Where can I see the DOE having a stake in Cymer?

Department of ENERGY not the DoD.

The EUV IP which Cymer owns was originally part of EUV LLC, which was an LLNL [0], Sandia [1], and Intel [2] initiative as part of a CRADA. Cymer eventually began working on EUV as well building on EUV LLC and SEMATECH's work, and was eventually purchased by ASML after the Dept of Commerce and DoE backed their acquisition in 2013 [4], but included Cymer in additional CRADAs [3], ongoing projects [5], and maintain Cymer as a separate operating unit [6] within ASML.

It is these CRADAs that allow the DoE to exert its muscle on IP ownership and knowhow, as they are essentially a license of IP and personnel on US DoE terms [7] while also allowing for private partners to commercialize.

[0] - https://www.llnl.gov/article/27641/euvl-partnership-makes-it...

[1] - https://newsreleases.sandia.gov/partners-unveil-first-extrem...

[2] - https://www.intel.com/pressroom/archive/speeches/EUV91197.HT...

[3] - https://www.osti.gov/biblio/1413999

[4] - https://www.asml.com/en/news/press-releases/2013/asml-comple...

[5] - https://www.llnl.gov/article/52226/llnl-selected-lead-next-g...

[6] - https://bidenwhitehouse.archives.gov/wp-content/uploads/2021...

[7] - https://www.directives.doe.gov/directives-documents/400-seri...