To what extent can we identify subsets of the system corresponding to "that which is being optimized" and "that which is doing the optimization"?

The information theoretic measure of individuality attempts to answer exactly this type of question.

From this view, a set of components (the system) is decomposed into two subsets (subsystem + environment). The proposed subsystem is assigned a degree of individuality by measuring the amount of information it shares with its future state, optionally conditioned on its environment. This leads to 2 types of individuality. The first type says that a proposed subsystem is individualistic to the degree that the subsystem is predictive of its future state after accounting for the information in the environment. The second type captures the notion of inseparability by assigning a high degree of individuality to subsystems that are strongly coupled with their environment in such a way that neither the subsystem nor environment alone are predictive of the next state of the subsystem.

For example, considering the set of atoms making up the space containing the robot-optimizer and vase, the set of robot-atoms retains the desired properties of an optimizer, and is also highly individualistic in the first sense since knowing the state of the robot atoms tells you a lot about their next state, but knowing about the set of non-robot atoms tells you very little about the state of the robot. On the other hand, considering the set of atoms making up the tree, the system as a whole is an optimizing system, but no individual subset of atoms accomplishes the target of the larger optimizing system.

Truly a joy to read! Thank you.

The information theoretic measure of individuality attempts to answer exactly this type of question.

From this view, a set of components (the system) is decomposed into two subsets (subsystem + environment). The proposed subsystem is assigned a degree of individuality by measuring the amount of information it shares with its future state, optionally conditioned on its environment. This leads to 2 types of individuality. The first type says that a proposed subsystem is individualistic to the degree that the subsystem is predictive of its future state after accounting for the information in the environment. The second type captures the notion of inseparability by assigning a high degree of individuality to subsystems that are strongly coupled with their environment in such a way that neither the subsystem nor environment alone are predictive of the next state of the subsystem.

For example, considering the set of atoms making up the space containing the robot-optimizer and vase, the set of robot-atoms retains the desired properties of an optimizer, and is also highly individualistic in the first sense since knowing the state of the robot atoms tells you a lot about their next state, but knowing about the set of non-robot atoms tells you very little about the state of the robot. On the other hand, considering the set of atoms making up the tree, the system as a whole is an optimizing system, but no individual subset of atoms accomplishes the target of the larger optimizing system.