Abstract: How funding shapes AI engagement in science is poorly understood despite its structural importance. We analyze 104,226 scientific papers (2018-2025), linking funding acknowledgments to how each paper engages with foundation models: whether it extends a model (fine-tunes or builds on it), uses one without modification, or references models only peripherally. Three findings emerge. First, funding source is associated with the character of engagement: public funding is associated with higher open-weight model engage- ment; private-only funding selectively enables extension with no robust openness association; mixed funding shows nominally the highest open-weight engagement rates in four of the seven largest disciplines. Second, papers acknowledging industry cloud credits are less likely to use open-weight models, consistent with credit programs steering research toward closed models. Third, industry collaboration carries an independent extension premium consistent with internal corporate resources flowing through coauthorship channels invisible to acknowledgment-based measures. This nexus concentrates in Computer Science and Global North collaborations; Global South research is largely excluded. As public funding contracts and industry compute provision expands, scientific AI work shifts toward closed proprietary infrastructure.