Global Constraints Beyond Finite Modal Data

Can all finite descriptions of a history determine whether that history is physically admissible? This question matters for theories whose laws constrain complete histories: a candidate may pass every finite test yet violate a condition on the whole history. We establish an exact criterion for when finite modal data suffice. Under the stated separation and extendibility assumptions, admissible histories form a dense subset of the space of coherent complete descriptions. Finite data determine admissibility exactly when that subset is closed; equivalently, every excluded history must have a finite witness to its exclusion. An explicit binary example shows the failure: two history spaces have the same finite descriptions and the same actual history, but admit different complete histories. We then extend the criterion to concurrent configurations, identifying alternative orders of independent events that describe the same completed configuration. The result identifies the global information a faithful reformulation must preserve. It applies to Arborescent Eternalism, an enrichment of the tenseless block with an objective structure of admissible alternatives, while leaving the interpretation of that modal structure open.