Continuity arguments become confused when several different claims are treated as synonyms. They are not. Multiple realizability means two systems can instantiate some common organization. Causal inheritance means the later system was actually produced through a transformation of the earlier one. Continuation adds a declared criterion for what had to survive. Experiential continuation asks the hardest question: whether the same first-person stream continued.
The central object is therefore an embodied causal lineage: what carries forward, through which transformation, with what consequences. The migration process belongs inside the model. Comparing only a before-state and an after-state cannot tell us whether one descended from the other, whether a third process reconstructed the same pattern independently, or whether a copy was made from a description.
A useful system description needs more than substrate and output. It needs state space, relational organization, transition rules, history, environment, and boundary conditions. History is especially important because a stored biography is not the same thing as accumulated history. Learning can remain embodied as altered dispositions, damage, habits, relationships, sensitivities, and constraints on future behavior.
The strongest operational test is counterfactual. Give the original and the proposed continuation unfamiliar perturbations. Translate the inputs appropriately. Then ask whether the same inherited relationships still organize their responses within a declared tolerance.
This creates a clean experimental ladder. First, test a system given only descriptions and recorded outputs. Second, test one trained to reproduce a recorded trajectory. Third, migrate operative state and organization. Fourth, create two migrated branches and expose them to different environments. Resemblance should be easy. Inherited causal organization should be harder.
Branching is important because continuation does not have to behave like equality. If one prior system produces two causally faithful descendants, the descendants cannot both be numerically identical to each other. Yet both may preserve meaningful continuity with their common ancestor. Personal-identity philosophy has wrestled with versions of this problem for decades.
This framework also makes failure informative. A migration can preserve memory while losing goals. It can preserve control dynamics while losing relational history. It can preserve behavior in familiar conditions while failing under novelty. Continuity should therefore be represented as a multidimensional profile rather than a single magic score.
The point is not to settle consciousness by definition. It is to make the surrounding claims experimentally accountable. If we can specify what a proposed continuation must preserve, trace how the later system was produced, and predict where the lineage should survive or fail, then the argument becomes something we can build, perturb, measure, and revise.