Penumbra: closed-loop observation under measurement gaps in cross-border trade
Ian Helfrich
The umbrella theory paper for the Penumbra program. Official trade and production statistics rest on a counterparty-declaration assumption that is increasingly violated by sanctioned-corridor activity (Russian seaborne crude under the 2022 oil cap, Iranian oil since 2018, Sahelian gold smuggled to the UAE, gray-market rare earths). The paper formalizes the measurement gap as a closed-loop problem on Wasserstein space: an equilibrium measure whose cost depends on its own remote-sensing image. Existence under standard assumptions; quantitative uniqueness under a feedback-contractivity hypothesis. The framework is the basis for Paper 1 (theory of adaptive-regularization observation equilibria) and Paper 2 (empirical Russian-crude application).
Section drafts complete; main.tex compiles cleanly in the working tree. Sections 1–10 cover the measurement gap, the taxonomy of failure modes, the equilibrium-on-Wasserstein-space mechanism, the observation operator, and the contraction argument.
Empirical evidence cited in the introduction includes the Hilgenstock-Babina-Itskhoki seaborne-crude work on the Russian oil cap, the SwissAid 2024 estimate of the ~$30B/year African export-to-UAE-import gold gap, and the broader satellite-economics literature on the Lucas-critique problem in nightlight-to-GDP elasticities.
Headed to SSRN in late spring 2026 after the empirical companion (Paper 2) lands a complete first-draft pass on the Russian seaborne-crude corridor.