Target Determinability under Partial Causal Observation: A Faithful Reduction Framework

Many distributed, scientific, autonomous, and auditing systems face a structural problem: global facts are generated by causal event histories, while observers usually access only partial projections of those histories. Logs, explanations, measurements, or traces therefore do not by themselves guarantee that a target fact is determinable. This paper names this structural limitation the causal observability gap and develops a formal framework for studying target determinability under partial causal observation.

An external target-determination problem is written as P = (ℋ, O, Q), where ℋ is the set of possible real histories, O is the real observation, and Q is the target fact. A faithful reduction maps such a problem to a formal triple (ℱ, Ω, D), where ℱ is a family of finite causal configurations, Ω is an observation function, and D is a formal target. Faithfulness requires preservation of histories, observations, and targets. The adequacy theorem proves that, under these conditions, Q is zero-error determinable from O if and only if D is zero-error determinable from Ω.

The core mathematical criterion is quotient factorization: D is determinable from Ω exactly when D is constant on every observational equivalence class, equivalently when there exists g such that D = g ∘ Ω, or equivalently when the observation partition refines the target partition, Π_Ω ≼ Π_D. If two configurations have the same observation but different target values, no decision procedure depending only on the observation can solve the target problem with zero error; the pair itself is a publicly checkable certificate of non-determinability.

The paper further develops reconstruction complexity, refinement monotonicity, finite ambiguity bounds, constrained causal evidence as conflict-edge coverage, dynamic configuration streams, approximate determinability, adversarial observation, observation composition, and privacy-observability tradeoffs. The contribution is a falsifiable and reproducible mathematical framework for target-fact determinability under partial causal observation, together with an engineering checker interface.