With the exception of this note, the entire document is machine-written. We have taken care to supervise this writing to ensure that the proofs meet a minimum standard of readability and that previous literature is appropriately cited. 

The manuscript began as an attempt to pursue the program of Magnen, Rivasseau, and S{\'e}n{\'e}or~\cite{MagnenRivasseauSeneor1993} using large language models. Our hope was that understanding the problem on the torus might offer some insight into the general case. The construction that resulted from this starting point is three-dimensional and lattice-based and draws extensively on Ba{\l}aban’s renormalization group methods.

As we were verifying these results, we learned that Alp\"oge and Sussman announced that they appear to have a YM4 existence argument, following Ba{\l}aban’s program (\url{https://github.com/ethanwsussman/Yang--Mills-LLM-experiment}). We are thrilled and look forward to learning more about these exciting developments. We hope that this manuscript nonetheless contributes to understanding constructive approaches to three-dimensional Yang--Mills theory. The first version of these results was posted to the first author's GitHub on October 17, 2026 (\url{https://github.com/pmaronow/3d-ym-torus}). Prior to posting the first version of this work, we did not inspect the ongoing work of Alp\"oge and Sussman. Further, to the best of our knowledge, the LLMs we used did not access their work during the mathematical development and drafting of this manuscript. This version contains additional clarifications relative to the first version; it has also been partitioned into five volumes.

We welcome suggestions concerning mathematical corrections, omitted citations, and attribution of ideas.

P.\ L.\ was partially supported by NSF grant DMS-2450004.