slowly dissipating to nothing? Maybe not for our reality (as current observations may be very far from the true state), but in a mathematical formalism it would be interesting to map out the information-theoretic and thermodynamic trajectory.
Philosophical
There is no reason why reality as we observe it should exist or be anything like we observe it. All science has done so far is infer things from observations, even if some of those go way beyond human intuition and direct observation. But, importantly, even if they go beyond, fundamentally everything is translated into something that is observable and understandable through our eyes, brains, and language (including math and physics). Science has not ventured outside of this, because to venture outside of oneself is impossible. Even if we think our inferences about the universe are general, they are still from a human frame of reference. A 2D entity can construct amazing tools to observe and understand the vast plane it is embedded in, but it will never be able to go outside its flatness.
Imagine yourself as a single cell trying to predict the kind of control and understanding you will have by the time you are a human. This is how we should regard ourselves now, and therefore the future is unimaginable to us. We are already way more than a single brain-body through our various tools (including language). Nothing at the frontier of science will help us in projecting this unimaginable future. Only better tools. Science as we do it now is incredibly local in the grand scheme of things. Nothing is fundamental, not science, not math, but we have to use the best tools available.