Abstract
Endpoint-control analysis can compress a temporally factored intervention history into a compound-input channel that preserves aggregate causal capacity while discarding the temporal product structure needed for fixed-coordinate intervention questions. We distinguish arbitrary relabeling of complete intervention histories from relabelings that preserve temporal factorization and show that the coarser abstract equivalence cannot, in general, identify the factor-sensitive support target. An explicit three-step deterministic witness gives abstractly equivalent endpoint channels with different first-coordinate intervention support, and the distinction persists under a common stochastic post-channel when the relevant endpoint difference remains detectable. Exhaustive finite enumeration further shows substantial structural ambiguity within matched endpoint-capacity classes. The underlying orbit/invariance logic is established mathematics; the contribution is the bounded endpoint-control application, with explicit deterministic and stochastic witnesses, finite ambiguity analysis, and a general existence family.
In plain language
A system can have the same amount of possible influence over a future endpoint while differing in where that influence came from. If an entire sequence of interventions is compressed into one compound input, the resulting endpoint-control measure can preserve how much causal capacity exists while losing which earlier time coordinate still makes a difference.
The paper demonstrates that this is not merely a verbal concern. It constructs systems that are equivalent under the endpoint abstraction but differ when the question is tied to a particular intervention coordinate in the earlier history.
Why it matters
Summary measures are valuable precisely because they discard detail. The problem appears when a scientific question later depends on one of the details the summary was designed to ignore. In this case, an endpoint capacity can answer a question like “how much causal influence is available?” without being able to answer “which earlier intervention still contributes to that influence?”
That distinction matters for causal attribution, control analysis, and abstraction. Two systems can be identical according to an endpoint-capacity statistic while supporting different conclusions about temporal provenance. If the research question concerns persistence of an earlier causal distinction, then the temporal factorization has to be retained or recovered through a more temporally resolved representation.
The result does not invalidate endpoint-control measures. It identifies a boundary on what they can establish: aggregate causal capacity and the temporal organization of that capacity are different inferential targets.
Publication record
The manuscript is being prepared for submission. No public preprint has yet been posted. A repository record or final journal DOI will be linked here when available.