The Counterfactual Reconstruction Theorem
When Local Economies Form One Global Economy
Research question
When do locally compatible experimental and administrative records arise from one global economic history, and what information repairs the remaining gap?
Abstract
Economic counterfactuals are often learned in pieces. Each experiment can yield a minimal state on its own policy domain, and overlaps can make the states locally compatible. A common economic history imposes a further requirement: the compatible pieces must admit one global lift. This paper represents local histories and their declared future responses by a morphism of sheaves. The counterfactual reconstruction theorem identifies the minimal global economy and a connecting map whose image is the complete obstruction to a global lift. Every valid linear specification restriction factors uniquely through this map. Its dimension is the gap between local and global state counts. A baseline archive can reveal part of the obstruction; its kernel determines the remaining bridge problem. A downstream counterfactual selects a canonical target bridge state. Records identify the target precisely when they carry this state, equivalently when they preserve every finite decision problem whose payoff depends on the target. The theorem gives a sharp identified set, a necessary and attainable record count, and the exact amplification of bounded record error. Across several targets, record requirements form a polymatroid whose intersections measure information reuse. The residual target spectrum gives exact frontiers for record counts, normalized measurement noise, and continuous precision budgets. Singular-value brackets and Gaussian projection inference retain coverage across rank changes. An Oregon Medicaid application reconstructs approval rates across separate reports and gives bridge counts zero, one, and four for three declared targets.
In one sentence
Separate studies can describe compatible pieces of an economy while still failing to form one global history; the missing bridge depends on the counterfactual being asked.